Number Bases

Base 10

We use "Base 10" every day ... it is our Decimal Number System.

It has 10 digits:

0   1   2   3   4   5   6   7   8   9

We count like this:

 0 Start at 0 • 1 Then 1 •• 2 Then 2 ⋮ ••••••••• 9 Up to 9 •••••••••• 10 Start back at 0 again, but add 1 on the left •••••••••• • 11 •••••••••• •• 12 ⋮ •••••••••• ••••••••• 19 •••••••••• •••••••••• 20 Start back at 0 again, but add 1 on the left •••••••••• •••••••••• • 21 And so on!

But there are other bases!

Binary (Base 2) has only 2 digits: 0 and 1

We count like this:

 0 Start at 0 • 1 Then 1 •• 10 Start back at 0 again, but add 1 on the left ••• 11 •••• 100 start back at 0 again, and add one to the number on the left... ... but that number is already at 1 so it also goes back to 0 ... ... and 1 is added to the next position on the left ••••• 101 •••••• 110 ••••••• 111 •••••••• 1000 Start back at 0 again (for all 3 digits), add 1 on the left ••••••••• 1001 And so on!

See how it is done in this little demonstration (press play):

Also try Decimal, and try other bases like 3 or 4.

Ternary (Base 3) has 3 digits: 0, 1 and 2

We count like this:

 0 Start at 0 • 1 Then 1 •• 2 ••• 10 Start back at 0 again, but add 1 on the left •••• 11 ••••• 12 •••••• 20 Start back at 0 again, but add 1 on the left ••••••• 21 •••••••• 22 ••••••••• 100 start back at 0 again, and add one to the number on the left... ... but that number is already at 2 so it also goes back to 0 ... ... and 1 is added to the next position on the left •••••••••• 101 And so on!

Quaternary (Base 4) has 4 digits: 0, 1, 2 and 3

We count like this:

 0 Start at 0 • 1 Then 1 •• 2 ••• 3 •••• 10 Start back at 0 again, but add 1 on the left ••••• 11 •••••• 12 ••••••• 13 •••••••• 20 Start back at 0 again, but add 1 on the left ••••••••• 21 And so on!

Quinary (Base 5) has 5 digits: 0, 1, 2, 3 and 4

We count like this:

 0 Start at 0 • 1 Then 1 •• 2 ••• 3 •••• 4 ••••• 10 Start back at 0 again, but add 1 on the left •••••• 11 ••••••• 12 •••••••• 13 ••••••••• 14 •••••••••• 20 Start back at 0 again, but add 1 on the left •••••••••• • 21 And so on!

Senary (Base 6) has 6 digits: 0, 1, 2, 3, 4 and 5

We count like this:

 0 Start at 0 • 1 Then 1 •• 2 ••• 3 •••• 4 ••••• 5 •••••• 10 Start back at 0 again, but add 1 on the left ••••••• 11 •••••••• 12 ••••••••• 13 •••••••••• 14 •••••••••• • 15 •••••••••• •• 20 Start back at 0 again, but add 1 on the left •••••••••• ••• 21 And so on!

Septenary (Base 7) has 7 digits: 0, 1, 2, 3, 4 5 and 6

We count like this:

 0 Start at 0 • 1 Then 1 •• 2 Then 2 ⋮ •••••• 6 Up to 6 ••••••• 10 Start back at 0 again, but add 1 on the left •••••••• 11 ••••••••• 12 ⋮ •••••••••• ••• 16 •••••••••• •••• 20 Start back at 0 again, but add 1 on the left •••••••••• ••••• 21 And so on!

Octal (Base 8) has 8 digits: 0, 1, 2, 3, 4, 5, 6 and 7

We count like this:

 0 Start at 0 • 1 Then 1 •• 2 Then 2 ⋮ ••••••• 7 Up to 7 •••••••• 10 Start back at 0 again, but add 1 on the left ••••••••• 11 •••••••••• 12 ⋮ •••••••••• ••••• 17 •••••••••• •••••• 20 Start back at 0 again, but add 1 on the left •••••••••• ••••••• 21 And so on!

Nonary (Base 9) has 9 digits: 0, 1, 2, 3, 4, 5, 6, 7 and 8

We count like this:

 0 Start at 0 • 1 Then 1 •• 2 Then 2 ⋮ •••••••• 8 Up to 8 ••••••••• 10 Start back at 0 again, but add 1 on the left •••••••••• 11 •••••••••• • 12 ⋮ •••••••••• ••••••• 18 •••••••••• •••••••• 20 Start back at 0 again, but add 1 on the left •••••••••• ••••••••• 21 And so on!

Decimal (Base 10) has 10 digits: 0, 1, 2, 3, 4, 5, 6, 7, 8 and 9

 0 Start at 0 • 1 Then 1 •• 2 Then 2 ⋮ ••••••••• 9 Up to 9 •••••••••• 10 Start back at 0 again, but add 1 on the left •••••••••• • 11 •••••••••• •• 12 ⋮ •••••••••• ••••••••• 19 •••••••••• •••••••••• 20 Start back at 0 again, but add 1 on the left •••••••••• •••••••••• • 21 And so on!

Undecimal (Base 11)

Undecimal (Base 11) needs one more digit than Decimal, so "A" is used, like this:

 Decimal: Undecimal: 0 1 2 3 4 5 6 7 8 9 10 11 12 ... 0 1 2 3 4 5 6 7 8 9 A 10 11 ...

Duodecimal (Base 12)

Duodecimal (Base 12) needs two more digits than Decimal, so "A" and "B" are used:

 Decimal: Duodecimal: 0 1 2 3 4 5 6 7 8 9 10 11 12 13 ... 0 1 2 3 4 5 6 7 8 9 A B 10 11 ...

Because there are more than 10 digits, hexadecimal is written using letters as well, like this:

 Decimal: Hexadecimal: 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 ... 0 1 2 3 4 5 6 7 8 9 A B C D E F 10 11 ...

Vigesimal (Base 20)

With vigesimal, the convention is that I is not used because it looks like 1, so J=18 and K=19, as in this table:

 Decimal: Vigesimal: 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 ... 0 1 2 3 4 5 6 7 8 9 A B C D E F G H J K 10 ...

Note: the Number Base is also called the Radix

How to Show the Base

To show what base a number has, put the base in the lower right like this:

1012
This shows that is in Base 2 (Binary)

3148
This shows that is in Base 8 (Octal)