Number SystemsNumber Systems Modified By: AM. Sihan (Hardware Engineering)

TYPE OF NUMBER SYSTEM -Binary Number System -Octal Number System -Decimal Number System -Hexadecimal Number System -Binary to Decimal -Octal to Decimal -Hexadecimal to Decimal -Decimal to Binary -Hexadecimal to Binary -Octal to Binary -Decimal to Octal -Binary to Octal -Decimal to Hexadecimal -Binary to Hexadecim

Binary :0,1 –Base 2 Decimal :0,1,2,3,4,5,6,7,8,9 – Base 10 Octal :0,1,2,3,4,5,6,7 – Base 8 Hexadecimal :0,1,2,3,4,5,6,7,8,9,A,B,C,D,E,F – Base For Ex: Binary : 0,1,1,10,11,100,101,110,111,1000,1001,1010,1011, … Decimal: 0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20… Octal : 0,1,2,3,4,5,6,7,10,11,12,13,14,15,16,17,20,21,22,23… Hexadecimal : 0,1,2,3,4,5,6,7,8,9,A,B,C,D,E,F,20,21,22,23,24,25,26,27,28,29,A,B,C,D,E,F30…

Common Number Systems SystemBaseSymbols Used by humans? Used in computers? Decimal100, 1, … 9YesNo Binary20, 1NoYes Octal80, 1, … 7No Hexa- decimal 160, 1, … 9, A, B, … F No

Quantities/Counting (1 of 3) DecimalBinaryOctal Hexa- decimal

Quantities/Counting (2 of 3) DecimalBinaryOctal Hexa- decimal A B C D E F

Quick Example = = 31 8 = Base DecimalBinary Octal Hexadecimal

Binary to Decimal Technique ◦ Multiply each bit by 2 n, where n is the “weight” of the bit ◦ The weight is the position of the bit, starting from 0 on the right ◦ Add the results

Example => 1 x 2 0 = 1 1 x 2 1 = 2 0 x 2 2 = 0 1 x 2 3 = 8 0 x 2 4 = 0 1 x 2 5 = Bit “0”

Octal to Decimal Technique ◦ Multiply each bit by 8 n, where n is the “weight” of the bit ◦ The weight is the position of the bit, starting from 0 on the right ◦ Add the results

Example => 4 x 8 0 = 4 2 x 8 1 = 16 7 x 8 2 =

Hexadecimal to Decimal Technique ◦ Multiply each bit by 16 n, where n is the “weight” of the bit ◦ The weight is the position of the bit, starting from 0 on the right ◦ Add the results

Example ABC 16 =>C x 16 0 = 12 x 1 = 12 B x 16 1 = 11 x 16 = 176 A x 16 2 = 10 x 256 =

Decimal to Binary Technique ◦ Divide by two, keep track of the remainder ◦ First remainder is bit 0 (LSB, least-significant bit) ◦ Second remainder is bit 1 ◦ Etc.

Example = ? =

Octal to Binary Technique ◦ Convert each octal digit to a 3-bit equivalent binary representation

Example = ? =

Hexadecimal to Binary Technique ◦ Convert each hexadecimal digit to a 4-bit equivalent binary representation

Example 10AF 16 = ? A F AF 16 =

Decimal to Octal Technique ◦ Divide by 8 ◦ Keep track of the remainder

Example = ? =

Decimal to Hexadecimal Technique ◦ Divide by 16 ◦ Keep track of the remainder

Example = ? = 4D = D

Binary to Octal Technique ◦ Group bits in threes, starting on right ◦ Convert to octal digits

Example = ? =

Binary to Hexadecimal Technique ◦ Group bits in fours, starting on right ◦ Convert to hexadecimal digits

Example = ? B B = 2BB 16

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