“Note that, in general, as the base gets smaller, the representation of a value requires more digits and looks bigger.”  Why?

Expert Answers
justaguide eNotes educator| Certified Educator

A value can be represented by a number using any positive number as base.

If a base B is chosen, to represent N as a number in base B it has to be written as a sum of powers of B.

`N = n_1*B^0 + n_2*B^1 + n_3*B^2 + ...`

Here, n can take on a value from 0 to B - 1.

For example, in base 10 a value `124_10` is represented as:

`124 = 4*10^0 + 2*10^1 + 1*10^2`

In base 2, the same would be written as:

`124 = 0*2^0 + 0*2^1 + 1*2^2 + 1*2^3+1*2^4+1*2^5+1*2^6`

= `1111100_2`

As the base gets smaller the value requires more digits to be represented as the value of the powers of the base decreases. For instance 2^n < 10^n, as a result, to represent the same value the sum, as described above, has to be extended to a higher power of the base.