Number Bases

Base 10

We are all used to using "Base 10" ... 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.
It will help you understand how all these different bases work.

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

Well ... we talked about this at the start but here it is again:

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

Hexadecimal (Base 16)

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

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

More About Bases

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)