More Simpsons trivia team names

I occasionally peruse the team names at the Woo Hoo! Classic Simpsons Trivia page because…what better way to spend my free time during my work breaks and such? Here are my recent favorites:

Uh, Dan, sir, people are becoming a bit…. confused by the way you and your co host are well, constantly holding hands

Because Woo Hoo Classic Trivia Brooklyn couldn’t exist without six white stripes, seven red stripes, and a hell of a lot of Dans!

We’re here, we’re queer, we dont want any more dans

Our theory is: Simon likes dog food!

Dan’s Moms say they’re cool. [On the night when both Dans' parents attended]

Christmas Ape Goes to Trivia Night

Are “poo” and “ass” taken?

The Non-Giving-Up Trivia Guys

Jeremy’s I. Ron Butterfly

You Have 30 Minutes to Name Your Team. You Have 10 Minutes to Name Your Team. Your Team Has Been Impounded. Your Team Has Been Crushed Into a Cube.

A Little Team Called “Love Is” – They Are Two Naked 8 Year Olds Who Are Married

You know those trivia nights where the two Dans with annoying voices yammer back and forth? We invented those!

Dan = White, Dan = White

The story of how two Dans and five other men parlayed a small business loan into a thriving trivia concern is a long and interesting one. And here it is.

Welcome to an Evening of Trivia and Picking up After Yourselves

Harry Shearer’s Non-Union Mexican Equivalents

The Only Monster Here is the Trivia Monster Who Has Enslaved This Bar. I Call Him Trivior! And It’s Time To Snatch This Bar From His Neon Claws!

To find the Dans, I just have to think like the Dans. I’m a big trivia host wannabe and i make the same stupid jokes every month … Berry Park!

♫ I hate every Dan I see, from Dan Mulhall to Dan Ozzi. No, you’ll never make a Daniel out of me… ♫

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Proof that if f and g are continuous functions, then f/g is also continuous (as long as g(x) ≠ 0)

In almost any calculus or analysis textbook, in the chapter on continuity of functions, you’ll encounter four theorems about the operations on functions that preserve continuity: multiplying a continuous function by a scalar (real number), adding two continuous functions, multiplying two continuous functions, and dividing two continuous functions. But no textbooks that I have seen, nor websites nor lecture notes nor study guides nor homework solutions nor Stack Exchange questions nor anything else I’ve found online, have the proof to the last one! On the chance that I can add something new to the internet for the first time, here is the proof that Professor Yuri Ledyaev did in my Advanced Calculus (introductory real analysis) class at Western Michigan University:

Let \(f, g\) be two continuous functions with domain \(A \subset \mathbb{R}\) and let \(a \in A\). Let \(g(a) \neq 0\). Then \(f/g\) is also continuous at point \(a\). The proof is nearly identical if \(A \subset \mathbb{R}^n\) —which is in fact the way we did the proof, in the first chapter on functions of multiple variables—but there’s no way I’m typing every single \(x\) and \(a\) in vector form in Latex. Just imagine they’re all bold or they have a little line over them, and imagine that the \(|x-a|<\delta\)'s are all \(\vec{x} \in B_\delta (\vec{a})\)'s.

Proof: For the function \(f/g\) to be continuous, this means
\lim_{x \to a} \frac{f(x)}{g(x)} = \frac{f(a)}{g(a)}.

The way to prove this equality is to apply the Cauchy definition of continuity, i.e., \(\forall \epsilon > 0\), \(\exists \delta > 0\) such that if \(|x – a| < \delta\), then \(\left|\frac{f(x)}{g(x)} - \frac{f(a)}{g(a)}\right| < \epsilon\). To obtain this inequality, we start with the latter absolute-value expression, get a common denominator, use the ol' add-and-subtract-the-same-thing trick, and apply the fact that both \(f\) and \(g\) are individually continuous. Observe:

\left|\frac{f(x)}{g(x)} - \frac{f(a)}{g(a)}\right| &=& \frac{|f(x)g(a) - f(a)g(x)|}{|g(x)||g(a)|} \nonumber\\[13pt]
&=& \frac{|f(x)g(a) - f(a)g(a) + f(a)g(a) - f(a)g(x)|}{|g(x)||g(a)|} \nonumber \\[13pt]
&\leq& \frac{|f(x)g(a) - f(a)g(a)| + |f(a)g(a) - f(a)g(x)|}{|g(x)||g(a)|} \nonumber \\[13pt]
&=& \frac{|g(a)||f(x) - f(a)| + |f(a)||g(x) - g(a)|}{|g(x)||g(a)|} \nonumber \\

(The \(\leq\) comes from the Triangle Inequality.)

Now, we have to interrupt our equation display to make use of the fact that \(g\) is continuous. Since \(g\) is continuous, we can let \(\epsilon_{\small{1}} = \frac{1}{2} |g(a)| > 0\). Then there exists \(\delta_{\small{1}}\) such that whenever \(|x-a|<\delta_{\small{1}}\),
|g(a)| - |g(x)| &\leq& |g(x) - g(a)| < \epsilon_{\small{1}} \nonumber \\[13pt]
|g(a)| - \epsilon_{\small{1}} &<& |g(x)| \nonumber \\[13pt]
\frac{1}{2} |g(a)| &<& |g(x)| \nonumber \\[13pt]
\frac{2}{|g(a)|} &>& \frac{1}{|g(x)|}
(Again the \(\leq\) in this equation display is due to (one form of) the Triangle Inequality.)

Applying this inequality to the \(|g(x)|\) in the denominator up above gives us
\frac{|g(a)||f(x) – f(a)| + |f(a)||g(x) – g(a)|}{|g(x)||g(a)|} &<& \frac{2}{|g(a)|^2} \frac{|g(a)||f(x) - f(a)| + |f(a)||g(x) - g(a)|}{|g(x)||g(a)|} \nonumber \\[13pt]
&=& \frac{2}{|g(a)|} |f(x) - f(a)| + \frac{2|f(a)|}{|g(a)|^2} |g(x) - g(a)| \nonumber \\[13pt]

Now, since \(f\) is continuous, we know that for any \(\frac{\epsilon}{2}\), there exists \(\delta_2\) such that making \(|x - a| < \delta_2\) will make \(|f(x) - f(a)| < \frac{\epsilon}{2}\cdot\frac{|g(a)|}{2}\).

Similarly, since \(g\) is continuous, there exists \(\delta_3\) such that making \(|x - a| < \delta_3\) will make \(|g(x) - g(a)| < \frac{\epsilon}{2}\cdot\frac{|g(a)|^2}{2|f(a)| + 1}\). (The \(+1\) must be added in case \(f(a) = 0\).)

Finally, we simply choose \(\delta = \min{\{\delta_1, \delta_2, \delta_3\}}\), and this will give us
\left|\frac{f(x)}{g(x)} - \frac{f(a)}{g(a)}\right| &<& \frac{2}{|g(a)|} |f(x) - f(a)| + \frac{2|f(a)|}{|g(a)|^2} |g(x) - g(a)| \nonumber\\[13pt]
&<& \frac{\epsilon}{2} + \frac{\epsilon}{2} = \epsilon
To summarize, this proof showed that if \(f\) and \(g\) are continuous, then for any \(\epsilon > 0\), letting \(|x-a|<\delta\) will make \(\left|\frac{f(x)}{g(x)} – \frac{f(a)}{g(a)}\right| < \epsilon\), meaning \(\frac{f(x)}{g(x)}\) is continuous at \(x=a\).

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Convergence of a difficult integral using the limit comparison test

Here’s a great problem from an exam in my second-semester Advanced Calculus (introductory real analysis) course taught by Yuri Ledyaev at Western Michigan University:

Find the values of \(p\) for which the integral converges:
\int_{1}^{\infty} \frac{\left(\tan\frac{1}{x}\right)^p}{x+x^2}

To determine what test to use, it is best to recall that
\lim_{\theta \to 0} \tan\theta \sim \theta

\lim_{\theta \to 0} \frac{\tan\theta}{\theta} = 1

which is a fancy way of saying that at very small values of \(\theta\), \(\tan\theta\) behaves like \(\theta\). (In case you forget this, it is easy to recall by remembering that \(\tan\theta = \frac{\sin\theta}{\cos\theta}\), whose denominator approaches \(1\) as \(\theta\) approaches \(0\), so \(\tan\theta\) behaves like \(\sin\theta\) for very small values of \(\theta\). All semi-advanced math students should remember the small-angle approximation rule, i.e., for small values of \(\theta\), \(\sin\theta \approx \theta\).)

We can substitute \(1/x\) for \(\theta\) to see that
\lim_{x \to \infty} \tan\frac{1}{x} \sim \lim_{x \to \infty} \frac{1}{x}

The reason we’re interested in this is that we need to know what \(\frac{\left(\tan\frac{1}{x}\right)^p}{x+x^2}\) behaves like or looks like as \(x \rightarrow \infty\). Whatever our integrand looks like is what we’ll compare it to in the limit comparison test. So:

\lim_{x \to \infty} \frac{\left(\tan\frac{1}{x}\right)^p}{x+x^2} \sim \lim_{x \to \infty} \frac{\left(\frac{1}{x}\right)^p}{x+x^2} \nonumber = \lim_{x \to \infty} \frac{1}{x^{p+1} + x^{p+2}} \nonumber \\

In this case, it is a good bet to choose the term in the denominator with the greater exponent rather than the term with the lesser exponent for use in the limit comparison test, so we’ll choose \(x^{p+2}\). That is, for the limit comparison test, let \(f(x) = \frac{\left(tan\frac{1}{x}\right)^p}{x+x^2}\) and \(g(x) = \frac{1}{x^{p+2}}\).

The limit comparison test for integrals says that if \(f\) and \(g\) are both defined and positive on \([a, \infty)\) and integrable on \([a, b]\) for all \(b \geq a\), and if \(\lim_{x \to \infty} \frac{f(x)}{g(x)}\) exists and is not equal to \(0\), then the integrals \(\int_{a}^{\infty} f(x) dx\) and \(\int_{a}^{\infty} g(x) dx\) are equiconvergent.


\lim_{x \to \infty} \frac{f(x)}{g(x)} &=& \lim_{x \to \infty} \frac{\frac{\left(\tan\frac{1}{x}\right)^p}{x+x^2}}{\frac{1}{x^{p+2}}} \\[3pt] \nonumber \\
&=& \lim_{x \to \infty} \frac{\left(\tan\frac{1}{x}\right)^p}{x+x^2} \cdot x^{p+2} \\[3pt] \nonumber \\
&\sim& \lim_{x \to \infty} \frac{\frac{1}{x^p}}{x+x^2} \cdot x^{p+2} \\[3pt] \nonumber \\
&=& \lim_{x \to \infty} \frac{x^2}{x+x^2} \\[3pt] \nonumber \\
&=& 1

Thus, our choices of \(f(x)\) and \(g(x)\) satisfy the limit comparison test, meaning \(\int_{a}^{\infty} f(x) dx\) converges when \(\int_{a}^{\infty} g(x) dx\) converges.

When does \(\int_{a}^{\infty} g(x) dx\) converge? When \(p+2 > 1 \) (by the p-series rule). Thus, both \(\int_{a}^{\infty} f(x) dx\) and \(\int_{a}^{\infty} g(x) dx\) converge when \(p>-1\) and diverge when \(p \leq -1\).

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My favorite Simpsons trivia team names

I’ve wasted several hours this week reading through the names of teams at the classic Simpsons trivia nights that are held in certain restaurants in Chicago, Vancouver, Brooklyn, Toronto, and Hamilton, Ontario. Many of them are hilariously clever. Naturally, I started thinking up some of my own that I’d like to use, even though there is approximately no chance I will ever be able to attend one of these trivia nights.

Below are two lists of names: the ones I’ve thought of and the ones I’ve seen at the website above. My first one listed is the only one I’ve noticed among the actually used team names (after I thought of it! great minds think alike!). I often refer to Neil Arsenty or Chicago’s Pizzeria Serio in my hypothetical names because I live a few hours from Chicago and can at least dream of visiting Chicago sometime and attending their trivia night. Also I love the Simpsons Mixtape podcast, whose hosts regularly attend the one in Chicago, and the Worst Episode Ever podcast, whose hosts regularly attend and host the one in Brooklyn, respectively.

My hypothetical team names:

Oh, Neil. I’d be lying if I said my team wasn’t committing crimes [Alternative: Oh, Neil. I'd be lying if I said my team wasn't cheating.]

We told you: we’re not Xena!

Mister Moe

Or, in honor of its discoverer, the Teamahedron, hm-hey, hm-hey

No. No. No. No. No. No. No. No. No. No. No. Yes—I mean no, no.

Tie good. You like trivia?

Ooh! Look at me! I’m making people happy! I’m the trivia man, from Trivia Land, in a gumdrop house on Lollipop Laaaaaane!

Losin’ at trivia? Oh, you better believe that’s a paddlin’

Whaddya meeeeeean, the pizzeria’s out of money?

In America, forst you get the trivia, then you get the donuts, then you get the women.

There’s William Henry Harrison, “We lost by 30 points!”

Why must you turn this pizzeria into a house of LIES?

Tell you what: we drive all the way to Chicago and we lose miserably, I owe you a Coke.

“Okay, Mr. Burns, uh, what’s your team’s name?” “I don’t know…”

I can’t believe it’s a trivia team

You know, I’ve had a lot of jobs: boxer, mascot, astronaut, imitation Krusty, baby proofer, trucker, hippie, plow driver, food critic, conceptual artist, grease salesman, carny, mayor, drifter, bodyguard for the mayor, country-western manager, garbage commissioner, mountain climber, farmer, inventor, Smithers, Poochie, celebrity assistant, power plant worker, fortune cookie writer, beer baron, Kwik-E-Mart clerk, homophobe, and missionary, but hosting Simpsons trivia, that gives me the best feeling of all.

We’re going out for trivia! If we don’t come back, avenge our deaths!

“We got first prize!” “You won first place at trivia?” “No, but we got it…….. Stealing is wrong.”

Ow, my eye! I’m not supposed to get trivia in it!

We watched all the classic Simpsons episodes really closely, so when we came to Classic Simpsons Trivia, the answers were stuck in our heads. It was like a whole different kind of cheating!

More testicles mean more iron

There’s very little meat in these trivia cards

I have had it with these trivia nights, Neil! The low score totals, team after team of ugly, ugly people.

If you want to play Simpsons trivia, and I mean really play it, you want the Carnivale

D’oh! A deer! A female deer!

No one would PRETEND to be a last-place Simpsons trivia team

“Johnny Tightlips, do you know the answer?” “Eh, I know a lot of things.”

Neil, this circle is you.

Mona Stevens, Penelope Olsen, Martha Stewart, and Muddie Mae Suggins

My favorites from the official trivia organization’s web page above:

Tiffany, Heather, Cody, Dylan, Dermott, Jacob, Taylor, Brittany, Wesley, Rumer, Scout, Cassidy, Zoe, Chloe, Max, Hunter, Kendall, Caitlin, Noah, Sasha, Morgan, Kyra, Ian, Lauren, Qbert, Phil, Neil

Trivia at This Time of Year? At This Time of Day? In This Part of the Country? Localized Entirely Within This Pizzeria?

Mr. Arsenty? We All Have Nosebleeds

More Winnin’, Les Winen

Our Team May Be Ugly and Hate-filled, But Wait, What’s the Third Thing You Said?

Our Team No Function Beer Well Without

Our team’s low score is the result of an unrelated alcohol problem

They Slept, They Stole, They Were Rude To The Other Players. But Still, There Goes The Best Damn Team A Trivia Night Ever Saw


This Team Engaged in Intercourse with Your Spouse or Significant Other. Now THAT’S Trivia!

Our Team Name is Agnes. It Means Lamb! Lamb of God!

Remember Our Team Name? We’re Back In Pog Form!

The bottom rung of society now that that cold snap killed all those hobos

I for one would like to see the trivia questions in advance, I dont like the idea of the same team winning two months in a row

Looks like Rusty’s team got a discipline problem. Maybe that’s why we beat them at Simpsons Trivia nearly half the time…

This teams got a hot date… a date… dinner with friends… dinner alone… watching tv alone… ok ok, we’re gonna go to berry park general knowledge trivia. Buzz.. Simpsons trivia.. Ding. We don’t deserve this kinda shabby treatment

“You’re always trying to give me long trivia names. What is it with you?” “I just think they’re neat.”

Of course we could make the questions more challenging, but then the stupider teams will be in here furrowing their brow in a vain attempt to understand the situation

Your older, balder, fatter team

Das Trivia Team Ist Ein Nuisance Team!

We Wouldn’t Have Thought We Could Put a Price on Neil Arsenty’s Life, But Here We Are

Especially Lisa! But ESPECIALLY Bart

This one team seems to love the speedo man!

Chris, When You Participate in Simpsons Trivia, It’s Not Whether You Win or Lose, It’s How Drunk You Get

Family. Religion. Friendship. These are the 3 Demons You Must Slay if You Wish To Succeed In Trivia

Can I Borrow a Team Name?

Remember When We Went to Simpsons Trivia and We Forgot How to Drive? [I like this one a lot because I can imagine the whole bar saying, "That's because you were drunk!" and the team responding, "And how!"]

Can I Have the Keys to the Car, Lover? I Want to Change Teams


The Seat Moisteners from Sector 7G

Evergreen Terrorist

Trivia Involves Being a Bit Underhanded, a Bit Devious, a Bit—as the French Say—Bartesque

We’re a Family Team. A Happy Family. Maybe Single People Play Trivia. We Don’t Know. Frankly, We Don’t Want To Know. That’s One Market We Can Do Without

I’m Sorry If You Heard Disneyland, But I Distinctly Said Simpsons Trivia Night

Too Crazy for Trivia Town, Too Much Trivia for Crazy Town!

Forwards, Not Backwards! Upwards, Not Forwards! And Always Twirling, Twirling, Twirling Towards First Place!

Excuse me, our team is also named Bort

The Bort Identity

Stupid team name. Be more funny!

The Team From Kua…Kual Lam…France!

It was the best of teams, it was the blurst of teams?!

Go Ahead, First Place Team… Enjoy Your Donuts. Little Do You Know You’re Getting Closer to the Poison Donut!

Team ‘You Know Who’ Playing The Secret ‘Wink Wink’ At The ‘You Know What’

You Don’t Win Trivia With Salad

The Following Answers Are Lies, But They’re Entertaining Lies, And Isn’t That The Real Truth? The Answer Is No.

Which Two of these Popular Trivia Team Members Died in the Last Year? If You Guessed Kelly and Brian, You’d Be Wrong. They Were Never Popular

And I Come Before You Good People Tonight with a team name. Probably the greatest… oh it’s not for you, It’s more of a Shelbyville Team Name

Don’t Make Me Run, I’m Full of Pizza

Die Team Die

Ah Yes. Shake it, Dan. Capital Knockers

They Said They Made the Team Themselves… from a Bigger Team

Stupid Teams Need The Most Attention

This Team Must Be Good. They Don’t Need A Lot of Players, Or Even Correct Spelling

Union Rule 26: This Team Must Win Trivia at Least Once Regardless of Gross Incompetence, Obesity or Rank Odor

Doesn’t This Team Know Any Songs That Aren’t Commericals?

“Oh, Simpsons Trivia, That’s Cool” “Are You Being Sarcastic, Dude?” “I Don’t Even Know Anymore”

You Want Us To Show This Question To The Cat, And Have The Cat Tell You What It Is? ’Cuz The Cat’s Going To Get It!

The Greatest Team Ever Hula’ed

A Shiny New Donkey For The Team That Brings Us The Head of Colonel Montoya

There Are Too Many Teams Nowadays. Please Eliminate Three.

Why Would They Come To Simpsons Trivia Just To Boo Us?

Only Who Can Win at Trivia? You Have Selected “You”, Referring to Our Team. The Correct Answer is “You”

Persephone? People Don’t Want Trivia Teams Named After Hungry Old Greek Broads

The Extra B is For BYOBB. What’s The Second B For? Best Team Ever!

On This Team, We Obey The Laws Of Thermodynamics!

The Team That Was Eventually Rescued By…Oh, Let’s Say Moe

Our Team is Hatless, Repeat, Hatless

I hate every Dan I see, from Dan Mulhall to Dan Ozzi, no you’ll never make a Daniel out of me!!

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Fascinating result of the Intermediate Value Theorem

This is problem #1 from chapter 3.9 in Advanced Calculus: Theory and Practice, my introductory real analysis textbook at Western Michigan University:

Suppose that \(f\) is continuous on \(\left[0, 2\right]\) and \(f(0) = f(2)\). Prove that there exist \(x_1\), \(x_2 \in \left[0, 2\right]\) such that \(x_2 – x_1 = 1\) and \(f(x_1) = f(x_2)\).

Informally, this says there are two \(x\)-values exactly \(1\) unit apart whose \(f\) values are equal. This result isn’t at all intuitive, and I liked it so much because it’s a great example of the type of abstract, theoretical result you learn to prove in mathematical analysis.

Recall that the Intermediate Value Theorem states that if \(f\) is a continuous function on an interval \(\left[a, b\right]\) and \(f(a) \neq f(b)\), then for every \(C\) between \(f(a)\) and \(f(b)\), there exists \(c \in (a, b)\) such that \(f(c) = C\). Often the Intermediate Value Theorem is stated as a specific case, where \(f(a) < 0\) and \(f(b) > 0\), in which case there exists \(c \in (a, b)\) such that \(f(c) = 0\). This is the case that will be relevant here. Now the solution to the problem:

Proof: Let \(g(x) = f(x+1) ~- f(x)\), defined on \(\left[0, 1\right]\). The function \(g\) is continuous, and $$g(0) = f(1) ~- f(0)$$ and $$g(1) = f(2) ~- f(1) = f(0) ~- f(1) = -g(0).$$

If \(g(0) = 0\), then \(f(0+1) ~- f(0) = 0\), so \(f(1) = f(0)\) and the solution is to take \(x_1 = 0\) and \(x_2 = 1\). If \(g(0) \neq 0\), then \(g(1)\) and \(g(0)\) are nonzero numbers of equal magnitude but opposite sign. By the Intermediate Value Theorem, there exists \(c \in (0, 1)\) such that \(g(c) = 0\). Now the solution is to define \(x_1 = c\) and \(x_2 = c+1\). This makes \(g(c) = f(c+1) ~- f(c) = f(x_2) ~- f(x_1) = 0\), so \(f(x_2) = f(x_1)\). \(\blacksquare\)

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Proofs of some trigonometric identities

Remember all those trigonometric identities in the front cover of your calculus book that were too hard to memorize and you didn’t have to anyway? Not the simple ones like \(\)\(\sin^2 x + \cos^2 x = 1\) or \(\tan^2 x + 1 = \sec^2 x\). I mean the angle-addition and -subtraction formulas and the like. We proved them pretty easily in my Advanced Calculus class at Western Michigan University, starting with some assumed knowledge of vector calculus. I don’t know what other ways there are to prove all of them, but this way starts with \(\cos (\alpha – \beta) \) and derives all of them from there.

\( \boldsymbol{ 1. \cos(\alpha – \beta) = \cos \alpha \cos \beta + \sin \alpha \sin \beta } \)

First, imagine two unit vectors with their tails at the origin and their heads on the unit circle. Vector \(\vec{a}\) makes an angle of \(\alpha\) with the horizontal axis and vector \(\vec{b}\) makes an angle of \(\beta\). Thus, the angle between them is \(\alpha – \beta\). And each vector written in component form is \(\vec{a} = \langle \cos \alpha, \sin \alpha \rangle \) and \(\vec{b} = \langle \cos \beta, \sin \beta \rangle \). Recall that their dot product is
\langle \cos \alpha, \sin \alpha \rangle \cdot \langle \cos \beta, \sin \beta \rangle &=& \| \vec{a} \| \| \vec{b} \| \cos(\alpha – \beta) \nonumber \\
\cos \alpha \cos \beta + \sin \alpha \sin \beta &=& 1 \cdot 1 \cdot \cos(\alpha – \beta) \nonumber \\

\( \boldsymbol{ 2. \cos (\alpha + \beta) = \cos \alpha \cos \beta – \sin \alpha \sin \beta\ } \)

The next one is easy because we can just replace \(\beta\) with \(-\beta\):
\cos(\alpha + \beta) &=& \cos(\alpha – (-\beta)) \nonumber \\
&=& \cos \alpha \cos (-\beta) + \sin \alpha \sin(-\beta) \nonumber \\
&=& \cos \alpha \cos \beta + \sin \alpha (-\sin \beta) \nonumber \\
&=& \cos \alpha \cos \beta – \sin \alpha \sin \beta \nonumber \\

That one relies on the fact that \(\cos (-\alpha) = \cos \alpha\) and \(\sin (-\alpha) = -\sin \alpha\), so…I guess you have to know that. The way we “proved” them is to simply draw an angle into the fourth quadrant that was the same magnitude as \(+\alpha\) and observe that the cosine (horizontal distance) was the same and the sine (vertical distance) was the same in magnitude but opposite in sign. I don’t know what other, more rigorous ways there are to prove that \( \cos(-\alpha) = \cos \alpha\) and \(\sin(-\alpha) = -\sin \alpha\), but there are only so many things you can spend time proving in a semester of Analysis.

Before we do the equivalent \(\sin\) identities, it is easiest to do the following two:

\( \boldsymbol{ 3. \cos \left(\frac{\pi}{2} – \alpha \right) = \sin \alpha } \)

\cos \left(\frac{\pi}{2} – \alpha \right) &=& \cos \frac{\pi}{2} \cos \alpha + \sin \frac{\pi}{2} \sin \alpha \nonumber \\
&=& 0 \cdot \cos \alpha + 1 \cdot \sin \alpha \nonumber \\
&=& \sin \alpha

\( \boldsymbol{ 4. \sin \left(\frac{\pi}{2} – \alpha \right) = \cos \alpha} \)

From #3, since the sine of an angle equals the cosine of \(\frac{\pi}{2} \) minus that angle, we can easily transform \(\sin \left(\frac{\pi}{2} – \alpha \right)\):
\sin \left(\frac{\pi}{2} – \alpha \right) &=& \cos \left(\frac{\pi}{2} – \left(\frac{\pi}{2} – \alpha \right) \right) \nonumber \\
&=& \cos \left(\frac{\pi}{2} – \frac{\pi}{2} + \alpha \right) \nonumber \\
&=& \cos \alpha

Now we can do the \(\sin\) angle-addition and angle-subtraction identities:

\( \boldsymbol{ 5. \sin(\alpha + \beta) = \sin \alpha \cos \beta + \sin \beta \cos \alpha} \)

\sin(\alpha + \beta) &=& \cos \left(\frac{\pi}{2} – (\alpha + \beta) \right) \nonumber \\
&=& \cos \left( \left(\frac{\pi}{2} – \alpha \right) – \beta \right) \nonumber \\
&=& \cos \left(\frac{\pi}{2} – \alpha \right) \cos \beta + \sin \left(\frac{\pi}{2} – \alpha \right) \sin \beta \nonumber \\
&=& \sin \alpha \cos \beta + \cos \alpha \sin \beta \nonumber \\

\( \boldsymbol{ 6. \sin(\alpha – \beta) = \sin \alpha \cos \beta – \sin \beta \cos \alpha} \)

\sin(\alpha – \beta) &=& \cos \left(\frac{\pi}{2} – (\alpha – \beta) \right) \nonumber \\
&=& \cos \left( \left(\frac{\pi}{2} – \alpha \right) + \beta \right) \nonumber \\
&=& \cos \left(\frac{\pi}{2} – \alpha \right) \cos \beta – \sin \left(\frac{\pi}{2} – \alpha \right) \sin \beta \nonumber \\
&=& \sin \alpha \cos \beta – \cos \alpha \sin \beta

And now we can do the double-angle identities:

\( \boldsymbol{ 7. \cos 2\alpha = \cos^2 \alpha – \sin^2 \alpha} \)
\cos 2 \alpha &=& \cos(\alpha + \alpha) \nonumber \\
&=& \cos \alpha \cos \alpha – \sin \alpha \sin \alpha \nonumber \\
&=& \cos^2 \alpha – \sin^2 \alpha

\( \boldsymbol{ 8. \sin 2\alpha = 2\sin \alpha \cos \alpha} \)
\sin 2\alpha &=& \sin(\alpha + \alpha) \nonumber \\
&=& \sin \alpha \cos \alpha + \sin \alpha \cos \alpha \nonumber \\
&=& 2\sin \alpha \cos \alpha

\( \boldsymbol{ 9. \cos \alpha – \cos \beta = -2 \sin\left(\frac{\alpha + \beta}{2}\right) \sin\left(\frac{\alpha – \beta}{2}\right)} \)
\cos \alpha – \cos \beta = & ~ \cos \left(\frac{\alpha + \beta}{2} + \frac{\alpha – \beta}{2}\right) – \cos \left(\frac{\alpha + \beta}{2} – \frac{\alpha – \beta}{2}\right) \nonumber \\
= & ~ \cos\left(\frac{\alpha + \beta}{2}\right) \cos \left(\frac{\alpha – \beta}{2}\right) – \sin \left(\frac{\alpha + \beta}{2}\right) \sin \left(\frac{\alpha – \beta}{2}\right) – \\ & \left[\cos \left(\frac{\alpha + \beta}{2}\right) \cos \left(\frac{\alpha - \beta}{2}\right) + \sin \left(\frac{\alpha + \beta}{2}\right) \sin \left(\frac{\alpha - \beta}{2}\right) \right] \nonumber \\
= & ~ -2 \sin\left(\frac{\alpha + \beta}{2}\right) \sin\left(\frac{\alpha – \beta}{2}\right)

\( \boldsymbol{ 10. \cos \alpha + \cos \beta = 2 \cos\left(\frac{\alpha + \beta}{2}\right) \cos\left(\frac{\alpha – \beta}{2}\right)} \)
\cos \alpha + \cos \beta = & ~ \cos \left(\frac{\alpha + \beta}{2} + \frac{\alpha – \beta}{2}\right) + \cos \left(\frac{\alpha + \beta}{2} – \frac{\alpha – \beta}{2}\right) \nonumber \\
= & ~ \cos\left(\frac{\alpha + \beta}{2}\right) \cos \left(\frac{\alpha – \beta}{2}\right) – \sin \left(\frac{\alpha + \beta}{2}\right) \sin \left(\frac{\alpha – \beta}{2}\right) + \\ & \cos \left(\frac{\alpha + \beta}{2}\right) \cos \left(\frac{\alpha – \beta}{2}\right) + \sin \left(\frac{\alpha + \beta}{2}\right) \sin \left(\frac{\alpha – \beta}{2}\right) \nonumber \\
= & ~ 2 \cos\left(\frac{\alpha + \beta}{2}\right) \cos\left(\frac{\alpha – \beta}{2}\right)

\( \boldsymbol{ 11. \sin \alpha – \sin \beta = 2 \sin \left(\frac{\alpha – \beta}{2}\right) \cos \left(\frac{\alpha + \beta}{2}\right)} \)
\sin \alpha – \sin \beta = & ~ \sin \left(\frac{\alpha + \beta}{2} + \frac{\alpha – \beta}{2}\right) – \sin \left(\frac{\alpha + \beta}{2} – \frac{\alpha – \beta}{2}\right) \\
= & ~ \sin \left(\frac{\alpha + \beta}{2}\right) \cos \left(\frac{\alpha – \beta}{2}\right) + \sin \left(\frac{\alpha – \beta}{2} \right) \cos \left(\frac{\alpha + \beta}{2}\right) – \\
& ~ \left[ \sin \left(\frac{\alpha + \beta}{2}\right) \cos \left(\frac{\alpha - \beta}{2}\right) - \sin \left(\frac{\alpha - \beta}{2}\right) \cos \left(\frac{\alpha + \beta}{2}\right) \right] \\
= & ~ 2 \sin \left(\frac{\alpha – \beta}{2}\right) \cos \left(\frac{\alpha + \beta}{2}\right)

\( \boldsymbol{ 12. \sin \alpha + \sin \beta = 2 \sin \left(\frac{\alpha – \beta}{2}\right) \cos \left(\frac{\alpha + \beta}{2}\right)} \)
\sin \alpha + \sin \beta = & ~ \sin \left(\frac{\alpha + \beta}{2} + \frac{\alpha – \beta}{2}\right) + \sin \left(\frac{\alpha + \beta}{2} – \frac{\alpha – \beta}{2}\right) \\
= & ~ \sin \left(\frac{\alpha + \beta}{2}\right) \cos \left(\frac{\alpha – \beta}{2}\right) + \sin \left(\frac{\alpha – \beta}{2} \right) \cos \left(\frac{\alpha + \beta}{2}\right) + \\
& ~ \sin \left(\frac{\alpha + \beta}{2}\right) \cos \left(\frac{\alpha – \beta}{2}\right) – \sin \left(\frac{\alpha – \beta}{2}\right) \cos \left(\frac{\alpha + \beta}{2}\right) \\
= & ~ 2 \sin \left(\frac{\alpha + \beta}{2}\right) \cos \left(\frac{\alpha – \beta}{2}\right)

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Proof that the limit as n approaches infinity of n^1/n = 1 (\(\lim_{n \to \infty} n^{1/n} = 1\))

\(\)Here’s an important limit from real analysis that gives quite a few people, including myself, a lot of trouble:
\lim_{n \to \infty}n^{1/n} = 1

Here is the proof that my Advanced Calculus professor at Western Michigan University, Dr. Ledyaev, gave in class. It uses the binomial expansion.

Proof: Since \(n \in \mathbb{N} \), for all \(n \geq 2\) we can write
n^{1/n} &=& 1 + \alpha ~ [where ~ \alpha \in \mathbb{R}^+] \nonumber \\
(n^{1/n})^n &=& (1 + \alpha)^n \nonumber \\
n &=& (1 + \alpha)^n \nonumber \\

We want to estimate \( \alpha \). If \( \alpha\) is, say, \(0\), then we’ll have \(n^{1/n} = 1+0\), meaning the limit we’re after will be \(1\). The binomial theorem says that
(a+b)^n &=& a^n + na^{n-1}b^1 + \frac{n(n-1)}{2}a^{n-2}b^2 + … \nonumber \\
&=& \sum\limits_{k=0}^n \binom{n}{k} a^{n-k}b^k\nonumber \\
(1+\alpha)^n &=& 1^n + \binom{n}{1}1^{n-1}\alpha + \binom{n}{2}1^{n-2}\alpha^2 + … \\[3pt] \nonumber \\
&>& 1 + n\alpha + \frac{n(n-1)}{2}\alpha^2 \\[3pt] \nonumber \\
&>& 1 + \frac{n(n-1)}{2}\alpha^2 \nonumber \\

So we have
1+\frac{n(n-1)}{2}\alpha^2 &<& (1+\alpha)^n = n \\[3pt] \nonumber \\
\frac{n(n-1)}{2}\alpha^2 &<& n – 1 < n \\[3pt] \nonumber \\
\alpha^2 &<& \frac{n}{\frac{n(n-1)}{2}} \\[3pt] \nonumber \\
\alpha^2 &<& \frac{2}{n-1} \\[3pt] \nonumber \\
\alpha &<& \sqrt{\frac{2}{n-1}} \nonumber \\

Thus, \(\lim_{n \to \infty}\alpha = 0 \), and \(\lim_{n \to \infty}n^{1/n} = \lim_{n \to \infty}(1+\alpha) = 1+0=1\). \(\blacksquare\)

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Interesting limit from real analysis: lim n!/n^n

In my Advanced Calculus (introductory real analysis) course at Western Michigan University, Dr. Ledyaev gave us this limit as a bonus homework problem to turn in:

\lim_{n \to \infty}\frac{n!}{n^n} = ~?

The answer is that the sequence converges and its limit is 0. Here is how I showed this:

Claim: The sequence \( \{a_n: a_n = \frac{n!}{n^n}\}\) is monotonically decreasing and bounded below.

Proof: Note that all terms of both the numerator \(n!\) and the denominator \(n^n\) are positive for all \(n \in \mathbb{N}\), so \(\{a_n\}\) is bounded below (by \(0\)). To determine whether the sequence is increasing or decreasing, we can use the ratio test:
\frac{a_{n+1}}{a_n} &=& \frac{\frac{(n+1)!}{(n+1)^{n+1}}}{\frac{n!}{n^n}} \\[3pt] \nonumber \\
&=& \frac{(n+1)!}{(n+1)^{n+1}} \cdot \frac{n^n}{n!} \\[3pt] \nonumber \\
&=& \frac{n!(n+1)}{(n+1)^{n+1}} \cdot \frac{n^n}{n!} \\[3pt] \nonumber \\
&=& \frac{n+1}{(n+1)(n+1)^n} \cdot n^n \\[3pt] \nonumber \\
&=& \frac{1}{(n+1)^n} \cdot n^n \\[3pt] \nonumber \\
&=& \frac{n^n}{(n+1)^n} < 1 \nonumber \\

The ratio \(\frac{a_{n+1}}{a_n} < 1\) for all \(n\), meaning the sequence is monotonically decreasing. Since it is bounded below and monotonically decreasing, it converges to a limit. \( \blacksquare \)

Claim: \( \lim_{n\rightarrow\infty} \frac{n!}{n^n} = 0. \)

Proof: Since \( \{a_n\} \) converges to a limit, call the limit \(L\). An important theorem states that \(\{a_{n+1}\}\) also converges to \(L\). From above,
\frac{a_{n+1}}{a_n} &=& \frac{n^n}{(n+1)^n} \\[3pt] \nonumber \\
{a_{n+1}} &=& \frac{n^n}{(n+1)^n} \cdot {a_n} \\[3pt] \nonumber \\

And using the product rule for limits,
\lim_{n \to \infty}{a_{n+1}} &=& \lim_{n \to \infty}\frac{n^n}{(n+1)^n} \cdot \lim_{n \to \infty}{a_n} \\[3pt] \nonumber \\
L &=& \lim_{n \to \infty}\frac{n^n}{(n+1)^n} \cdot L \nonumber \\

If we can show that \( \frac{n^n}{(n+1)^n} \) converges to some real number \( r \), then we will have \( L = r \cdot L \). If \( r \neq 1 \), then the only solution to that equation is \( L = 0 \). In fact, we will show that \( \frac{n^n}{(n+1)^n} \) converges to \( \frac{1}{e} \). Observe,

\frac{n^n}{(n+1)^n} = \left(\frac{n}{n+1}\right)^n = \left(\frac{1}{\frac{n+1}{n}}\right)^n = \left(\frac{1}{1+\frac{1}{n}}\right)^n = \frac{1}{\left(1+\frac{1}{n}\right)^n}

Recall that \( \lim_{}(1+\frac{1}{n})^n = e \), so by the quotient rule for limits, \( \lim{}\frac{1}{(1+\frac{1}{n})^n} = \frac{1}{e} \). Substituting \( \frac{1}{e} \) for \( r \) above, we have \( L = \frac{1}{e} \cdot L \), so \( L = \lim_{}a_n = 0 \). \( \blacksquare \)

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Cool theorem about midpoints and parallel vectors from multivariable calculus

This is a cool theorem from multivariable calculus that my professor at Western Michigan University, Steve Mackey, showed us during lecture one day early in the semester.

Theorem: Let \(a\), \(b\), \(c\), and \(d\) be any four points in \(\mathbb{R}^{3}\). Let \(M\), \(N\), \(P\), and \(Q\) be the midpoints between the adjacent points. Then vector MQ must always be identical to vector NP.

I have inserted a picture to help visualize it. I made the picture in PowerPoint, so it’s about as accurate as a hand-drawn picture.

It looks like the points are all in the same plane, but that’s just because it’s easier to draw them that way. They can be any four points in three-dimensional space.

Proof: Start with vector MQ. By the basic rules of vector addition,

MQ = MA + AQ
= 1/2 BA + 1/2 AD
= 1/2 (BA + AD)
= 1/2 BD

Now do the same with vector NP:

NP = NC + CP
= 1/2 (BC + CD)
= 1/2 BD

Thus, MQ = NP = 1/2 BD. ■

The reason this is so cool is because it holds true for any four points in \(\mathbb{R}^{3}\), which makes it very unexpected. You can rearrange the names of the points and midpoints so that the vector labeled MQ isn’t identical to the vector labeled NP, but then two other vectors will be identical. The point is that given any four points in three-dimensional space, some pair of midpoint-connecting vectors will be identical.

After I wrote this post, I realized something that Dr. Mackey didn’t mention (or at least, I didn’t write in my notes): two other vectors must also be identical. Can you show which ones?

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For some reason, I really liked The Host

Last weekend Kathy and I watched the movie The Host on Netflix. It’s based on the novel by Stephenie Meyer, whose name I just found out has no A’s in it. This movie is yet another example of why you (or at least I) shouldn’t read other people’s opinions of a movie, TV show, or book, or even peek at the average rating at a place like, IMDb, or Rotten Tomatoes, before checking it out yourself. Luckily, I didn’t, so I had no idea how down on this movie most people were, even though I knew it was based on a Stephenie Meyer novel that Kathy quit reading early on and that her friend finished but disliked. Sometimes a low rating can lower your expectations so much that you enjoy it more than you expect, but other times it can make you expect badness and notice it more acutely than you might have. This is especially true if you read negative reviews first and hear what specific criticisms people have.

I love science fiction more than any other genre, whereas Kathy couldn’t even finish Hyperion. (I mean, seriously, Hyperion! An all-time masterpiece of science fiction! Everyone should like that! At least she liked Ender’s Game, though I still haven’t been able to talk her into reading Speaker for the Dead.) Even so, this movie was her choice. We tend to take turns choosing what we watch, and she chose The Host this time. For obvious, Stephenie Meyer–related reasons, this was more of a “her” movie in our Netflix queue, though given its premise and the fact that it is science fiction and not fantasy, it should have been a movie that I’d be expected to like more than she would. It’s kind of funny, though, and a good thing, that we both end up liking most of the movies that are chosen by only one of us. Some recent examples are Moneyball (mine), What To Expect When You’re Expecting (hers), and The Perks of Being a Wallflower (hers). A good example of a movie only the chooser liked is The Messenger (mine). Oddly enough, I think two other Saoirse Ronan movies were liked less by the chooser than the other person: I didn’t think The Lovely Bones (hers) was all that bad, though I certainly have no desire to watch it again or buy it; and she might have liked Hanna (mine) a little more than I did, though I don’t think either of us will want to see it again. I also seem to remember Kathy liking Rare Exports (mine) more than I did. In these three cases and possibly others I can’t remember, the chooser’s relative dislike of a movie was probably related to their high expectations, which was why they chose it. See? Always have low expectations!

Many movie critics and fans are probably still waiting for the deeply talented Saoirse Ronan to headline a high-quality movie, but I think The Host fits the bill. I understand the widespread criticism that the movie is slow, plodding, and low on action, but I was not bored or indifferent during a single scene. It isn’t an action-packed movie, but I think that’s fine because that’s not what it was meant to be and not what it needs to be. The movie didn’t feel too long or dragged out at all.

A second common criticism is Stephenie Meyer–related: the love rectangle is not compelling, it’s too young adult-y and teenage girl-y, it’s too infected with Nicholas Sparks sappiness, and the two boys are not given enough depth or characterization to make us feel strongly about it. I also understand this criticism but disagree with it more strongly than with the first one. I didn’t think it was too pandering to a teenage-girl audience; I merely thought it was depicting what a teenage girl in Melanie’s situation might go through. I should mention that the four characters in this love rectangle are Melanie, the alien that inhabits her and controls her body (“Wanderer”), and the two aforementioned boys. I did think that Melanie’s reasons for wanting Wanderer to do this and not wanting Wanderer to say that to the two boys, as well as her brother and uncle, were not explained and fleshed out as fully they could have been, causing a little frustration and confusion in me, but this was the only aspect of the movie I found frustrating.

The third common criticism I encountered in reading the reviews after I saw the movie was that the dialog between Melanie (from inside her own mind) and Wanderer (using Melanie’s actual voice) was unintentionally funny and atrociously written. I strongly disagree. I’m no professional movie critic and know nothing about how to write movie dialog, but I found the dual-personality aspect of Melanie/Wanderer well written and expertly performed. I thought Ronan’s acting, the script, and the directing perfectly depicted the conflicted nature of a mind struggling to assert itself—to exist—and an alien struggling to justify its actions and reconcile them with its sense of morals. Other than the overall science-fiction storyline, Ronan’s portrayal of this inner struggle was the highlight of the movie for me. But maybe it could have been even better if that struggle was less about boys and more about deeper ethical and psychological issues.

The main reason I’m even writing this post, now 800-plus words in, is to respond to a truly vacuous, clueless, bafflingly stupid statement by Claudia Puig in her review of the movie for USA Today. The premise of The Host is that an alien species has invaded and populated the Earth by taking over our bodies and our minds. When an alien does this, the human host is effectively killed; their mind ceases to function if not exist altogether, and the body is controlled by the alien. The alien can access all of the host’s memories, which is especially useful for finding rebels who would prefer not to be killed and their species exterminated. The thing is, a rare human will have a strong enough psyche to rebel against its possessor and stay alive, as Melanie does. Usually when this happens, the other aliens just remove their comrade and kill the rebelious host or possess the host with a stronger, more ruthless alien. Only a few small pockets of living humans remain, in hiding or on the run. But according to Claudia Puig, these rebelious hosts and the insurgents who have avoided parasitization altogether are being irrational and primitive, because look at all the progress the aliens have created!

Like Twilight, the action is slowed by too many dull-eyed stares meant to be smoldering. A bigger problem is that the aliens are an exceedingly pleasant bunch who have rid the world of its problems. What’s not to like? The human rebellion comes off like a bunch of hillbillies angry for no justifiable reason.

I’ll repeat that in case your mind was too blindsided and dumbfounded by such idiocy to process it: An alien race wants to exterminate the human race and is damn close to doing it, and the humans who resist this eventuality are “hillbillies” who are “angry for no justifiable reason.” It boggles the mind. One is liable to sit agape in horror and depression at the psyche that could conjure such an opinion—at the types of real-world leaders, ideas, and solutions Claudia Puig would endorse and the horrors we would have to inflict upon our fellow humans to achieve her ideal order. It’s like she perceives “progress” and “peace” as some nearly tangible, identifiable things that have value on their own and should be strived for at all costs, regardless of who is doing the striving and who is benefitting from them. She must have had the same scornful reaction to all that pesky resistance the Borg face from all those hillbilly humanoids who like their species they way they are. She must not have objected to the Borg’s assimilation of the human race at the beginning of Star Trek: First Contact and must have been equally annoyed and confused at the Enterprise for going back in time and foolishly trying to stop it. There is literally no difference between the Borg and the aliens of The Host, except superficially. I never thought I could lose all respect for someone as a person from reading a mere movie review, but I never thought I’d read anything so contradictory, so insulting, to rational thought in a mere movie review.

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Probability problem from Star Trek: The Next Generation

In the first episode of season 7 of Star Trek: TNG, “Descent, part II”, a certain character (no spoilers from me!) tells another character that a medical experiment has a 60% chance of failing, meaning it will kill the subject. But, this evil character says, since he has three captives to perform the experiment on, “the odds are that at least one of the procedures will be successful.”

Is he right? Is there a >50% chance that at least one of the procedures will be successful? With a 40% chance of succeeding and three trials to get it right, it seems obvious at an intuitive level that at least one of them will succeed. But because I last watched this episode shortly after my first semester of Statistics, I thought it’d be fun to calculate the exact probability that at least one of the procedures will be successful.

From introductory Statistics, we can see that this is a relatively simple binomial experiment, with \(p\) (the probability of success) \(= .4\) and \(n = 3\). As is often the case when you need to calculate the probability that something will happen at least once, it is easiest to calculate the probability that it won’t happen, and subtract that from \(1\).


\(P(all~three~procedures~fail) = .6^3 = .216. \\
P(at~least~one~procedure~succeeds) = 1 – .216 = .784\).

There are two other ways to compute this probability. Hopefully, they yield the same result!

From an important binomial probability theorem,

\(b(x; n, p) = {n \choose x} p^x (1 – p)^{n-x}\)

where \(b\) is the probability mass function (pmf) of a binomial experiment, meaning the probability of a single outcome (as opposed to the cumulative density function, which measures the collective probability of multiple outcomes), \(x\) is the number of successes, \(n\) is the total number of trials, and \(p\) is the probability of success. The notation \({n \choose x}\) is pronounced “n choose x” and means the total number of ways to choose \(x\) outcomes out of \(n\) possible outcomes. This is a good introduction to combinations (and permutations).

First, let’s use the binomial pmf to calculate the probability of zero survivors among the three procedures:

\(b(0; 3, .4) = {3 \choose 0} (.4)^0 (1 – .4)^3 = .216\)

As it turns out in this simple example, the above computation is just \(1\cdot 1\cdot .6^3\), so basically the same as the original high-school-level computation we did first. I’ll go out on a limb and assume that subtracting this from \(1\) will give the same result as it did above.

We can also use that binomial pmf to calculate the probability that one procedure will succeed plus the probability that two will succeed plus the probability that all three will succeed. This calculation would ignore the reality that the evil experimenter will stop after the first success, but to calculate the probability that at least one procedure will succeed, we need to include all three of them.

\(b(1; 3, .4) + b(2; 3, .4) + b(3; 3, .4) \\
= {3 \choose 1} (.4)^1 (1 – .4)^2 + {3 \choose 2} (.4)^2 (1 – .4)^1 + {3 \choose 3} (.4)^3 (1 – .4)^0 \\
= .432 + .288 + .064 = .784\)

I know of one final way to calculate the probability that at least one procedure will succeed: use the TI-83’s binomcdf function. It is located under the DISTR menu, which is the 2nd option on the VARS key. The syntax is


and this tells you the cumulative probability of all outcomes in a binomial experiment from \(0\) to \(x\) successes. In this case, we are interested in the cumulative probability from \(x=1\) to \(x=3\), not \(x=0\) to \(x=3\). Therefore, in the TI-83 we can type either

\(binomcdf(3,.4,3) – binomcdf(3,.4,0)\)
\(binomcdf(3,.4,3) – binompdf(3,.4,0)\)

Both commands tell us the cumulative probability of zero successes through three successes minus the probability of zero successes, and both give \(.784\).

So we can see that our common-sense intuition was right: with a 40% chance of success, the chances are very favorable that at least one of the first three trials will produce a success.

At what point does the probability of success surpass 50%? My guess is two trials. This can be easily confirmed by changing \(n\) from \(3\) to \(2\) and calculating the binomial probability:

\(P(getting~at~least~one~success~out~of~the~first~two~trials) \\
= b(2; .4, 2) + b(2; .4, 1) = {2 \choose 2} .4^2 .6^0 + {2 \choose 1} .4^1 .6^1 \\
= .64 \\
(= 1 – b(2; .4, 0) = 1 – .36 = .64)\)

Another, more high-school-ish way to verify the probability of succeeding within the first two trials is to realize there are only two ways this could happen: succeed on the first trial, or fail on the first trial and succeed on the second:

\(P(succeed~on~the~first~trial) + P(fail~first~and~then~succeed)\\
= .4 + .6\cdot .4\\
= .64\)

Another thing our evil experimenter might be interested in is the expected value of the number of captives he will need to achieve success. Expected value is basically a weighted average. This is a good beginner’s summary of expected value. One of the first things that strikes any Statistics/Probability student about expected value is that you should hardly ever actually expect to get the expected value in an experiment, because often the expected value is impossible to achieve. For instance, your experiment only produces integer outcomes, but the expected value, being a (weighted) average, is a decimal. This is the case with many binomial experiments. The number of captives our evil experimenter will perform the procedure on is \(1\), \(2\), or \(3\), but I bet the expected value of this binomial experiment will be between \(1\) and \(2\).

The definition of expected value as a weighted average is more apt for random variables than binomial variables, but you can still calculate expected value for binomial distributions. In fact, in this case we can calculate two different expected values.

First, the simple, standard expected value of a binomial distribution: \(E(X) = np\). That is, the expected number of successes from \(n\) trials is \(n\) times the probability of success. Pretty simple, huh? So

\(E(X) = np = 3\cdot .4 = 1.2\)

So if he performed the procedure on all three captives, he should expect \(1.2\) successes. Similarly, the expected number of successes after the first two trials is \(.8\), and the expected number of successes after the first trial is \(.4\).

But that’s not the expected value I originally referred to. I said the experimenter might be interested in the expected number of procedures he’d have to perform to reach one successful procedure. I can’t find any definitive theorem or formula that tells how to calculate such an expected value in my Statistics textbook or the few places I’ve looked online, but I think it’s this:

\(1 = n(.4) \\
1/.4 = n\\
2.5 = n\)

In other words, since each experiment has a \(.4\) chance of succeeding, how many experiments do you expect to need to reach \(1\) success? What times \(.4\) equals \(1\)? It’s \(2.5\).

That’s higher than I expected. That was the number I expected to be between \(1\) and \(2\). This seems incongruent with our result above that the probability of success surpasses 50% after two trials. If the probability of success becomes better than even after two trials, shouldn’t you expect to reach one success in \(\leq 2\) trials? And shouldn’t the expected number of successes after two trials be something greater than \(1\), instead of \(.8\), then? I know both sets of calculations are correct, so this is either one of those counterintuitive results you often get in probability, or I’m framing one of the questions wrong…

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Yes, that useless pseudo-rule against ending sentences with a preposition is a useless pseudo-rule

It always has been, and it always will be.

One of the many problems with arguing with strangers on the internet is that people insist upon things that they are not sure about. They argue, often heatedly and passionately, in favor of a point that they are not 100% sure is correct. Arguing about the value of one opinion over another is OK, though clearly millions of people get way too vicious and hateful about it every day. But if you’re arguing about some issue that is reasonably easily verified as true or false, then you aren’t really arguing: you’re either explaining why someone else is wrong or proving yourself ignorant.

Believe it or not, this happens fairly often between people arguing about language and grammar on the internet. The refusal of some people to understand that they are wrong on matters of fact or on matters of terminology or definitions, and their continued insistence that they are right about something that they haven’t even bothered to look up or cite any sources on, is the reason I no longer contribute to any discussion about language and grammar on any website, even when it seems like a simple matter of fact vs. falsehood.

For example, a couple years ago I commented in a flippant, dismissive way that “Never end a sentence with a preposition” is a useless pseudo-rule up with which I shall not put, etc. Someone responded with something like this:

I would argue there is no such thing as a “useless pseudo-rule” – there is only how people speak. In certain settings, avoiding ending your clauses with a preposition can make you sound erudite and cultured, and in some settings it can make your sound like a pompous twat with a stick up your ass. The only rule is that one should be aware of how best to communicate with one’s audience.

Clearly he or she did not understand what the word “rule” means. The reason some over-zealous schoolteachers and grammarians created that rule is because they considered it wrong to violate it; that’s what rule means. My point in calling it a pseudo-rule is that it was never actually a rule of grammar because it has never been wrong to violate it; they just concocted it to enforce stiff formality in school students; nothing about the history or evolution of English suggested that it was ungrammatical or even unwise to end sentences with prepositions. The reason I called it useless is because, well, if it is neither ungrammatical nor harmful to break it, then it doesn’t help and we shouldn’t pay attention to it at all!

Notice the internal contradictions in that person’s comment:

there is no such thing as a “useless pseudo-rule” – there is only how people speak

So you agree with me.

The only rule is that one should be aware of how best to communicate with one’s audience.

So you agree with me! If it isn’t wrong to break it, then it isn’t a rule! That’s what rule means! But since people tried to pass it off as a rule for decades, it is apt to call it a pseudo-rule.

It seems accurate to say that my use of both the words “useless” and “pseudo-rule” is redundant. Since “Never end a sentence with a preposition” is not and never was a rule of English grammar, that makes it useless as a rule per se.

But I also contend that it is useless as style advice. I think my interlocutor even agreed with that, although they were so busy being indignant and contradictory that they probably wouldn’t realize it:

In certain settings, avoiding ending your clauses with a preposition can make you sound erudite and cultured, and in some settings it can make your sound like a pompous twat with a stick up your ass.

That’s true of like a hundred different words, phrases, and grammatical constructs! There are no rules for or against using them! If it is true that X can be good or bad, then “It is a good idea to do X” is bad style advice, because it might in fact be a bad idea to do X. If you want to change your “rule” to “If the situation calls for it, then do X”, then, first, that’s different from “Always do X” or “Never do X”, which is the subject of this post and my original comment that spurred this idiocy, and second, it’s unhelpful (useless!) advice anyway because it doesn’t help writers determine when X is called for.

It is grammatical to end a sentence with a preposition or not. It can be good style to end a sentence with a preposition or not. If you go out of your way to write in an unnatural, stuffy way to put the preposition at the end of your clause, it will probably make you sound like a pompous twat who doesn’t recognize good style. That’s the whole of it! It’s not a rule and never was. It is not (often) good style advice. There is nothing to argue over. In some cases you can call it bad style, in some cases you can call it good style, and in other cases it doesn’t really matter. That’s why it’s not a rule at all, and that’s why it’s useless to even make an issue out of it.

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The infinitude of prime numbers—Euclid’s proof in my own words

Euclid is believed to be the first mathematician to prove that there are infinitely many prime numbers. Most of us learn only that Euclid established and codified the framework of two- and three-dimensional geometry, but he accomplished far more than that. I think middle-schoolers and high-schoolers are given the impression that Elements was merely a book about the rules and characteristics of geometric shapes, angles, and relationships, but it contained important theorems about number theory, arithmetic, and algebra as well. He was a superb number-theorist and second only to Archimedes among mathematicians of antiquity.

In contradiction with his pigeonholing by school students as a mere geometer, Euclid’s most famous and perhaps most important individual contribution to mathematics is his proof that there are infinitely many prime numbers. This proof also appears in Elements. In my non-scientific experience browsing math websites and blags, Euclid’s proof of the infinitude of primes is almost universally considered one of the most beautiful, elegant (they love that word) proofs in the history of mathematics.

When I read the proof at Wikipedia, I understood it pretty well after reading it a few times, but I found that several months later, I couldn’t remember it or restate it. So I sought out a few more versions of his proof and found a couple that explained it even better. Now I know it for good because I understand it even better than I did the first time—naturally, I understand it best when stated in my own words—and I’ll publish it here for anyone who might be helped by seeing my version:

First, remember that every integer greater than 1 has a unique prime factorization, i.e., it can be written as a unique combination (product) of prime numbers or is prime itself. This is called the fundamental theorem of arithmetic and was also first proved by Euclid. For instance, 12 is factored as 2 x 2 x 3, and 36 is factored as 2 x 2 x 3 x 3. (These examples make it clear that the number of repetitions of each prime is part of the uniqueness, though their order doesn’t matter.)

Now to the meat of the proof. It is basically a proof by contradiction: assume the conjecture (“there are infinitely many prime numbers”) is false, and see that it cannot actually be false. So assume there are finitely many primes, or that you have gone reeeeeaalllllly far out along the number line and have reached the last prime number. Now multiply all of the prime numbers together. Call their product N. Obviously N is not prime because it has a prime factorization: all of the prime numbers. (As a side note, observe that N must be even because it has 2 as a factor. In fact, you know its last digit is 0 because it is divisible by both 5 and 2.)

Add 1 to N to get a new number, N + 1. There are exactly two possibilities for the type of number that N + 1 is: it is either prime or composite. If it is prime, then our assumption was false, because this prime is larger than the supposed largest prime.

If N + 1 is composite, then it has a unique prime factorization. What is that factorization? Well, we don’t know what it is, but we know what it isn’t: it does not include any of the prime factors of N. This is because N + 1 would give a remainder of 1 if divided by any one of the primes, or any combination of the primes, that produced N—by the nature of addition and multiplication, you would have to add 2 to N to get a number that is even divisible by the lowest prime, 2. And you would have to add 3 to get the next number divisible by 3, add 5 to get the next number divisible by 5, etc. Since N is exactly divisible by any and all combinations of those primes, N + 1 would give a remainder of 1 (as opposed to some other remainder) when divided by any combination of them. This means N + 1 is not divisible by any prime factor of N. Therefore, composite N + 1 has some prime factor that was not included in the factors of N, so the supposed list of all primes missed at least one.

This process can be repeated ad infinitum for any N and N + 1 and their prime factors, meaning any finite list of primes cannot include them all, so there are infinitely many primes.

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Stupid, pedantic formalism fetishists are turning however into a conjunction that is replacing mid-sentence but

The more I hear people speak and read their writing in all kinds of situations, in real life or in TV/movies, in blog posts, essays, or informal discussion threads, in scripted dialogue or narration or while ad-libbing, the more often I notice however being used as a direct replacement for but, with the same intonation of the voice, cadence of speech, and use of commas as with but. The reason for this is easy to guess: English-speaking students across the world have been misinformed about the appropriateness of various words by at least two generations of pedantic turgidity/formalism fetishists, mainly in English class. Hundreds of millions of people have been taught that But (and its sister conjunction And) absolutely should not, under any circumstances, be used at the beginning of sentences in anything approaching a formal, academic context.

But this admonition is simply wrong. It is pure fantasy. It has no basis in grammar, history, or the traits of polished, edited, formal English prose. It appears in no English style guides in existence. Some person(s) just made it up out of the blue decades ago, presumably to train young students away from starting every other sentence with And. There are editors and entire editing companies (I work for one) that replace every sentence-initial But with However,—and they don’t even do it manually; they run a macro in Microsoft Word that automatically makes this replacement. This is how categorically, inexcusably wrong they see sentence-initial conjunctions. And based on what? It is literally a rule that they, or someone they learned “English” from, conjured up from their own imaginations. Bryan Garner explains why this superstition is completely unfounded. Sentence-initial conjunctions are not less formal, less educated, or less proper or appropriate in any imaginable context or register. Some people just need to get that through their heads and un-learn the misinformation they were inculcated with as youngsters.

In the meantime, until the tide of superstition turns away from this love of sentence-initial However and intolerance of sentence-initial conjunctions, we will continue to see however creep into the proper territory of but in the middle of sentences as well.

I think it’s ironic that this widespread mis-use of however has been caused by the formalism pedants themselves. They hate the mis-use of punctuation around mid-sentence however (so do I), and they hate the appropriation of new meanings or functions by existing words (I do sometimes as well). But their zealotry against sentence-initial conjunctions has had such an effect on self-conscious, unconfident, under-educated English speakers that millions of them now use however as a coordinating conjunction in direct replacement of but, presumably because they want to sound more smart, formal, or fancy. But they don’t. They sound less educated and less competent. It makes it clear that they don’t know the difference between different types of conjunctions (even at a subconscious level), don’t understand the difference between the adverb however and the subordinating conjunction however, don’t know how to use commas and semicolons correctly, and need to rely on turgid formalism instead of the content of their message to have the desired impact on their audience.

Posted in Grammar, Language | Leave a comment

Summer miscellany

Typical Redditor: Reads seven Harry Potter books that are about love, friendship, family, innocence, loyalty, cooperation, the triumph of good over evil, the corrupting influence of power, and the good that will come to good-hearted people, and all he can focus on is the inconsistencies in magical spells, magical items, and magical abilities.

Today I learned: The garbanzo bean and the chickpea are the same thing!

It’s a shame we can’t leave Christmas lights up all year. They’re so pretty and fun and festive. I guess your neighbors would really get sick of them, and I personally wouldn’t want to leave them up all year because that would decrease the specialness of Christmas lights at Christmas time. But I wish that wouldn’t happen. I wish our homes and cities could look like that all the time and remain special.

Ira and Asa sound like women’s names. Every time I hear the name Ira Gershwin, I think of a female blues or jazz singer, and every time I hear the name Asa, such as Asa Phelps in an awesome episode of The Simpsons, it makes me think of a woman’s name. How many other men’s names end in A? It’s just confusing.

When I was a young child, the first car I remember my parents owning was a 1987 or 1988 Ford Taurus. One of the main things I remember about it was the digital speedometer display. I think I’ve only seen a digital speedometer in one other car since our Taurus. I remember every time the car started up, all of the pixels or bars of the digital display would temporarily light up, so that the speed said 88 for a second, and every time I saw that, I thought of the Delorean in Back to the Future going 88 mph. I’d say to my mom, “Oh, no, Mom, we’re going to go back in time!”

Are there even any toilet paper rolls that aren’t “double” rolls anymore? At what point does it become pointless to continue referring to a toilet paper roll that is now standard-size as a “double roll”? The same can be asked of laundry detergent: Almost all of them are now 2x- or 3x-concentrated detergents now, so is any company ever going to drop the “2x” or “3x” label? They’re probably hesitant to because then it would seem like theirs is less concentrated than other brands and that you’d get fewer washes out of the same-size bottle, even though the number of loads is clearly indicated on every bottle. We could be stuck with meaningless “double” and “2x” and “3x” on these products forever, or at least as long as they are made.

I’m making Kathy watch Star Trek: The Next Generation with me on Netflix, and we’re already on the sixth season. I remember wishing, when I watched TNG as a child and then a college student, that Geordi would say the line, “But you don’t have to take my word for it,” or at least, “You don’t have to take my word for it,” as the actor LeVar Burton did on Reading Rainbow. I guess serious dramas can’t cheapen themselves with allusionary humor like that, but I think it would have been worth it. Castle has made all kinds of Firefly references, and it’s no worse for it.

If there were a hell and each person’s hell were individualized to their darkest fear or worst way to spend eternity, my personal hell would be forever being on the verge of a sneeze but never being able to sneeze.

Posted in Books, Entertainment, Food, Life, TV | 2 Comments

The coins in the dark puzzle

In the New York Times’ Numberplay column yesterday, Gary Antonick presents an old but good logic puzzle:

There are twenty-six coins lying on a table in a totally dark room. Ten are heads and sixteen are tails. In the dark you cannot feel or see if a coin is heads up or tails up but you may move them or turn any of them over. Separate the coins into two groups so that each group has the same number of coins heads up as the other group. (No tricks are involved.)

I know it’s an old (and good) puzzle because I had read it before, and for this reason, some inkling of the solution was lurking in the back of my brain somewhere. I remember it being relatively simple but elegant. (It seems like the solution to every logic puzzle and half of mathematical proofs are elegant, and that “simple” solutions and “elegant” solutions are heavily overlapping subsets. But on second thought, maybe this one is merely cool or neat.)

The solution
My mostly faded memory of that solution involved moving the coins into two piles and turning over all of one pile, or 10 of one (or both) piles, or 16 of one (or both) piles, or maybe half of the coins, so I tried a few strategies in my head until I came across the right one.

You can’t see the coins, but you can feel them and count them as you move them and flip them. Clearly, you need to separate them into two groups before flipping them, because if you try a strategy of flipping and then grouping them, you won’t be able to tell which ones you’re moving!

Separate the 26 coins into a left and a right group of 10 and 16, respectively. The left group has between 0 and 10 coins that are heads up, and the right group has 10 minus that number heads up. Assume the left group has nothing but 10 heads-up coins, meaning the right group has 0 heads-up coins. Flip over all 10 coins in the left group, resulting in 0 heads-up coins, the same as the right group.

Or assume the left group has 9 heads-up coins and 1 tails-up coin. Flip all 10 over, resulting in 1 heads-up coin, the same as the right group.

Or assume the left group has 8 heads-up coins and 2 tails-up coins. Flip all 10 over, resulting in 2 heads-up coins, the same as the right group.

You can verify that this pattern continues all the way down to 0 heads-up coins in the left group, which, when flipped over, become 10 heads-up coins, the same as the right group.

So no matter how many heads-up coins start in the left group of 10, after flipping them all over, this number becomes the same as the number of heads-up coins on the right side!

The math behind it
I did not arrive at that solution by deduction or the use of any math or equations. Rather, I arrived at it by drawing on my vague memory of reading the solution two or three years ago and by performing a lot of trial and error in my head. When I pose this logic puzzle to my children several years from now, I will tell them what I think is the best way to solve this puzzle: to try a bunch of things until you come up with the winning strategy and explain why it works afterwards, not to come up with a theory to explain what should work and then verify it with several iterations.

But there is, of course, algebra to explain why that solution works:

In the left group of 10 coins, there are n heads and 10–n tails. In the right group, there are 10–n heads and 16–(10–n)=6+n tails. Of those four quantities, it is easy to notice that two of them are represented by the same expression: there are 10–n tails in the left group and 10–n heads in the second group. (This happens because we have chosen the group sizes wisely.) To take advantage of this fact, we need to perform some action that makes the two 10–n expressions refer to two sets of heads instead of one set of heads and one set of tails. As we now know, we must flip over all 10 coins in the left group. This converts the n heads to n tails (not entirely relevant) and the 10–n tails to 10–n heads. Now there are 10–n heads in both groups!

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Farkers are antisocial, maladjusted creeps

Yet again, immediately upon ignoring my better judgment and revisiting one lazy afternoon, I was reminded why I avoid even reading its discussion threads for years at a time. By and large, its members seem to be oblivious, ignorant, sheltered, myopic, antisocial creeps and douchebags who retreat to the safe confines of websites populated by others like them because their real lives are so pathetic and dysfunctional, so disconnected from the real world.

The discussion thread that most recently reminded me of these facts was this one, which was about this column written by a mother who is concerned about the violent, debasing, disturbing porn video her 11-year-old son was shown by schoolmates. It is an interesting, well-written, sensible, level-headed column, so I’ll quote it at length:

Last week my son told me he had watched something horrible online. Something sexual where the young women involved seemed coerced into an act that was brutal and disgusting, not just to an uninitiated 11-year-old, prone to anxiety, but to anyone with a shred of humanity. …

He watched it because one of his new friends told him he should – because it was “funny”. …

He said he had been horrified watching a short video online but was unable to stop thinking about it. He told me he couldn’t “unsee” it, and how he felt his childhood was effectively over. He had not told me anything as he thought I’d be angry with him.

So I’m left cuddling my son, who is strung between childhood and adolescence. He tells me that everything is moving too fast. We talk about his observation that you can’t “unsee” stuff. We talk about how you can’t go backwards. And we talk about the importance of moving forward. I tell him how he needs to grow older so that the world can have a great man in their midst.

Then we talk about the porn industry and how often it portrays women as passive beings. We talk about how women in the video he saw are real people, forced into very unpleasant situations – perhaps mums and sisters, certainly daughters – and we talk about how very far from “funny” videos like these really are. We also talk about how sometimes women choose to go into the sex industry and that when the work is on their terms, that’s OK.

We talk about why people might access porn. That being curious is completely natural. We talk about the difference between what he watched that was brutal and violent and something that the majority of people might find titillating.

I am looking at this through the eyes of my 11-year-old. He can see that there are gradations of porn. Some of it, though an unrealistic view of sex between two consenting adults, is bearable and allows you to retain a basic positive belief in the world. But then there is the degrading, shockingly violent porn that showed him a dark underbelly of an online world that until that moment was largely populated by Minecraft and Harry Potter. Faced with this hideous new information, he simply doesn’t know where to file it.

After watching the video, he changed his settings on his phone to strict. He was the last in his year to get a phone. I held out giving him one, not due to fear of him having access to porn, but because I question why someone his age needs a phone.

A month ago, however, I caved in to his peer pressure. I want him, for his sake, to fit in where he can.
I use the internet all the time. I am very active on social media. I’ve seen porn – most of us have. But I recognise that this time the internet has crept up and slapped me right in the face.

This week, one of this country’s major teaching unions published research suggesting that 90 per cent of eight to 16-year-olds had at some stage accessed pornography on the internet – many without meaning to – and asked for training in how to deliver lessons warning of the dangers of pornography. This is not about censorship but education. It’s about having frank discussions about the content that our generation has created and giving it a context for the younger generations who are consuming and replicating it.

Children have always found ways to discover the world on their own and that’s essential and it’s important that adults don’t interfere with that discovery and self-education. But it’s our adult world that is increasingly seeping into their childhood, at the touch of a button. And when the mark of fitting in with your mates becomes watching a “funny” video, which is essentially violent porn that changes your world in an instant, then I think we, as a society, need to reassess things.

The Farkers reveal their ugly true colors by jumping all over the author for being a “bad parent” by failing to prepare her 11-year-old (!) for seeing violent, debasing, abusive, coerced pornography, and assuming that what he saw wasn’t so bad, and saying that all normal kids see that stuff sooner or later so 11 is just fine, and the child will never adjust to the real world with his parents sheltering him so much, and they’re obviously cheap, oppressive bastards for waiting all the way until age 11 to buy him a smart phone, and this mother clearly just wants to censor the internet. It is hard to imagine such a large proportion of an entire online community missing the point so badly and failing to address a single issue that the column was actually about. See for yourself:

Yes, oh great and knowledgeable parent person, shelter your son more. That way, he’ll be guaranteed to grow up to become a well-adjusted member of society. There will definitely not be any negative repercussions from trying to protect him from things which he does not understand, and there will be zero chance of your little snowflake having sex-related psychological issues in his adult relationships with women.

The mother was not trying to shelter her son or protect him from sexual content; she was trying to protect him from violent, disturbing, debasing pornography—that means rape! (Only one contributor to that discussion thread even mentions rape, so that says something about Farkers’ understanding of the column and the issue in question.) It is hard to imagine this ignoramus being more wrong about the mother’s point or about what types of experiences will poison the boy’s future adult relationships. The way children get screwed up psychologically and lose the ability to have happy, healthy sexual relationships is by seeing and experiencing the exact things this boy claims to have seen. The people who will have sex-related psychological issues are the ones who think debasing, dehumanizing pornography is “funny” or arousing in any way. Until they are probably in their mid- to late teens, children cannot cope with or understand certain sights and experiences, violent sexual assault among them. Many people who go into pornography, who suffer from dysfunctional intimate relationships, or who become sex offenders have a disturbed conception of sex, intimacy, violence, abuse, and interpersonal relationships, which often results from exposure to something sexual and/or violent at too young an age to process it and cope with it. True, most of these were probably victims of actual abuse and not of an unexpected porn video, but exposure to any sexual content at a very young age and exposure to this type of disturbing sexual abuse at a pre-teen age can very well cause long-term psychological harm. Much more than “sheltering” a boy from footage that, according to the mother, would disgust anyone with a shred of humanity. I trust her assessment of the video more than this basement-dwelling sociopath who probably is aroused by that type of thing.

wait, the mom in the article was rational while taking to her son. at least she was able to talk to him AFTER the FACT. (why she didnt talk to him BEFORE the FACT was actual the cause of the problem. talk to your kids. NOW! Whoops too late.)

“educating pupils to the dangers of viewing internet pornography ”
Yup, the author of the article agrees with the crazies. PORN KILLS!!!!
YES we should all be talking to kids about sex and porn. (teachers and parents, probably not farkers…)
YEs they are going to see it either way, no matter what you do. His not having a phone just meant that he would see it on his friends phone.

The sex talks that parents are supposed to have with their children do not involve describing the types of disgusting, dehumanizing pornography that sick fucks find stimulating or arousing. The author of the column seems to imply that she and her husband have talked to their son about sex to some degree, so they have a mature enough relationship to be able to talk about porn at this time. This doesn’t sound like the first sex-related talk the author has had with her son. (If parents haven’t even broached the topic of sex with their children, then I guarantee they can’t all of a sudden have a calm, rational, fruitful discussion about dehumanizing sexual abuse one night.)

So this Farker’s “point”, if you want to call it that, that the parents were negligent for not talking to their son about sex yet, is almost certainly nullified by the facts. If this Farker’s point was that the parents should have talked to their son about the violent, disturbing, dehumanizing rape-pornography that’s out there, and described it in detail, possibly by finding examples to play for him, so that he wouldn’t be shocked by it when he found it on his own, then this Farker’s disconnection with the real world and human decency is self-evident. If this Farker’s point is that the parents should have already talked to their son about the existence of violent, disgusting pornography but without going into any detail or description, then I don’t see how that would have helped anything in this case. If this Farker’s point is that parents should talk to their children about sex and include some information about the basics of pornography (its purpose, the fact that it’s acceptable and hurts no one as long as its consensual), but not mention violent, dehumanizing pornography, then that also wouldn’t help anything.

In summary, this Farker doesn’t have any discernible point except to lash out at a parent because it makes him feel good to get on his high horse about over-protective parents and the pussification of children, when neither of these factors is relevant.

Several other Farkers were guilty of the same basic kind of misunderstanding: thinking that the issue at hand is talking to children about sex (and even pornography) and that this mother’s failing was that she waited too long, until after her son had been horrified by an online video, to talk to him about sex. Here are the four other such comments I noticed:

1. If you can’t take five lousey minutes to talk to you kids about sex. . .

2. If only there were a way for parents to help their children understand such complex issues.

/then again ,I suppose I’m asking too much as their are innumerable adults walking around with childish notions about sex

3. The problem is shiatty parents. You need to teach your kids when it’s acceptable to view porn, drink, and curse.

4. People are naive to think that their “children” are not having or thinking about sex. Humans are naturally curious. Unless you use the fear of god to fark them up.

Let me make this clear: This column is not about sex or talking to children about sex. It is about an 11-year-old child, who is not even an adolescent and is three or four years away from even starting high school (or whatever they call it across the pond), who was disturbed, troubled, and disgusted by a violent, debasing, dehumanizing video of sexual abuse that he was tricked into watching by his peers, who called it “funny”. (Whether they called it such to trick him into thinking it was a comedic video or they actually found it funny, I don’t know. Probably the former.) No parents’ sex talk with their children should include this type of abusive, coercive rape-pornography, except to warn their children to stay away from it and to remind them that there are some very bad people in the world who don’t respect others and who need to hurt others to feel good about themselves. People should not be exposed to certain things at all in their early childhood years and should only be exposed to palatable, non-disturbing sex and violence as they grow into their pre-teen and teenage years. This mother is rightly concerned that her son and millions of other children could be and are being irrevocably damaged by violent, unsettling images of involuntary abuse and debasement. (Even if the video was fiction staged by voluntary actors, it clearly sounds like way too much for an 11-year-old to see. He didn’t think it was fiction; that’s enough. The author opines that no one with a shred of humanity should react with anything but repulsion to it, and I trust her opinion much more than maladjusted Farkers’.)

No human being should ever perform any violent or coercive act, especially a dehumanizing and abusive act like the one in question; they certainly shouldn’t film it; it shouldn’t exist as pornography, whether staged or real, because no one should be aroused or in any other way turned on by violence and abuse; and if you are aroused by rape and debasement, then you are a psychopath who is unfit for human society.

The purpose of pornography is to sexually arouse the viewers to enhance their sexual experience either alone or with their partner(s). If you are sexually aroused by brutality, coercion, and debasement, then you have a mental illness and need professional help, possibly institutionalization. You are the one who is maladjusted and dysfunctional. I am not talking about insistent or even forceful persuasion in which one person is reluctant but then gives in to carnal desire voluntarily. I am not talking about objectification, which is fundamentally different from dehumanization. I am not talking about the entertainment value of violence and bloodshed in video games, police dramas, and war movies. I am talking about rape, forceful and involuntary, whether it is all fiction or not. The human brain should not be wired to be sexually aroused by any type of violence or coercion, and if yours is, then it is abnormal, and not in the Albert Einstein/Leonardo da Vinci kind of way. The purpose of brutal, debasing rape-pornography is not to add a level of rawness or realism to a story about crime, or to comment on our violent society, or to depict how evil the rapist is for getting satisfaction out of that act, or to make us sympathize with the victim, or to provide an entertaining, bloody fight of good guys vs. bad guys; it is to arouse the viewer by showing brutal rape, as well as depicting the arousal and satisfaction of the rapist. Therefore, just as the occurrence of any violent act is disheartening to any decent human, there mere existence of brutal rape-pornography is something all decent humans should oppose and keep children from seeing, because we shouldn’t want any fellow human to be so disturbed as to be aroused by it. We should never want to see it—we should be disgusted and disappointed it even exists, for the reasons above—and we should be doubly opposed to our children seeing it.

That does not mean we should pretend it doesn’t exist. That does not mean we shouldn’t warn our children about miswired psychopaths who are sexually aroused by violence. That does not mean we shouldn’t teach our children about evil and violence. It simply means children who are barely out of elementary school and cannot possibly understand the complexities of intimacy, sex, and the psychological and physical abuse of rape should not be exposed to disturbing rape-pornography that is more likely to scar them than enlighten them. At least let them get into puberty before exposing them to such overwhelming stuff.

It is bad enough that brutal atrocities have been committed by murderers, rapists, generals, dictators, and other psychopaths throughout history; at least the genocides and wars and serial killings and individual acts of abuse, rape, and murder are universally seen as deplorable acts of violence. But what is even worse is when deplorable acts of violence, whether fictional or real, are depicted as serving the sexual pleasure of the abusers and are filmed for the sexual enjoyment of viewers. It is not healthy for anyone to be aroused by documentary footage or reenactments of wars, genocides, murders, or rapes, and it is equally unhealthy for anyone to be aroused by pornography that is brutal, coercive, and dehumanizing, whether fictional or real. That is why children should never see it and why adults probably shouldn’t, either.

“Too young to have a phone.”
For 8 billion years parents have been using this moronic chestnut to keep from having to spend money and to punish their children because when they were kids they had to walk up hill both ways in the snow.

But most 11-year-olds don’t need a cell phone. What do they need it for? They can’t drive yet. It is extremely unlikely that they will go anywhere or be in any situation that the parents don’t know about. Their parents or other parents chauffeur them almost everywhere they go. Most households with children have a land line that the children can use to talk to their friends. The one and only reason any child wants a cell phone is as a status symbol to compare with the other children’s. The only good reason to give a middle-schooler a cell phone is to use in emergencies where using a land line is impossible, which this Farker doesn’t bring up. He assumes parents deny their children cell phones and other gadgets to avoid spending any money above the bare necessities of life and to make their children suffer through all the parents had to suffer through at that age. Wrong and wrong.

Has any kid, ever, said they want to remain a child?

Um, yes, plenty. I have a feeling most children, at least most children who are relatively happy, healthy, well-adjusted children with comfortable, pleasant home lives have wanted to avoid growing up at some point in their adolescence. This feeling was echoed by three or four other Farkers who responded to that comment.

I blame the parent. The kid was smart enough, after viewing said pron, to change the settings on the phone to ‘strict’…the parent should of done that immediately prior to giving their kid the phone. The author’s stated how they saw pron all over the internet and still gave the phone to their kid with unlimited viewing ability.

Parental FAIL.

You don’t need to blame the parent, the child, or the internet that produced the pornography. If anyone, you should blame the child’s degenerate classmates for fooling him into viewing a video by describing it as “funny” and by finding it the slightest bit enjoyable or entertaining. Maybe the parent didn’t know the phone had a “strict” setting (I’m almost certain mine doesn’t), or maybe they didn’t want to shelter the child by being over-protective, which this Farker probably would have objected to if the parent had originally taken those precautions. I don’t want to put words in his mouth, though, so suffice it to say that there’s no need to blame the parent or child or the technology at all.

From the article: He told me he couldn’t “unsee” it, and how he felt his childhood was effectively over.

It’s pretty damn obvious that the author is putting words into her kid’s mouth here. I don’t doubt that he was shocked by seeing something extreme, but there’s no way in hell an 11 year old actually said that.

It’s pretty damn obvious that clueless Farkers will grab onto any person, situation, or story that doesn’t fit their myopic worldview and rail against it with whatever comes to mind, regardless of the validity of their argument.

This Farker seems skeptical that an 11-year-old child would use the word “unsee” or say he felt his childhood was effectively over. First, where have you encountered the word “unsee” in your travels? Mainly on the internet, of course! So this Farker is saying he finds it unlikely that a child of the internet age would use internet terminology in real life? And not only is that unlikely, but the person who put that word into the child’s mouth was the mother, who, while she describes herself as an active, frequent internet user, is not a child of the internet age, is probably in her late 30’s or early 40’s, and is therefore not in the demographic group of most frequent users of internet jargon. No, between the two of them, it is far more likely that the child is the one who used the word “unsee”, rendering yet another Farker’s “logic” completely invalid.

As for the unlikelihood that the child actually said he felt like his childhood was effectively over, that wasn’t quoted and so was obviously the mother’s words. She was paraphrasing the gist of his feelings in her own words. Any half-literate simpleton could deduce that from this thing we call punctuation.

Yet again, a Farker demonstrates his incapacity for a considered, sensible evaluation of the mother’s position, preferring instead to lash out at what he considers an easy target, only to fail to make a single valid point.

But, if there were an instructional on the best way for a mom to not give her son weird sexual hangups for life…this would be the opposite of it.

This mother (and, probably, the father or her partner) has talked to her son about sex and is capable of having a thoughtful, sensible, mature discussion with him about pornography and porno actors. This is exactly what the parents of an 11-year-old should be doing to raise a mature, composed, sexually healthy adult. In contrast, an example of something that would give a person “weird sexual hangups for life” would be seeing shocking, disgusting, dehumanizing, coercive sexual abuse as a pre-teen and being surrounded by peers who think it has any redeeming qualities. These are facts that are obvious to anyone who has actually been through all of childhood, grown up into an adult, had normal, healthy relationships, developed a sense of respect for women, learned that we should be outraged at any and all violence and abuse, and acquired something resembling a decent moral compass.

I am of the opinion that the incident she is describing… never actually happened. Makes for a convenient excuse for a porn-hating article.

The ethics of porn are complex. Some people think everything is simple. Therefore, they choose an opinion that lets them think a complex subject is not complex. This opinion is usually wrong.

This Farker completely ignores an important and long passage of the column. I’ll quote it again:

Then we talk about the porn industry and how often it portrays women as passive beings. We talk about how women in the video he saw are real people, forced into very unpleasant situations – perhaps mums and sisters, certainly daughters – and we talk about how very far from “funny” videos like these really are. We also talk about how sometimes women choose to go into the sex industry and that when the work is on their terms, that’s OK.

We talk about why people might access porn. That being curious is completely natural. We talk about the difference between what he watched that was brutal and violent and something that the majority of people might find titillating.

I am looking at this through the eyes of my 11-year-old. He can see that there are gradations of porn. Some of it, though an unrealistic view of sex between two consenting adults, is bearable and allows you to retain a basic positive belief in the world. But then there is the degrading, shockingly violent porn that showed him a dark underbelly of an online world that until that moment was largely populated by Minecraft and Harry Potter. Faced with this hideous new information, he simply doesn’t know where to file it.

That’s three paragraphs of the mother displaying an understanding that the entire issue of pornography is complex (especially as relates to children) and that it’s not possible to demonize all porn or shun it in a black-and-white manner. And she probably passed at least some of that mature understanding of this complex issue on to her child. She discussed it with him like a mature, responsible parent. This is the polar opposite of what this Farker and most others in this discussion thread have done. This Farker’s conclusion is: “Nope, she made it up. She just wants to demonize porn. Her thought processes and conclusions are wrong.” That is not complex or nuanced but rather jumps straight to conclusions that are not only unsupported by the column but are in fact directly and specifically contradicted by the column. So in fairness to this commenter, he probably didn’t even read the whole thing; that’s why his points are so stupid and vapid.

Yeah, for all we know it was just some video of a chick taking a jizzblast to the face. Hardly violent, but to Pruneface McUptight in the article it’s all ZOMG, VIOLENCE AGAINST WIMMINZ!!!!

Translation: “I didn’t read much of the mother’s column, so I’m just going to take the lazy approach of assuming what I want so that I can bash her as an uptight prude, because this fits more nicely into my myopic worldview, viz., that everyone in the history of the world who isn’t a Farker or other basement dweller who agrees with me most of the time is worthy of scorn and condescension.”

Jesus christ, the kid is 11 so
a) he should already be familiar with porn
b) should be infatuated with it-not scarred by it.
c) might have teh gheys
d) is a pussy

When I was 11 I was smoking unfiltered lucky strikes (quit @ 16), smoking pot (no comment), drinking (never stopped) and looking for a connection for acid & coke (came around a year or two later). Violent porn and Faces of death were old news.

Kids these days are never allowed to grow up or make mistakes, thats why they are all pussies.

Again, this is not about porn or sex, it is about disgusting violence and debasement. It is about the depiction of an immoral, inhuman, illegal violation as sexually enjoyable, filmed for sexual enjoyment. Kids who miss out on seeing violent, debasing rape-porn don’t grow up to be sissies who are scared of sex because of it. In contrast, kids who see violent rape-pornography might very well be more likely to grow up to be sexual abusers and rapists. I don’t have any data from longitudinal or retrospective studies to back that up, but my main evidence is that anyone who is aroused by that already has a disturbed psyche, and only someone with a disturbed psyche can become a sexual abuser.

*sigh* And the quest to censor the internet continues.

There is nothing explicit or implicit in the mother’s column about censoring the internet. Again, instead of addressing the issues the mother brings up—she doesn’t actually propose any concrete solutions to the problem of children viewing porn at ever-younger ages; that isn’t really the point of the column—this Farker just jumps on a simple issue (censorship) that he feels strongly about and that no sensible person could possibly oppose him on and bashes the author for her (imagined) wrongheadedness. Lazy and stupid—par for the course for

That was the most pretentious “think about the children” article I have ever read.

Poor kid.

It was the opposite of pretentious. It was sober and sensible. This Farker probably didn’t actually read the column. If he did, I feel sorry for him that that’s his best guess at the meaning of the column.

Finally, I should mention that eventually several Farkers did chime in with sensible viewpoints and facts that contradicted the sociopaths above, though none of them was very forceful or eloquent about it. That’s the way it goes with online communities like the hivemind is antisocial, ignorant, puerile, myopic, and fervently, crusadingly intolerant of differing opinions—precisely the reason I abandoned Fark and haven’t even logged in since about 2007—and the minority thinkers have to tread lightly to avoid offending too many sheep and starting flame wars all the time.

Posted in Children, Interwebs, Morans | 1 Comment

Seeing Brave in the theater

Shortly after it came out, Kathy and I went to see the Brave in the theater. It was a good movie and all, but one thing I’ll never forget is seeing a mother and two children walking back into the theater during the middle of the movie—in fact, I think it was an important, revealing scene with Merida and that old bear—and the three of them couldn’t care less about what was going on on screen. This was a huge, momentous, plot-altering scene of the movie, and during their whole trip into the theater room, up the stairs, and back into their seats, not even the mother acted the slightest bit interested in turning her head occasionally to the screen to see what was going on, trying to piece together what she had missed, or shushing her kids so they wouldn’t have to miss any more.

It wasn’t that they were being rude; in fact, her kids might not have been making any noise, though that seems doubtful, because they’re kids. What struck me was how little—none, it seemed—this woman cared about a good, interesting, well-told movie and what she had missed of it and how she could catch up with the plot after missing several minutes. Paying attention to and enjoying the movie just weren’t among her goals for this movie outing. Such concerns weren’t even on her radar. Her purpose in taking her kids to see Brave was to go to a public place for a relatively easy, sedentary activity, to avoid the summer heat, and to put her kids in front of some big, colorful, moving pictures for an hour and a half.

I made remarks along these lines, in much briefer terms, either during or after the movie to Kathy, and she agreed it was kind of funny or weird. Unrelatable, at least. We couldn’t relate to someone who would go to a movie, even if it’s a kids’ movie, and have no interest in following and enjoying the whole thing. Now, I fully expect to take our children to some movies or other events where my main goal is to entertain and distract them for a couple hours to ease the burden of caring for them once in a while, but I also expect to at least pay attention to the whole movie and care about following it all. And let’s not forget that kids’ movies these days are on average better than ever, with the possible exception of the wonderful early- to mid-1990’s Disney movies (Brave won the Oscar for best animated film). So anyone who was paying attention to Brave would have become interested and invested in the movie almost immediately. But once the duty of taking her charges to the bathroom or the concession stand arose, this mother’s interest in the movie apparently disappeared. I hope Kathy and I never get like that with our children in the theater. Or even at home when we’re actively watching a movie with our children.

Posted in Children, Life | Leave a comment

Sentences I like

This is my second post about sentences I’ve encountered that struck me as very well-worded, poignant, impactful sentences that I would have been proud to write. (Here is the first post.)

His teeth felt strange in his head, tiny tombstones set in pink moist earth.
—Stephen King, The Gunslinger

Those gods might not punish at once, but sooner or later the penance would have to be paid…and the longer the wait, the greater the weight.
—Stephen King, The Waste Lands

It was only this clearing that had heard the full and painful measure of her grief; to the stream she had spoken it, and the stream had carried it away.
—Stephen King, Wizard and Glass

And beneath them as the night latened and the moon set, this borderland world turned like a dying clock.
—Stephen King, Wolves of the Calla

We spread the time as we can, but in the end the world takes it all back.
—Stephen King, Wolves of the Calla

A mist hung over the Devar-tet Whye like the river’s own spent breath.
—Stephen King, Song of Susannah

Outside, the wind gusted. The old horse whinnied as if in protest to the sound. Beyond the frost-rimmed window, the falling snow was beginning to twist and dance.
—Stephen King, The Dark Tower

The Earth is a very small stage in a vast cosmic arena. … Our planet is a lonely speck in the great enveloping cosmic dark. In our obscurity, in all this vastness, there is no hint that help will come from elsewhere to save us from ourselves.
—Carl Sagan, Pale Blue Dot

On the northern faces and higher ground of the rolling hills in the valley divide, the wind combed billowing fields of gray standing hay with rhythmic strokes, while dark evergreen boughs of spruce and pine swayed and shivered in erratic gusts that found their way around to the protected south-facing sides.
—Jean M. Auel, The Plains of Passage

The Wheel of Time turns, and Ages come and pass, leaving memories that become legend. Legend fades to myth, and even myth is long forgotten when the Age that gave it birth comes again.
—Robert Jordan, The Eye of the World

There was a steaming mist in all the hollows, and it had roamed in its forlornness up the hill, like an evil spirit, seeking rest and finding none. A clammy and intensely cold mist, it made its slow way through the air in ripples that visibly followed and overspread one another, as the waves of an unwholesome sea might do.
—Charles Dickens, A Tale of Two Cities

Thus it had come to pass, that Tellson’s was the triumphant perfection of inconvenience. After bursting open a door of idiotic obstinacy with a weak rattle in its throat, you fell into Tellson’s down two steps, and came to your senses in a miserable little shop, with two little counters, where the oldest of men made your cheque shake as if the wind rustled it, while they examined the signature by the dingiest of windows, which were always under a shower-bath of mud from Fleet-street, and which were made the dingier by their own iron bars proper, and the heavy shadow of Temple Bar.
—Charles Dickens, A Tale of Two Cities

No vivacious Bacchanalian flame leaped out of the pressed grape of Monsieur Defarge; but, a smoldering fire that burnt in the dark, lay hidden in the dregs of it.
—Charles Dickens, A Tale of Two Cities

…nobody wondered to see only Madame Defarge in her seat, presiding over the distribution of wine, with a bowl of battered small coins before her, as much defaced and beaten out of their original impress as the small coinage of humanity from whose ragged pockets they had come.
—Charles Dickens, A Tale of Two Cities

Troubled as the future was, it was the unknown future, and in its obscurity there was ignorant hope.
—Charles Dickens, A Tale of Two Cities

When the dream of vengeance in which we joined begins to drown all innocence in the blood tide, we are forced to look at the mingling of innocence and violence in ourselves.
—Stephen Koch, Afterword to A Tale of Two Cities

Sixteenth Street traffic moves in frustrated inches and headlong stampedes.
—David Mitchell, Cloud Atlas

Sometimes the fluffy bunny of incredulity zooms around the bend so rapidly that the greyhound of language is left, agog, in the starting cage.
—David Mitchell, Cloud Atlas

The cold sank its fangs into my exposed neck and frisked me for uninsulated patches.
—David Mitchell, Cloud Atlas

He chiseled open the fault lines in the others’ personalities.
—David Mitchell, Cloud Atlas

I lived with them on Montague Street in a basement down the stairs
There was music in the cafes at night and revolution in the air.
—Bob Dylan, “Tangled Up In Blue”

There is no greener green than the green of a ball field in spring.
Buster Olney

Tall green plants, possibly corn, grew in softly sighing ranks that stretched to the distant horizon where the last arc of a huge red sun was setting.
—Dan Simmons, The Fall of Hyperion

Posted in Books, Writing | 2 Comments

Weird Al wrote “Talk Soup” at the request of E! for their show Talk Soup

Holy crap, I never knew this before: Weird Al Yankovic wrote his 1993 song “Talk Soup” at the request of the E! television network to be used on their show Talk Soup.

Well, to put it bluntly, they kind of jerked me around. The producers of the show approached me, asking me to do a new theme song for the show. I wrote the lyrics (which they approved) and then recorded the song (which they said they “loved”). And then they never used it. Go figure.

I’ve always liked that song since I first heard it on my cassette tape of his album Alapalooza (most famous for “Jurassic Park” and “Bedrock Anthem”) back in the mid-90’s. I thought it was kind of funny and/or curious that the “E!” audio logo appeared at the end of the song, which apparently is the same exact clip used by E! and not an imitation. I guess I assumed Weird Al put that there as more of a reference or homage to the TV show Talk Soup, which made fun of daytime talk shows like his song did (before his song did), or maybe something akin to Jim Morrison’s “Stronger Than Dirt” at the end of “Touch Me” (a reference to the Ajax household cleaner commercials of the time, which had a four-note melody similar to the last four notes of “Touch Me”). But it turns out the name of Weird Al’s song, its content, and the “E!” audio logo at the end were made for/came from the TV network itself. You learn something new every day…

Posted in Entertainment, Music | Leave a comment