Where the Laws No Longer Hold

third in a finite series on infinity
(see posts 1 and 2)

Somehow, I suspect I wouldn’t survive long on the frontier.

Drop me in the American West, circa 1850, and I fear my math-blogging and bad-drawing skills might not carry me far. I need indoor plumbing. I need the rule of law. I need chain coffee shops. I’m not cut out for the frontier.

And yet the frontier is exactly where I found myself the other day, when I came across this formula in the wonderful Penguin Book of Curious and Interesting Numbers, by David Wells:


I decided to play around with this product a bit. After all, what are products for, if not playing around?

(Go ahead and play with your Apple products. I’ll play with my infinite ones. We’ll see who has more fun.)

I felt like there should be an easier way to write this expression, exploiting the repetition of factors, so I gave it a shot, and created this:


Then my brain exploded and the universe dissolved around me, because I had just punched logic in the face, and it had punched me back.

The left side of that equation is π/2. It’s roughly 1.57.

The right side of that equation, however, is a product of many numbers—all of them below 1.

What happens when you multiply two numbers smaller than 1? You get another number smaller than 1.

How the heck could that equal 1.57?


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The Strange Music of the Harmonic Series

part 2 in a finite series on infinity
(see also part 1)

A few weeks ago, the webcomic Saturday Morning Breakfast Cereal posted a cartoon about the harmonic series.

(Obviously it’s a mistake to post an actual cartoonist’s work alongside my own second-grade-quality scrawl, but hey, maybe I’ll benefit from a math humor cheerleader effect.)

Now, what is the harmonic series? It’s this:

The sum never stops. It goes on forever and ever. Lovely, yes, but does it—in any meaningful sense—“equal” anything? Continue reading

Everything Is Linear (Or, the Ballad of the Symbol Pushers)

What is the biggest problem facing humanity this week?

  • A. The threat of Grexit
  • B. The bittersweet knowledge that someday, when all of this has passed, we’ll have fewer opportunities to use the amazing word “Grexit”
  • C. People thinking functions are linear when they’re SO NOT LINEAR
  • D. Other (e.g., cat bites)

If you answered C, then congratulations! You are probably a teacher of math students ages 13 to 20, and we all share in your pain.

For everyone else (including you poor cat-bitten D folk), what are we talking about? We’re talking about errors like these (warning—mathematical profanity ahead):

20150708084840_00001What’s wrong with these statements? Well… everything. Continue reading

Infinity Plus One: Please Check Your Intuitions at the Front Desk

the first post in a finite series

If there’s one thing about math that people love—and to make it through the average day, I have to believe there’s at least one—it’s infinity.

Throw the word into a math lesson, and ears perk up. Infinity? Did he say infinity? It’s like a distant celebrity, the subject of endless gossip and rumor. “I heard infinity isn’t even a number!” “Only the universe is really infinite.” “My last teacher said infinity times two is the same as infinity.”  “I can use infinity to prove that 1 = 0!”

Infinity is a sound too high for our ears, a light too bright for our eyes, a taste so sweet that it would tear through our tongues like acid. Basically, it’s mathematical Mountain Dew.

Tellingly, all of our words for infinity define it by what it isn’t. Infinite: not finite. Unlimited: not limited. Boundless: without bounds. It’s hard to articulate what infinity does, so we settle for naming what it doesn’t: end. Infinity is the Anansi of mathematics, a trickster spider weaving baffling webs of paradox and contradiction.

Take this example: which has more numbers, List A or List B?

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US vs. UK: Who Grades Students?

When I began teaching in the UK, I discovered that the word “grade” belongs on that List of Words That Change Meaning Dramatically When You Cross the Pond, along with “vest,” “pants,” and “rubber.”

What’s the difference? Well, in the UK, a “rubber” is an eraser, and in the US…

Oh! You mean for “grades.” Well, in the US, “grades” are given by teachers. They aim to assess the quality of your work throughout an entire term. In the UK, your “grades”—the ones that matter to universities—are your scores on a handful of high-stakes exams.

In the US, the scores you get from your teachers form the bulk of your permanent academic record; in the UK, those scores don’t even appear.


In my experience, both American and British educators react with utter horror to the opposite system. And, weirdly, many objections are mirror images of one another.

Objection #1: Your System Drowns Students in Testing

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A Technique is Just a Trick That Went Viral


In February, I was throwing together a geometry test for my 12-year-olds. I wanted a standard angle-chasing problem, but – and here’s the trick – I’m lazy. So I grabbed a Google image result, checked that I could do it in my head, and pasted it into the document.


But when I started writing up an answer key, I ran into a wall. Wait… how did I solve this last time? I trotted out all the standard techniques. They weren’t enough. A rung of the logical ladder seemed to have vanished overnight, and now I was stuck, grasping at air.

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