MISSION · Number sense
Binary and Other Number Bases
Place value works the same way whatever number you group by
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LEVEL 2 · LEARN
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The idea in one sentence
Place value never depended on ten — each column is the base multiplied by the column to its right, and ten is just the base we happen to have fingers for.
What base ten is actually doing
Write 3207 and the columns mean
\[3 \times 10^3 + 2 \times 10^2 + 0 \times 10^1 + 7 \times 10^0\]Three facts are doing the work. Columns are powers of the base. Digits run from 0 up to one less than the base. And a zero holds an empty column open so the others keep their places.
Change the base and all three still hold. Only the number changes.
Binary
Base two has columns \(1, 2, 4, 8, 16, 32, \dots\) and exactly two digits, 0 and 1.
\[1011_2 = 1\times 8 + 0 \times 4 + 1 \times 2 + 1 \times 1 = 11\]To go the other way, work down from the largest power that fits. For 13: the biggest power of two under it is 8, leaving 5. Then 4 fits, leaving 1. Then 2 does not fit, so 0. Then 1 fits.
\[13 = 8 + 4 + 1 = 1101_2\]There is a mechanical version that needs no thinking: divide by 2 repeatedly and collect the remainders, then read them bottom to top.
\[13 \div 2 = 6 \text{ r } 1, \quad 6 \div 2 = 3 \text{ r } 0, \quad 3 \div 2 = 1 \text{ r } 1, \quad 1 \div 2 = 0 \text{ r } 1\]Reading upwards: \(1101\). The same method converts to any base — divide by that base instead.
Why computers use it
A wire is either carrying current or it is not. Two states are easy to build reliably and easy to keep distinct; ten voltage levels would be neither. So everything inside a machine is a string of two-state switches, and binary is the arithmetic that fits.
Each digit is a bit, eight of them make a byte, and a byte holds \(2^8 = 256\) different values — which is why so many limits in computing are 256, 1024, or 65536 rather than round decimal numbers.
Binary strings get long, though: 200 is \(11001000_2\). So programmers write in hexadecimal, base 16, which needs extra digits and borrows letters for them: \(A = 10\) through \(F = 15\). Its virtue is that 16 is \(2^4\), so each hex digit stands for exactly four bits and converting is regrouping rather than arithmetic.
\[200 = 12 \times 16 + 8 = \text{C8}_{16}\]Bases in ordinary life
Base ten is not as universal as it looks.
Time and angles run on base 60, inherited from Babylon, which is why an hour has 60 minutes and a circle has 360 degrees rather than 100 of anything. Sixty divides evenly by 2, 3, 4, 5, 6, 10, 12, 15, 20 and 30, so most fractions of an hour are whole numbers of minutes — a genuine advantage that has kept it alive for four thousand years.
Eggs and inches come in twelves for the same reason. And French quatre-vingts — “four twenties” for 80 — is a fossil of a base-twenty system.
Common mix-ups
- Using a digit as large as the base. Base 5 has digits 0 to 4; there is no digit 5, the same way base ten has no single digit for ten.
- Reading the remainders top to bottom. They come out backwards and must be reversed.
- Saying “ten” for \(10_2\). It is “one zero, base two”, and it equals 2. Reading \(1011_2\) as “one thousand and eleven” imports base ten by accident.
- Forgetting the subscript. \(101\) is a different number in every base, so mark which one you mean.
- Assuming the arithmetic changes. Addition, carrying and long division work exactly as they always did; you just carry at the base instead of at ten.
Check yourself
Convert \(1011_2\) to base ten, write 13 in binary, and name the largest single digit in base 5.
\(8 + 2 + 1 = 11\). For 13, the remainders read upwards give \(1101_2\). And base 5 runs 0 to 4, so the largest digit is 4.
LEVEL 3 · REFLECT
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