# Base b representation

When we write down sequences of digits in decimal (or base 10) notation, we interpret them according to a formula. For example, we interpret the number 1234 as a 1 in the "thousands place", a 2 in the "hundreds place", a 3 in the "tens place", and a 4 in the "ones place". In other words, we interpret the string of digits "1234" as .

It is occasionally useful to write a similar formula, but with a different set of possible digits. We have the following definitions:

If are all natural numbers satisfying for all , then the base b interpretation of , written is given by

For example, .

Definition: Digit
numbers satisfying are called base b digits.
If is a sequence of base digits, and if , then we say that is the Base b representation of

For example, the base 9 representation of 596 is the sequence of digits because .

Note that the base is not an intrinsic property of a natural number, but rather is a property of the way we write the number down. It does not make sense to say something like "let be a base b number", any more than it makes sense to say "let be a number in roman numerals". The object 'is' the same as the object (and also the object ); it is just written down in three different ways.