The main difference between base five numeration system and our familiar base 10 numeration system is that grouping is done in groups of 5 instead of 10.

For instance, to represent 24 in base 10 using sticks, you could use two groups of ten and 4 as shown below

Thus, 24

44

In like manner, if you group 17 in groups of five, you can get three groups of 5 and 2

Look at the following pattern:

19

18

17

16

15

What do you think 15 is in base 5?

Yes, 15

This is important. It is not 3 just because we have three fives in 15

In fact, what it means is three fives in the second place value and zero number in the first place value

The first ten base 5 numerals are shown below:

1

3

10

12

14

Notice that the first four numbers have the same representation in base 10 and in base 5

The second difference between base five and base ten is that in base ten, we use the 10 digits 0, 1, 2, 3 , 4 , 5, 6, 7, 8, and 9

In base 10, when you have 10 tens, you replace it by one hundred and starting from the right one hundred is in the third position

In general, place values are power of 10: 10

On the other hand,

In base 5, we use only 5 digits 0, 1, 2, 3 , and 4

In base five, when you have 5 fives, you replace it by one twenty five and starting from the right twenty five is in the third position

In this case, place values are power of 5: 5

Having said that, what is 57 in base 5

Group 57 in groups of 25 and 5

57 has 2 groups of 25, 1 group of 5 and 2, so 57