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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.

105104103102101100
Hundreds of ThousandsTens of ThousandsThousandsHundredsTensOnes
100000100001000100101

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

2726252423222120
128's64's32's16's8's4's2's1'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+1×2= 16+4+1= 21
101112 = 10111B = 1×24+0×23+1×22+1×21+1×2=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.


8⁰
512's64's8's1'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+7×8= 16+7 = 23
308 = 3×81+0×8= 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.

163162161160
4096's256's16's1'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+8×16= 40
2F16 = 2FH = 2×161+15×16= 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
0000
1111
21022
31133
410044
510155
611066
711177
81000108
91001119
10101012A
11101113B
12110014C
13110115D
14111016E
15111117F
16100002010
17100012111
18100102212
19100112313
20101002414
21101012515
22101102616
23101112717
24110003018
25110013119
2611010321A
2711011331B
2811100341C
2911101351D
3011110361E
3111111371F
321000004020
                             

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