computer fundamentals lecture 02 sri lanka institute of information technology (sliit)

04/08/23 Sri Lanka Institute of Information Technology

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Computer Fundamentals

Lecture 2: Number Systems


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Objectives After completing this lecture you will

be able to: Explain different positionally-waited

number systems Translate numbers between number

systems Appraise binary number system


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Agenda Number bases used with computers Why binary? Number Base Conversion Conversion of Fractions


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Number bases used with computers

Decimal, b=10a={0,1,2,3,4,5,6,7,8,9}

Binary, b=2a={0,1}

Octal, b=8a={0,1,2,3,4,5,6,7}

Hexadecimal, b=16a={0,1,2,3,4,5,6,7,8,9,A,B,C,D,E,F}


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Why Binary is used in Computers? Electronic components that represent

binary are better than those that represent binary because of; Easier to design Cost effective


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Number Base ConversionDecimal to Binary divide the number successively by 2, and after each

division record the remainder which is either 1 or 0. example,

12310 becomes 123/2 =61 r=161/2 = 30 r=130/2 = 15 r=015/2 = 7 r=17/2 =3 r=13/2 =1 r=11/2 =0 r=1

12310 = 11110112


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Decimal to Octal Divide the number by 8 remainder is a number in the range 0 to 7.

example, 462910 becomes

4629/8= 578 r=5578/8= 72 r=272/8= 9 r=09/8 = 1 r=11/8 = 0 r=1

462910 =110258

Number Base Conversion


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Decimal to Hexadecimal Divide the number by 16 remainder lies in the decimal range 0 to 15, corresponding to the

hexadecimal range 0 to F. example, 5324110 becomes

53241/16 = 3327 r=93327/16 = 207 r=15 = F207/16 =12 r=15 = F12/16 =0 r=12 = C

5324110=CFF916



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Binary to Decimal Take the left most none zero bit, double it and add it to the bit on its right. Now take this result, double it and add it to the next bit on

the right. Continue in this way until the least significant bit has been

added in.



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Binary to Decimal (cont’d)

For example, 10101112 becomes

Therefore, 10101112 = 8710



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Binary to Octal Form the bits into groups of three starting at the

binary point and moving leftwards. Replace each group of three bits with the

corresponding octal digit (0 to 7).For example,110010111012 becomes 11 001 011 101

3 1 3 5

Therefore,110010111012 = 31358



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Binary to Hexadecimal The binary number is formed into groups of four bits

starting at the decimal point. Each group is replaced by a hexadecimal digit from 0 to

9, A, B, C, D, E, and F.For example,

110010111012 becomes 110 01011101

6 5 D

Therefore, 110010111012 = 65D16



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Octal to Binary Each octal digit is simply replaced by its 3-bit binary equivalent. It is important to remember that (say) 3 must be replaced by 011

and not 11.For example,413578 becomes 4 1 3 5 7

100 001 011 101 111

Therefore,413578 = 1000010111011112.



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Hexadecimal to Binary Each hexadecimal digit is replaced by its 4-bit binary equivalent. For example AB4C16 becomes A B 4 C

1010 1011 0100 1100

Therefore, AB4C16 = 10101011010011002



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Octal to Decimal Take the left-most digit, Multiply it by eight and add it to the digit on its right. Then, multiply this subtotal by eight and add it to the next

digit on its right. The process ends when the left-most digit has been added to

the subtotal.For example, 64378 becomes 6 4 3 7

4852

416419

33523359

Therefore, 64378 = 335910



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Hexadecimal to Decimal The method is identical to the procedures for binary and

octal except that 16 is used as a multiplier.

For example, 1AC16 becomes 1 A C

1626

416428

Therefore, 1AC16 = 42810



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Conversion of FractionsConverting Binary Fractions to Decimal Fractions For example, consider the conversion of 0.011012 into decimal

form.0.0 0 1 1 0 1

1/21/2

1/45/4

5/813/8

13/1613/16

13/32Therefore, 0.011012 = 13/32.


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Converting Decimal Fractions to Binary Fractions For example, 0.687510 becomes 0.6875 x 2 1.3750

0.3750 x 2 0.75000.7500 x 2 1.50000.5000 x 2 1.0000

0.0000 x 2 ends the processTherefore, 0.687510 = 0.10112

Conversion of Fractions


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Thank YouLecture 03: Computer Arithmetic

computer fundamentals lecture 02 sri lanka institute of information technology (sliit)

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