Numeral System - Positional Systems in Detail

Positional Systems in Detail

In a positional base-b numeral system (with b a natural number greater than 1 known as the radix), b basic symbols (or digits) corresponding to the first b natural numbers including zero are used. To generate the rest of the numerals, the position of the symbol in the figure is used. The symbol in the last position has its own value, and as it moves to the left its value is multiplied by b.

For example, in the decimal system (base 10), the numeral 4327 means (4×103) + (3×102) + (2×101) + (7×100), noting that 100 = 1.

In general, if b is the base, one writes a number in the numeral system of base b by expressing it in the form a_nbn + a_{n − 1}bn − 1 + a_{n − 2}bn − 2 + ... + a₀b0 and writing the enumerated digits a_na_{n − 1}a_{n − 2} ... a₀ in descending order. The digits are natural numbers between 0 and b − 1, inclusive.

If a text (such as this one) discusses multiple bases, and if ambiguity exists, the base (itself represented in base 10) is added in subscript to the right of the number, like this: number_base. Unless specified by context, numbers without subscript are considered to be decimal.

By using a dot to divide the digits into two groups, one can also write fractions in the positional system. For example, the base-2 numeral 10.11 denotes 1×21 + 0×20 + 1×2−1 + 1×2−2 = 2.75.

In general, numbers in the base b system are of the form:

The numbers bk and b−k are the weights of the corresponding digits. The position k is the logarithm of the corresponding weight w, that is . The highest used position is close to the order of magnitude of the number.

The number of tally marks required in the unary numeral system for describing the weight would have been w. In the positional system, the number of digits required to describe it is only , for . E.g. to describe the weight 1000 then four digits are needed since . The number of digits required to describe the position is (in positions 1, 10, 100,... only for simplicity in the decimal example).

Note that a number has a terminating or repeating expansion if and only if it is rational; this does not depend on the base. A number that terminates in one base may repeat in another (thus 0.3₁₀ = 0.0100110011001...₂). An irrational number stays aperiodic (with an infinite number of non-repeating digits) in all integral bases. Thus, for example in base 2, π = 3.1415926...₁₀ can be written as the aperiodic 11.001001000011111...₂.

Putting overscores, n, or dots, ṅ, above the common digits is a convention used to represent repeating rational expansions. Thus:

14/11 = 1.272727272727... = 1.27 or 321.3217878787878... = 321.3217̇8̇.

If b = p is a prime number, one can define base-p numerals whose expansion to the left never stops; these are called the p-adic numbers.