## 3.1 NATURAL NUMBERS

### 3.1.1 Weighted Systems

Any natural number (nonnegative integer) can be represented, in a unique way, in the form of a sum of powers Bi of some natural number B greater than 1, each of them multiplied by a natural number smaller than B. The following theorem defines the base-B numeration system.

Theorem 3.1 Given a natural number B greater than 1, any natural number x smaller than Bn can be expressed in the form

where every coefficient xi is a natural number smaller than B. Furthermore, there is only one possible vector (xn−1 xn−2x0) representing x.

The following algorithm computes the coefficients xi:

Algorithm 3.1

`for i in 0..n − 1 loop x(i):=x mod B; x:=x/B; end loop;`

Definitions 3.1

1. The most commonly used values of B are 10 (decimal system), 2 (binary system), 16 (hexadecimal system), and 8 (octal system). The coefficients xi of the base-B representation of x are called the base-B digits of x. The binary digits are called bits. The hexadecimal digits 10, 11, 12, 13, 14, and 15 are usually replaced by letters: A, B, C, D, E, and F.
2. This type of representation is called positional as the weight Bi associated with the digit xi depends on i, that is, on the position of the digit within the vector (xn−1 xn−2x0).
3. The base-B digits could in turn be encoded in another base. As an example, if B = 10 and the decimal digits are represented in the form of 4-bit ...

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