Numeral Systems

• Numeral System
• Binary Numeral System - Base-2
• Octal Numeral System - Base-8
• Decimal Numeral System - Base-10
• Hex Numeral System - Base-16
• Numeral System Conversion Table

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.

Examples:

10101₂ = 10101B = 1×2⁴+0×2³+1×2²+0×2¹+1×2⁰ = 16+4+1= 21

10111₂ = 10111B = 1×2⁴+0×2³+1×2²+1×2¹+1×2⁰ = 16+4+2+1= 23

100011₂ = 100011B = 1×2⁵+0×2⁴+0×2³+0×2²+1×2¹+1×2⁰ =32+2+1= 35

Octal Numeral System - Base-8

Octal numbers uses digits from 0..7.

Examples:

27₈ = 2×8¹+7×8⁰ = 16+7 = 23

30₈ = 3×8¹+0×8⁰ = 24

4307₈ = 4×8³+3×8²+0×8¹+7×8⁰ = 2247

Decimal Numeral System - Base-10

Decimal numbers uses digits from 0..9.

These are the regular numbers that we use.

Example:

2538₁₀ = 2×10³+5×10²+3×10¹+8×10⁰

Hexadecimal Numeral System - Base-16

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

H denotes hex prefix.

Examples:

28₁₆ = 28H = 2×16¹+8×16⁰ = 40

2F₁₆ = 2FH = 2×16¹+15×16⁰ = 47

BC12₁₆ = BC12H = 11×16³+12×16²+1×16¹+2×16⁰ = 48146

Numeral Systems Conversion Table

Decimal Base-10	Binary Base-2	Octal Base-8	Hexadecimal Base-16
0	0	0	0
1	1	1	1
2	10	2	2
3	11	3	3
4	100	4	4
5	101	5	5
6	110	6	6
7	111	7	7
8	1000	10	8
9	1001	11	9
10	1010	12	A
11	1011	13	B
12	1100	14	C
13	1101	15	D
14	1110	16	E
15	1111	17	F
16	10000	20	10
17	10001	21	11
18	10010	22	12
19	10011	23	13
20	10100	24	14
21	10101	25	15
22	10110	26	16
23	10111	27	17
24	11000	30	18
25	11001	31	19
26	11010	32	1A
27	11011	33	1B
28	11100	34	1C
29	11101	35	1D
30	11110	36	1E
31	11111	37	1F
32	100000	40	20

 

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See also

• Number conversion
• Binary to decimal conversion
• Octal to decimal conversion
• Hex to decimal conversion