NUMBER SYSTEMS

4 TYPES OF NUMBER SYSTEM

1. Decimal

2. Binary

3. Octal

4. Hexadecimal

1. Base 10 (Decimal) — Represent any number using 10 digits      [0–9]

2. Base 2 (Binary) — Represent any nu⁴mber using 2 digits          [0–1]

3. Base 8 (Octal) — Represent any number using 8 digits [0–        7]

4. Base 16(Hexadecimal) — Represent any number using 10         digits and       6 characters [0–9, A, B, C, D, E, F]

* Decimal Number System

The number system we use every day, based on 10 digits (0,1,2,3,4,5,6,7,8,9). 
Position is important, with the first position being units, then next on the left being tens, then hundreds and so on.

105	104	103	102	101	100
Hundreds of Thousands	Tens of Thousands	Thousands	Hundreds	Tens	Ones
100000	10000	1000	100	10	1

For example 23456=2×10⁴+3×10³+4×10²+5×10¹+6×10⁰

Another Example:

2538₁₀ = 2×10³+5×10²+3×10¹+8×10⁰

* Binary Number System

The Binary number system has a base of 2.So that the 2 digits 0 and 1 are used. This number system is basically the same as the decimal  number system except only two digits are used. Binary numbers are made up of binary digits which are referred to as bits

27	26	25	24	23	22	21	20
128's	64's	32's	16's	8's	4's	2's	1's

For example ,to represent the decimal number 161 in binary would require the eight bit binary number 10100001that is 10100001=2⁷×1+2⁶×0+2⁵×1+2⁴×0+2³×0+2²×0+2¹×0+2⁰×1

               =128+0+32+0+0+0+0+1

               =161₁₀

More 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 Number System

Simillarly Octal number system has a base of 8 and uses digits from 0 to 7.

8³	8²	8¹	8⁰
512's	64's	8's	1's

For example ,to represent the decimal number 161 in octal would require the 2 digit octal number 241 

241₈=8²×2+8¹×4+8⁰×1

        =128+32+1

        =161₁₀

More 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

* Hexadecimal Number System

         

Hexadecimal numbers uses the base of 16 and include digits from 0 to 9 and alphabet from A to F.

H denotes hex prefix.

163	162	161	160
4096's	256's	16's	1's

For example ,to represent the decimal number 161 in octal would require the 2 digit hexadecimal number A1

A1₁₆=16¹×10+16⁰×1

    =160+1

    =161₁₀

       

More 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