Teach Number Bases Step by Step

Count out a pile

Count out 23 beans together, one at a time.

SayCounting one by one is slow. Let’s find a faster way to see how many there are.

Bundle by tens

Put beans into cups ten at a time until you can’t make another full cup. You’ll have 2 full cups and 3 loose beans.

SayEvery time we have ten, we bundle them into one cup. A cup means ten.

Write what you see

Draw two columns and label them Tens and Ones. Write 2 under Tens and 3 under Ones.

SayThe 2 tells how many cups of ten, and the 3 tells how many loose ones. Together they say twenty-three.

Make a bigger bundle

Ask what would happen with ten full cups. They’d make one bigger bundle: a hundred. Add a Hundreds column to the left.

SayEach place is worth ten of the place to its right: ones, tens, hundreds, thousands.

Notice the digits

Ask why we never write a single digit bigger than 9.

SayAs soon as we have ten of anything, we bundle it and move one place to the left. So each place only needs the digits 0 to 9. That’s why it’s called base 10.

Same game, new rule

Play the bundling game again, but now bundle by twos. Start with 5 beans: two make a pair, two pairs make a bundle of four.

SayIn binary, as soon as we have two of something, we bundle it. So each place can only hold 0 or 1.

Learn the places

Show the strip: 8, 4, 2, 1. Each place is double the one to its right, and you could keep going: 16, 32, 64, 128.

SayIn base 10 the places grow by tens. In binary they grow by twos.

Count to eight together

Use the coins on the strip, heads for 1 and tails for 0. Count: 0, 1, 10, 11, 100, 101, 110, 111, 1000.

SayWatch the right-hand coin. When it would go past 1, it flips back to 0 and we carry one to the left, like a car’s mileage counter.

Read a binary number

Lay out 1011 on the strip. Add up the places that have a 1: 8 + 2 + 1 = 11.

SayA 1 means take this place. A 0 means skip it. So 1011 is eight, plus two, plus one: eleven.

Write a number in binary

To write 13, start at the biggest place that fits. 8 fits, leaving 5, so put a 1 in the 8s. 4 fits, leaving 1: a 1 in the 4s. 2 doesn’t fit: a 0. 1 fits: a 1. So 13 is 1101.

SayBiggest place first. If it fits, write 1 and take it away. If it doesn’t, write 0 and move on.

Count on your fingers

Give each finger on one hand a place: thumb 1, pointer 2, middle 4, ring 8, little finger 16. A raised finger is a 1. Count up as high as you can.

SayWith five fingers you can count to 31, because 16 + 8 + 4 + 2 + 1 is 31.

Bundle by eights

Count out 19 beans and bundle them by eights: 2 full cups and 3 loose. So 19 is written 23 in octal.

SayIn octal we bundle by eight, so we only use the digits 0 to 7.

Learn the places

Write the places from right to left: 1, 8, 64, 512. Each is eight times the one before.

Count past seven

Count aloud and write as you go: …, 6, 7, 10, 11, …, 17, 20.

SayAfter 7 there’s no digit 8, so we write 10: one bundle of eight and nothing left over. Octal 17 is fifteen, and octal 20 is sixteen.

Octal to ordinary numbers

For 157 in octal, multiply each digit by its place and add: 1 × 64 + 5 × 8 + 7 × 1 = 64 + 40 + 7 = 111.

Ordinary numbers to octal

Divide by 8 and keep the remainder each time, then read the remainders from the bottom up. For 100: 100 ÷ 8 = 12 remainder 4; 12 ÷ 8 = 1 remainder 4; 1 ÷ 8 = 0 remainder 1. Reading up: 144.

SayThe first remainder is the ones digit, so it goes on the right.

The binary shortcut

Each octal digit is exactly three bits, because 8 is 2 × 2 × 2. So 57 in octal is 5 = 101 and 7 = 111: 101 111.

SayOctal is a short way to write binary, three bits at a time.

We need more digits

Ask: if we bundle by sixteen, each place can hold up to fifteen. What single symbol can we write for ten, eleven or fifteen?

SayHex borrows letters. A is ten, B is eleven, and so on up to F, which is fifteen.

Count past nine

Count and write: …, 8, 9, A, B, C, D, E, F, 10, 11, …, 19, 1A, …, 1F, 20.

SayAfter F comes 10: one bundle of sixteen and nothing left. Hex 1F is thirty-one, and hex 20 is thirty-two.

Learn the places

Write the places from right to left: 1, 16, 256, 4,096. Each is sixteen times the one before. Two hex digits go up to FF, which is 255.

Hex to ordinary numbers

For 2B: 2 × 16 + B × 1 = 32 + 11 = 43. For 1A0: 1 × 256 + 10 × 16 + 0 = 416.

SayChange any letters to their values first, then multiply by the places and add.

Ordinary numbers to hex

Divide by 16 and keep the remainders, writing 10 to 15 as A to F. For 300: 300 ÷ 16 = 18 remainder 12 (C); 18 ÷ 16 = 1 remainder 2; 1 ÷ 16 = 0 remainder 1. Reading up: 12C.

The binary shortcut

Each hex digit is exactly four bits, because 16 is 2 × 2 × 2 × 2. Split 1010 0111 into fours: 1010 is A and 0111 is 7, so it’s A7.

SayThat’s why programmers like hex: it’s binary, four bits at a time, and much easier to read.

The safe route: through base 10

To change from one base to another, first turn the number into an ordinary base 10 number (multiply each digit by its place and add). Then turn that into the new base (divide again and again, keeping remainders).

SayBase 10 is home base. You can always get anywhere by going home first.

The shortcuts: through binary

Between binary and hex, group bits in fours. Between binary and octal, group them in threes. Always group from the right, and add zeros on the left to fill the last group: 10111 becomes 0001 0111, which is 17 in hex.

Octal to hex

Go through binary: 57 in octal is 101 111. Regroup in fours from the right: 0010 1111. That’s 2F in hex. Check it in base 10: 57 in octal is 47, and 2F in hex is 2 × 16 + 15 = 47.

Always check

Convert your answer back the other way, or type it into the converter above.

SayIf you can get back to where you started, you know it’s right.