Numeral System Numbers Base Number Systems Positional Numerals

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4 Positional systems in detail

Also see Positional notation.

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.

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, we write a number in the numeral system of base b by expressing it in the form a₁b^k + a₂b^k-1 + a₃b^k-2 + ... + a_k+1b⁰ and writing the digits a₁a₂a₃ ... a_k+1 in 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 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:

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.

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.

5 Specific numeral systems

5.1 Positional systems

b=2: Binary
b=3: Ternary (or trinary)
b=4: Quaternary
b=5: Quinary
b=6: Senary
b=7: Septimal
b=8: Octal or Octonary
b=9: Novenary
b=10: Decimal or Denary (a single system, but there are both long scale and short scale naming variants in widespread use)
b=12: Duodecimal (used in the Chepang language of Nepal)
b=13: Tredecimal
b=16: Hexadecimal (properly sedecimal, but hexadecimal is standard)
b=20: Vigesimal (see Maya numerals)
b=25: Quinquevigesimal (see D'ni numerals)
b=27: Septemvigesimal
b=60: Sexagesimal (see Babylonian numerals)
b=64: Quattuorsexagesimal
b=120: Centivigesimal

5.2 Positional-like systems with non-standard bases

b=−3: Negaternary
b=−2: Negabinary
b= φ: Golden mean base
b= e: Natural base
b=2i: Quater-imaginary base
b=: Binary square-root-2 times i base
b=i − 1: Binary i−1 base
b= Fibonacci number: Fibonacci coding
b= ω: Cantor Normal Form for ordinal numbers
b varies: Mixed radix

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Home > Numeral system

4 Positional systems in detail

5 Specific numeral systems

5.1 Positional systems

5.2 Positional-like systems with non-standard bases

Topics: Numeral System Numbers Base Number Systems Positional Numerals...