Why Computers Only Know Two Numbers

Hold up one hand

Most people can count to five on one hand.

You are about to be able to count to thirty-one on the same hand, using the same five fingers, with no tricks and nothing hidden. And when you can do it, you will have understood the one idea that everything else in a computer is built on top of.

It starts with a slightly rude question: why does a computer only know two numbers?

The usual explanation is backwards

Ask around and someone will tell you: binary is a secret code, and the computer translates our real numbers into it.

That sounds reasonable. It has the direction completely reversed.

There is no real number sitting underneath waiting to be translated back. The switches are the number. What the computer shows on your screen — the shape `7`, the shape `42` — is the translation, done at the last possible moment, for your benefit.

And while we are at it: base ten is not natural either. Ten is not a special quantity. Nothing in the universe comes in tens on purpose. We count in tens because of an accident of anatomy — we have ten fingers, and we started by pointing at them.

Why two, and not ten

Inside a processor, a single piece of information is held by a tiny switch. Current is flowing, or it is not.

Now imagine you wanted a switch that could hold ten different amounts instead — a tenth full, two tenths full, and so on. You would have to measure how much current was flowing, precisely, billions of times a second, in something the width of a few atoms, while it is warm and the wire next door is doing the same thing.

The measurement would go wrong constantly. Is that four tenths, or five? Nobody could ever be sure.

With two states there is nothing to measure. It is on, or it is off, and the gap between those is enormous compared to any wobble. So the machine is never confused about what it is holding. Two is not a limitation somebody settled for. Two is the choice that makes the whole thing reliable.

One of these switches — one on-or-off answer — is called a bit. Eight of them together are called a byte.

Counting to 31 on one hand

Here is the whole trick. Instead of every finger being worth 1, each finger is worth double the one before. Put a finger up and you add its value; leave it down and you add nothing.

Start with a fist. Raise them one at a time and watch the running total.

How many switches would you need?

Every extra switch doubles how many different things you can say. So: 1 switch says 2 things, 2 switches say 4, 3 say 8, 4 say 16, and so on. Sort each job into the smallest number of switches that can do it.

Now the part that surprises people

Somewhere in a computer's memory sit eight switches reading:

0 1 0 0 0 0 0 1

Use the doubling trick and that is the number 65. Definitely, arithmetically 65.

Except that in the standard text code every computer agrees on, 65 is the letter A. So it is also the letter A.

And if those eight switches happened to be part of a black-and-white photograph, 65 is a shade of dark grey. So it is also a shade of dark grey.

Here is the uncomfortable bit: the switches do not say which. There is nothing else in there. No label, no note, no hint. All three readings are equally available and the memory is completely silent on the question.

What decides is the program that opens it. The meaning was never stored — a person chose it, wrote it into a program, and the program applies that choice every time it looks.

Four things to keep