Unit 1 of 61 hr learning time

Ten Fingers, Two Wires

Why we count in tens but the machine counts in twos — and how a row of on/off switches becomes any number you like, from a single bit up to a byte's 255.

17% of Numbers & Bits

You already know how to count. What you might not have noticed is why you count the way you do — and that the machine, given different equipment, counts a different way. Get that one idea straight and binary stops being strange.

Ten is a habit

We count in tens. Three hundred and sixty-five is 3 hundreds, 6 tens, and 5 ones — each place worth ten times the one to its right. We do this so naturally it feels like the only way numbers could work.

It isn’t. We count in tens for an accident of anatomy: we have ten fingers. Ten was never special to the numbers themselves — it was special to us. Change the hands and you change the habit.

The machine has two fingers

A computer’s “fingers” are wires, and a wire has just two states: current flowing, or not. On, or off. 1, or 0. That single on-or-off is the smallest piece of information there is, and it has a name — a bit.

One bit can’t say much. On or off; yes or no; 1 or 0. To count past one, the machine does exactly what you do when you run out of fingers on one hand: it uses more.

Counting past one

Line up several bits and give each a place — and here’s the only twist: because there are two states per place instead of ten, each place is worth twice the one to its right, not ten times. So the places go 1, 2, 4, 8, 16, and on up, doubling each step.

Read the switches 1 0 1: that’s one 4, no 2, and one 1 — which adds up to 5. Spectrum BASIC lets you type the switches straight in with the word BIN, and tells you the everyday number they make:

  10 PRINT BIN 101
A Spectrum screen showing the number 5 printed below the program PRINT BIN 101.
You typed the switches — 1, 0, 1 — and the computer told you the number they spell: 5. One four, no two, one one.

The pattern is the number. BIN doesn’t do a calculation you couldn’t do yourself; it just reads the switches the machine’s way and shows you the answer in yours.

Turn another switch on

Place value isn’t a rule to memorise — it’s something you can watch. Add a switch on the left, in the next place up (the 8s), and the number should jump by eight:

Add a switch in the 8s place
+1-1
1- 10 PRINT BIN 101
1+ 10 PRINT BIN 1101
22
A Spectrum screen showing the number 13 printed below the program PRINT BIN 1101.
One more switch on, in the 8s place — and 5 became 13. Eight, and four, and one. The new switch was worth exactly its place.

Five became thirteen — a jump of eight, because the switch we turned on sits in the 8s place. Every switch is worth its place and nothing else; turning one on adds that place’s value, every time.

Eight of them make a byte

Bits almost never travel alone. The machine handles them in groups of eight, and a group of eight bits has its own name you’ll meet on every page from here: a byte.

How high can eight switches count? Turn them all on:

  10 PRINT BIN 11111111
A Spectrum screen showing the number 255 printed below the program PRINT BIN 11111111.
Eight switches, every one on: 255. The largest number a single byte can hold — which is why 255 turns up absolutely everywhere once you start looking.

128, 64, 32, 16, 8, 4, 2, 1 — add them up and you get 255. That’s the most a single byte can hold, and it’s the reason 255 (and its near neighbour 256) seem to haunt computing: colour values, character counts, the highest score that “rolls over.” Now you know where the number comes from — it’s just eight switches, all on.

When it’s wrong, see why

  • BIN reports an error. It accepts only 0s and 1s — those are the only states a switch has. A stray 2 (or any other digit) in there isn’t a binary number, so the machine stops. Check that every digit after BIN is a 0 or a 1.
  • The number is far bigger or smaller than you expected. You may be reading the switches the wrong way round. The rightmost switch is the 1s, and the places grow as you move left. Read right-to-left, not left-to-right.
  • A leading zero seems to do nothing. That’s correct — BIN 0101 and BIN 101 are the same number, exactly as 0007 and 7 are. A zero on the far left adds an empty place worth nothing.

What you’ve learnt

  • We count in tens out of habit — ten fingers — not because numbers demand it.
  • A bit is one switch: on or off, 1 or 0, the smallest piece of information there is.
  • Line bits up and each place is worth twice the one to its right (1, 2, 4, 8…), so a pattern of switches spells an ordinary number.
  • Eight bits make a byte, which counts from 0 to 255 — the number behind a thousand limits you’ll meet later.
  • BIN in Spectrum BASIC is the universal idea made concrete: type the switches, read the number.

What’s next

You can now write a number in twos as well as tens. In Unit 2 we add the third way the machine’s world is written — hexadecimal — and see why programmers reach for it constantly: it’s the shorthand that makes a row of eight bits readable at a glance.