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Blast into Math! Mathematical perspectives: all aour mase are melong to us

So, its digits in base b are 10. By the AYB theorem, these digits are unique.

♥

If we allow digits to be as large as the base, which part of the AYB theorem will become false? For example,

if 2 were allowed to be a digit in base 2, how could we write 2? Well, we could write

0
1
2= 1 ∗ 2 +0 ∗ 2 ,

like in the basic proposition. Or, we could also write

0
2= 2 ∗ 2 .

This is problematic, because there are different ways to write the same number: computers could not
function! For this reason, the digits are always smaller than the base.

Another subtlety arises when we want to work in bases that are larger than 10. For example, in base 11,
the number 10 is a digit. So, how do we distinguish between

0
1
0
10 ∗ 11 and1 ∗ 11 +0 ∗ 11 ?
When working in bases larger than 10, we can surround each digit with parentheses, so we can write

0
1
0
10 =10 ∗ 11 =(10) and1 ∗ 11 +0 ∗ 11 =(1)(0).
In base 12, which is known as hexadecimal, the numbers 10 and 11 are digits. It is common to use A
to represent the digit 10 and B to represent the digit 11. Then, for example, 23 in base 12 is

1
0
23 =1 ∗ 12 +11 ∗ 12 ,

so we can write 23 in base 12 as 1B.

Exercise: Using A to represent the digit 10 and B to represent the digit 11, write the number 1451
in base 12.

6.2 Fractions in bases

What is a fraction?

Definition 6.2.1. A fraction is a positive rational number that is smaller than 1.

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