Number bases: counting in twos, fives, eights and more

We count in tens because we have ten fingers. But there is nothing special about ten. Computers count in twos, old clocks and egg trays use twelves and dozens, and a market seller counting tomatoes in heaps of five is very close to counting in base five.

By the end of these notes you should be able to:

• show numbers in different bases on an abacus;
• identify the place value of digits in any base;
• convert numbers from one base to another;
• add, subtract, multiply and divide numbers in different bases.

What is a number base?

The base of a number system is the number of different digits it uses, and the number of units that make one group of the next size.

• Base ten (denary) uses the ten digits 0, 1, 2, 3, 4, 5, 6, 7, 8, 9.
• Base two (binary) uses only 0 and 1.
• Base five uses 0, 1, 2, 3, 4.
• Base eight (octal) uses 0 to 7.

We show the base with a small number written below and after the number, called a subscript. For example, 234_five or 234₅ means "two three four in base five". If no base is written, the number is in base ten.

Place value in different bases

In base ten, each place is worth ten times the place to its right: ones, tens, hundreds, thousands. In any base, each place is worth base times the place to its right.

Using an abacus

An abacus has spikes, one for each place value. In base five, each spike can hold at most four beads, because when you reach five beads you remove them and put one bead on the next spike to the left.

To show 213_five on an abacus with spikes labelled 25s, 5s and 1s, put 2 beads on the 25s spike, 1 bead on the 5s spike and 3 beads on the 1s spike.

Converting from another base to base ten

Multiply each digit by its place value, then add.

Converting from base ten to another base

Divide by the new base again and again, writing down each remainder, until the quotient is 0. Then read the remainders from bottom to top.

Reading the remainders from the bottom up: 45 = 101101_two. Check: 32 + 8 + 4 + 1 = 45.

Converting between two bases that are not ten

Go through base ten: first convert to base ten, then convert from base ten to the new base.

Adding in other bases

Add column by column from the right, as in base ten. When a column total reaches the base or more, write down what is left over and carry the number of whole groups of the base.

Subtracting in other bases

When you need to borrow, you borrow one group of the base, not ten.

Multiplying in other bases

Multiply digit by digit from the right. Convert each product into the base, write the remainder and carry the groups.

Dividing in other bases

The easiest method at this level is to convert to base ten, divide, and convert the answer back.

Finding an unknown base

Sometimes you are told that a number in an unknown base n equals a number in base ten. Write the place values using n and solve the equation.

Where number bases are used

• Computers and phones store every photo, message and song as long strings of 0s and 1s, in base two.
• Base sixteen (hexadecimal) uses the digits 0–9 and the letters A–F for 10–15. Web designers use it to write colours: the dark green on this website is written #1F4D3E.
• Time uses base sixty: 60 seconds make a minute and 60 minutes make an hour.
• Dozens and grosses (12 and 144) are still used when buying eggs, pencils and exercise books in bulk.

Summary

• A number in base n uses the digits 0 to n − 1, and each place is worth n times the place to its right.
• To base ten: multiply each digit by its place value and add.
• From base ten: divide repeatedly by the new base and read the remainders from the bottom up.
• Between two other bases: go through base ten.
• When adding or multiplying, carry groups of the base. When subtracting, borrow one group of the base.
• Always check your answer by converting to base ten.

Practice questions

1. State the value of the digit 3 in 1342_five, giving your answer in base ten.
2. Convert to base ten: (a) 11010_two (b) 324_five (c) 207_eight
3. Convert 58 to (a) base two (b) base five (c) base eight.
4. Convert 1111_two to base five.
5. Work out, giving answers in the same base: (a) 342_five + 214_five (b) 10110_two − 1011_two (c) 34_five × 3_five
6. If 41_n = 21, find the value of n.
7. Activity of integration. A school canteen wants a simple code to label 30 lockers using only the digits 0, 1 and 2. Show how the lockers could be numbered in base three, and explain how many digits the largest locker number needs.

Answers: 1) 75 2) (a) 26 (b) 89 (c) 135 3) (a) 111010_two (b) 213_five (c) 72_eight 4) 30_five 5) (a) 1111_five (b) 1011_two (c) 212_five 6) n = 5 7) 30 = 1010_three, so four digits.