Numeral Systems

Numeral System

b – numeral system base

d_n – the n-th digit

n – can start from negative number if the number has a fraction part.

N+1 – the number of digits

Binary Numeral System – Base-2

Binary numbers uses only 0 and 1 digits.

B denotes binary prefix.

For example:

1110₂ = 1110B = 1×2⁴+1×2³+1×2²+0×2¹ = 16 + 8 + 4 = 28

11101₂ = 11101B = 1×2⁴+1×2³+1×2²+0×2¹+1×2⁰ = 16 + 8 + 4 + 1 = 29

101100₂ = 101100B = 1×2⁵+0×2⁴+1×2³+1×2²+0×2¹+0×2⁰= 32 + 8 + 4 = 44

Octal Numeral System – Base-8

Octal numbers uses digits from 0..7.

For examples:

36₈ = 3×8¹+6×8⁰ = 24 + 6 = 30
50₈ = 5×8¹+0×8⁰ = 40
2403₈ = 2×8³+4×8²+0×8¹+3×8⁰= 1024 + 256 + 24 = 1304

Decimal Numeral System – Base-10

Decimal numbers uses digits from 0..9.

These are the regular numbers that we use.

For example:

1142₁₀ = 1×10³+1×10²+4×10¹+2×10⁰ = 1000 + 100 + 40 + 2 = 1142

Hexadecimal Numeral System – Base-16

Hex numbers uses digits from 0..9 and A..F.

H denotes hex prefix.

For examples:

14₁₆ = 14H = 1×16¹+4×16⁰ = 16 + 4 = 20
3F₁₆ = 3FH = 3×16¹+15×16⁰ = 48 + 15 = 63
CB12₁₆ = CB12H = 12×16³+11×16²+1×16¹+2×16⁰= 49152 + 2816 + 16 + 2 = 51986

Numeral systems conversion table