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.
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
| 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 |
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