Numbers & Functions · 2 The Hole in the Number Line

2.5 What does an irrational number even look like?

Fair question. You can write a fraction down exactly: 3 8. How do you write √ 2?

One way is to write it as a decimal: √ 2 = 1.41421356237 . . .

and those dots are doing serious work. The decimal goes on forever, and — this is the signature of an irrational number — it never falls into a repeating pattern.

Compare. A fraction, written as a decimal, always either stops or repeats: 1 4 = 0.25 (stops), 1 3 = 0.3333 . . . (repeats), 1 7 = 0.142857 (repeats in sixes).

That’s what being a fraction means, written in decimals: eventually predictable. Stops, or loops. An irrational number does neither. Its decimal marches off forever with no loop, no pattern, no shortcut.

So here’s a clean way to hold the whole picture in your head: TERMINATING rational it stops 1 4 = 0.25 out of digits it just stops REPEATING rational a block loops forever 1 7 = 0.

1 4 2 8 5 7 1 4 2 8 5 7 one block (6 digits), repeating IRRATIONAL irrational never repeats p 2 = 1 . 4 1 4 2 1 3 5 6 no block ever repeats the digits never settle stops or repeats ⇒ rational • forever without repeating ⇒ irrational Figure 2.3. The decimal fingerprint. A decimal that stops or repeats is rational; one that runs on forever with no repeating block is irrational. Kind of number Its decimal Integer stops immediately (5 = 5.000 . . .)

Fraction (rational) stops, or repeats forever Irrational goes forever, never repeats All of them together the real numbers The Hole in the Number Line

2.6 A secret about ``decimal''

Did you notice the word decimal keeps showing up? It comes from decem, Latin for ten. When you write 1.414, every digit is secretly about powers of ten. We built our whole way of writing numbers around the number ten.

Why ten? Honestly, because you have ten fingers. That’s it. There’s nothing mathematically special about ten — it’s an accident of human anatomy. If we’d evolved with eight fingers, we’d probably count in eights.

A computer doesn’t have fingers. Deep down, a computer is made of tiny switches, and a switch has exactly two states: on or off. So computers count in twos. Their natural number system uses only two digits, 0 and 1, and it’s called binary. In binary the places double instead of multiplying by ten: a ones place, a twos place, a fours place, an eights place. So the number we write as 13 becomes 1101 in binary, because 1101(binary) = 8 + 4 + 0 + 1 = 13.

base ten 1 tens 3 base two 1 8 1 4 0 2 1 1 8+4+0+1 = 13 Figure 2.4. The same number, two costumes. In base ten the places go . . . , 100, 10, 1; in base two they double: . . . , 8, 4, 2, 1. The number 13 is 1101 in binary.

Every digit in a computer — every one of those 0s and 1s — is called a bit. Everything a computer has ever done is, underneath, an enormous pile of bits. The grandest machines humans have ever built are, all the way down, just patterns of on and off. Keep this in your back pocket: ten was never sacred. It was just convenient.

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