Lecture 7 How computers process data (Number Systems) PRESENTED BY MD. MAHBUBUL ALAM, PHD 1

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 PRESENTED BY MD. MAHBUBUL ALAM, PHD 2

Quantities/Counting (1 of 3) DecimalBinaryOctal Hexa- decimal PRESENTED BY MD. MAHBUBUL ALAM, PHD 3

Quantities/Counting (2 of 3) DecimalBinaryOctal Hexa- decimal A B C D E F PRESENTED BY MD. MAHBUBUL ALAM, PHD 4

Quantities/Counting (3 of 3) DecimalBinaryOctal Hexa- decimal Etc. PRESENTED BY MD. MAHBUBUL ALAM, PHD 5

Conversion Among Bases The possibilities: Hexadecimal DecimalOctal Binary PRESENTED BY MD. MAHBUBUL ALAM, PHD 6

Quick Example = = 31 8 = Base PRESENTED BY MD. MAHBUBUL ALAM, PHD 7

Decimal to Decimal (just for fun) Hexadecimal DecimalOctal Binary Next slide… PRESENTED BY MD. MAHBUBUL ALAM, PHD 8

=>5 x 10 0 = 5 2 x 10 1 = 20 1 x 10 2 = Base Weight PRESENTED BY MD. MAHBUBUL ALAM, PHD 9

Binary to Decimal Hexadecimal DecimalOctal Binary PRESENTED BY MD. MAHBUBUL ALAM, PHD 10

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 PRESENTED BY MD. MAHBUBUL ALAM, PHD 11

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” PRESENTED BY MD. MAHBUBUL ALAM, PHD 12

Octal to Decimal Hexadecimal DecimalOctal Binary PRESENTED BY MD. MAHBUBUL ALAM, PHD 13

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 PRESENTED BY MD. MAHBUBUL ALAM, PHD 14

Example => 4 x 8 0 = 4 2 x 8 1 = 16 7 x 8 2 = PRESENTED BY MD. MAHBUBUL ALAM, PHD 15

Hexadecimal to Decimal Hexadecimal DecimalOctal Binary PRESENTED BY MD. MAHBUBUL ALAM, PHD 16

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 PRESENTED BY MD. MAHBUBUL ALAM, PHD 17

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 = PRESENTED BY MD. MAHBUBUL ALAM, PHD 18

Decimal to Binary Hexadecimal DecimalOctal Binary PRESENTED BY MD. MAHBUBUL ALAM, PHD 19

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. PRESENTED BY MD. MAHBUBUL ALAM, PHD 20

Example = ? = PRESENTED BY MD. MAHBUBUL ALAM, PHD 21

Octal to Binary Hexadecimal DecimalOctal Binary PRESENTED BY MD. MAHBUBUL ALAM, PHD 22

Octal to Binary Technique ◦Convert each octal digit to a 3-bit equivalent binary representation PRESENTED BY MD. MAHBUBUL ALAM, PHD 23

Example = ? = PRESENTED BY MD. MAHBUBUL ALAM, PHD 24

Hexadecimal to Binary Hexadecimal DecimalOctal Binary PRESENTED BY MD. MAHBUBUL ALAM, PHD 25

Hexadecimal to Binary Technique ◦Convert each hexadecimal digit to a 4-bit equivalent binary representation PRESENTED BY MD. MAHBUBUL ALAM, PHD 26

Example 10AF 16 = ? A F AF 16 = PRESENTED BY MD. MAHBUBUL ALAM, PHD 27

Decimal to Octal Hexadecimal DecimalOctal Binary PRESENTED BY MD. MAHBUBUL ALAM, PHD 28

Decimal to Octal Technique ◦Divide by 8 ◦Keep track of the remainder PRESENTED BY MD. MAHBUBUL ALAM, PHD 29

Example = ? = PRESENTED BY MD. MAHBUBUL ALAM, PHD 30

Decimal to Hexadecimal Hexadecimal DecimalOctal Binary PRESENTED BY MD. MAHBUBUL ALAM, PHD 31

Decimal to Hexadecimal Technique ◦Divide by 16 ◦Keep track of the remainder PRESENTED BY MD. MAHBUBUL ALAM, PHD 32

Example = ? = 4D = D PRESENTED BY MD. MAHBUBUL ALAM, PHD 33

Binary to Octal Hexadecimal DecimalOctal Binary PRESENTED BY MD. MAHBUBUL ALAM, PHD 34

Binary to Octal Technique ◦Group bits in threes, starting on right ◦Convert to octal digits PRESENTED BY MD. MAHBUBUL ALAM, PHD 35

Example = ? = PRESENTED BY MD. MAHBUBUL ALAM, PHD 36

Binary to Hexadecimal Hexadecimal DecimalOctal Binary PRESENTED BY MD. MAHBUBUL ALAM, PHD 37

Binary to Hexadecimal Technique ◦Group bits in fours, starting on right ◦Convert to hexadecimal digits PRESENTED BY MD. MAHBUBUL ALAM, PHD 38

Example = ? B B = 2BB 16 PRESENTED BY MD. MAHBUBUL ALAM, PHD 39

Octal to Hexadecimal Hexadecimal DecimalOctal Binary PRESENTED BY MD. MAHBUBUL ALAM, PHD 40

Octal to Hexadecimal Technique ◦Use binary as an intermediary PRESENTED BY MD. MAHBUBUL ALAM, PHD 41

Example = ? E = 23E 16 PRESENTED BY MD. MAHBUBUL ALAM, PHD 42

Hexadecimal to Octal Hexadecimal DecimalOctal Binary PRESENTED BY MD. MAHBUBUL ALAM, PHD 43

Hexadecimal to Octal Technique ◦Use binary as an intermediary PRESENTED BY MD. MAHBUBUL ALAM, PHD 44

Example 1F0C 16 = ? 8 1 F 0 C F0C 16 = PRESENTED BY MD. MAHBUBUL ALAM, PHD 45

Exercise – Convert... Don’t use a calculator! DecimalBinaryOctal Hexa- decimal AF Skip answer Answer PRESENTED BY MD. MAHBUBUL ALAM, PHD 46

Exercise – Convert … DecimalBinaryOctal Hexa- decimal C AF Answer PRESENTED BY MD. MAHBUBUL ALAM, PHD 47

Binary Arithmetic: Addition & Subtraction XYX+Y PRESENTED BY MD. MAHBUBUL ALAM, PHD 48 XYX-Y

Binary Arithmetic: Multiplication & Division XYX*Y PRESENTED BY MD. MAHBUBUL ALAM, PHD 49

Boolean Algebra The digital circuits present in a digital computer are designed using a mathematical discipline known as Boolean Algebra. It describes the relationship between the inputs and outputs of a digital circuit. Boolean Algebra was named in honor of Gorge Boole, an English Mathematician, who had proposed the basic principles of this. Objective: Boolean Algebra is used mainly by design engineers in order to obtain the required output by using least number of logic gates. PRESENTED BY MD. MAHBUBUL ALAM, PHD 50

Components Like any other algebra, Boolean Algebra also uses variables and operations. ◦A Boolean variable has only two possible values which is either true (1) or false (0) ◦Basic Boolean operations are: AND, OR and NOT PRESENTED BY MD. MAHBUBUL ALAM, PHD 51

Basic Logical Operations All these three basic logical operations can be represented symbolically as ◦A AND B = A. B ◦A OR B = A + B ◦NOT A = A’ These operations can be defined in a form known as Truth Table, which s a list of all possible input values and the output for each input combination. PRESENTED BY MD. MAHBUBUL ALAM, PHD 52

Truth Table for AND Operator Truth Table for a 2-input AND Operator is as follows ABY = A. B PRESENTED BY MD. MAHBUBUL ALAM, PHD 53

Truth Table for OR Operator Truth Table for a 2-input OR Operator is as follows ABY = A + B PRESENTED BY MD. MAHBUBUL ALAM, PHD 54

Truth Table for NOT Operator Truth Table for NOT Operator is as follows AY = A’ PRESENTED BY MD. MAHBUBUL ALAM, PHD 55

Logic Gate In electronics, a logic gate is an idealized or physical device implementing a Boolean function; that is, it performs a logical operation on one or more logical inputs, and produces a single logical output. PRESENTED BY MD. MAHBUBUL ALAM, PHD 56

Thank you (Courtesy: Dept. of IT, York University) PRESENTED BY MD. MAHBUBUL ALAM, PHD 57