# Counting in different bases

As we humans have ten fingers we learn to count in tens ( base 10 ).  If we had three fingers on each hand we would count using base 6. The table below shows how counting in base 10 compares to counting in other bases.

The column down the left hand side are the decimal numbers 1 to 10. The numbers in the table are how they would be written in the different bases from base 2 to base 9.

[table id=2 /]

Examples

In Base 5, twenty is written as 40 because we are counting in groups of five. There are no units, so the right hand digit is zero, but there are four lots of five so the left hand digit is four. So the notation for twenty in Base 5 is 40.

In Base 3, seventeen is written as 122 because we are counting in groups of three. There are two units, so the right hand unit is two, there are two lots of three so the next digit is two and there is one group of nine so the left hand digit is one. So the notation for seventeen in Base 3 is 122.

## Notation

The table below shows the number sequence for counting in base 2 to base 5.  When writing the numbers down the units are on the right, numbers groups increasing to the left.

In the decimal system this notation would be:

Thousands       Hundreds     Tens    Units

[table id=3 /]

If we look at base 5 above. The first digit on the right is units. The next digit represents groups of five ( 1 x 5 ), then groups of twenty five ( 5 x 5 ), then groups of one hundred and twenty fives ( 5 x 25 ) and finally groups of six hundred and twenty fives ( 5 x 125 ) and so on.

Example

If we want to convert 493 to base 5 we could do it this way:

493 / 125 = 3 remainder 118

118 / 25 = 4 remainder 18

18 / 5 = 3 remainder 3

3 units.

Check the working out.

3 x 125 = 375

4 x 25 = 100

3 x 5 = 15

3

375 + 100 + 15 + 3 = 493

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