Numeral system

:Occasionally the term "number system" is used for this concept, but that is also the name of an altogether different concept; see number system.

Positional systems in detail

Also see Positional notation.

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In a positional base-b numeral system (with b a positive natural number 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.

Related Topics:
Natural number - Radix

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

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In general, if b is the base, we write a number in the numeral system of base b by expressing it in the form a1bk + a2bk-1 + a3bk-2 + ... + ak+1b0 and writing the digits a1a2a3 ... ak+1 in order. The digits are natural numbers between 0 and b-1, inclusive.

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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: numberbase. Unless specified by context, numbers without subscript are considered to be decimal.

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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.

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

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:

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(a_na_{n-1}...a_1a_0.c_1c_2c_3...)_b =

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sum_{k=0}^n a_kb^k + sum_{k=1}^infty c_kb^{-k}

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The numbers bk and b-k are the weights of the corresponding digits.

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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 unperiodic (infinite amount of unrepeating digits) in all integral bases. Thus, for example in base 2, π = 3.1415926...10 can be written down as the unperiodic 11.001001000011111...2.

Related Topics:
If and only if - Rational - π

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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.

Related Topics:
Prime number - P-adic number

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