A bit of number magic - engaging students and showing them beauty.

There’s nothing like a bit of number magic to get the attention of students. One of my favourite ‘tricks’ is squaring large numbers. It’s really fun when a calculation comes up in a maths problem and you can say, “Oh, 57 squared, that’s 3,249.” It tends to catch the students’ attention, makes them pause for a moment, and highlights a glimmer of the beauty and creativity in maths.

I first read about this in a wonderful book by James Gleick about the physicist Richard Feynman. There is a section where a conversation between Feynman and Hans Bethe, during the time they were working on the Manhattan project, is related:

When Bethe and Feynman went up against each other in games of calculating, they competed with special pleasure. Onlookers were often surprised, and not because the upstart Feynman bested his famous elder. On the contrary, more often the slow-speaking Bethe tended to outcompute Feynman. Early in the project they were working together on a formula that required the square of 48. Feynman reached across his desk for the Marchant mechanical calculator

Bethe said, "It's twenty-three hundred."

Feynman started to punch the keys anyway. "You want to know exactly?" Bethe said. "It's twenty-three hundred and four. Don't you know how to take squares of numbers near fifty?" He explained the trick. Fifty squared is 2,500 (no thinking needed). For numbers a few more or less than 50, the approximate square is that many hundreds more or less than 2,500. Because 48 is 2 less than 50, 48 squared is 200 less than 2,500-thus 2,300. To make a final tiny correction to the precise answer, just take that difference again-2-and square it. Thus 2,304.

This ‘trick’ is based on the simple expansion

(x + y)² = x² + 2xy + y²

In particular,

(50 + y)² = 50² + 2 × 50 × y + y²

                 = 2,500 + 100y + y²

So you just need to take the amount more (or less – it works perfectly well with negative numbers) than 50, and add (or subtract) that many hundreds to (or from) 2500. E.g. “57 squared is three thousand, two hundred, and …”

With a tiny bit of practice you can square the difference quickly whilst saying the first part, and finish with “… forty nine” – making it seem like no effort was required at all.

Of course, this expansion works for any number, but gets more complicated (the final squared term changes the hundreds, not just the tens and units) for differences greater than 10, so it’s especially simple for the squaring of numbers between 40 and 60.

And the best part is not having the students think that it’s magic, but explaining it to them afterwards when they’re hooked and engaged!

It’s also a nice calculation to bring up when you’re on the topic of binomial expansion.

References:

Gleick, J. (1992). Genius: The life and science of Richard Feynman. Pantheon Books.