Engines of Our Ingenuity

No. 2711
TWO

by Andrew Boyd

Today, two. The University of Houston’s College of Engineering presents this series about the machines that make our civilization run, and the people whose ingenuity created them.

the number 2

Two is an unassuming number. It can’t claim to be first, and anything less than three cheers just won’t do. Two is small. But we shouldn’t let that fool us. It can make big things happen.

Try folding a piece of notebook paper in half a few times. We actually see two at work in, well, two different ways. We’re reducing the overall size of the paper by two — it’s half as big as the original sheet. But we’re also doubling its thickness — it now has two layers.

How long can we keep this up? I can manage to fold the paper in half six times but the result’s pretty messy. It’s hard to get the corners lined up just right when folding thirty-two layers upon themselves. I’ve seen determined middle school boys armed with pliers manage seven folds, but never eight.

The simple process of doubling yields large numbers very quickly — and that’s useful. Computers do arithmetic using binary numbers — numbers expressed with only the two symbols zero and one. That makes two the biggest star in the industry. But computers couldn’t use binary numbers if it took long strings of zeroes and ones to write down the numbers we work with. Numbers like the size of the national debt. Fortunately, that’s not the case.

If our computer uses numbers that are three zeroes and ones long, it can count to seven. One symbol or bit longer and it can count to fifteen. Each additional bit doubles how high the computer can count. At a length of sixty-four bits — common for today’s personal computers — it can count as high as the national debt a million times over.

national debt in binary

Two can also quickly cut big numbers down to size. A copy of the Houston phone book has roughly two-thousand pages. Let’s look for someone’s name as follows. We’ll divide the book in two by looking for the name in the middle, on page one-thousand. If it’s there, we’re done. If not, we know which half it’s in because the book is alphabetized.

Suppose it’s in the first half. Now we’ll divide the first half in two and look in the middle on page five-hundred. Thanks to the power of two, we can find the page we’re looking for in no more than eleven steps. That’s the absolute worst case.

Of course, two’s bigness transcends mathematics. Husband and wife. Parent and child. Best friend and best friend. All special relationships between two. Creating life requires two.

And so many questions are framed in twos. We ask: True or false? Good or bad? Despair or hope? Hatred or love? Liberal or conservative? Two helps us neatly package a very complicated world. If only life were really that easy.

two people holding hands

I’m Andy Boyd at the University of Houston, where we’re interested in the way inventive minds work.

(Theme music)


Notes and references:

For a related episode, see THE POWER OF THREE.

Common wisdom is that the actual number of times something “flat” can be folded in half is limited to seven. The actual number depends on a variety of factors. See, for example, http://pomonahistorical.org/12times.htm.

With three bits a computer can hold the eight numbers zero through seven. With four bits it can hold the sixteen numbers zero through fifteen. The addition of a bit exactly doubles the number of numbers a computer register can hold: 1 bit, two numbers; two bits, four numbers, and so on. The computer can only “count” to one less than the number it can hold.

The picture of the holding hands is from Wikimedia Commons.



The Engines of Our Ingenuity is Copyright © 1988-2011 by John H. Lienhard.