Number System Mep

8/4/2019 Number System Mep

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NUMBER SYSTEM & COMPUTER

ARITHMETIC

CS 211 COMPUTER FUN DAMENTALSMEP_MIT-AQUINAS 1st term sy 2011-2012


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Objectives

Learn the following Positional Number system

Different number system

Conversion of number system Fractional numbers

Computer arithmethic



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Positional Number system

based on exactly where the numbers are in thesequence of numbers

A number is represented by a string of digits

where each digit position has an associated

weight.

The value of each digit in such a number isdetermined by three considerations:

1. The digit itself,

2. The position of the digit in the number, and

3. The base of the number system.

CS 211 COMPUTER FUN DAMENTALS


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Decimal (base 10) numbers are expressed in thepositional notation

4202 = 2x100 + 0x101 + 2x102 + 4x103

The right-most is the least significant digit

The left-most is the most significant digit

Positional Number system



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4202 = 2x100 + 0x101 + 2x102 + 4x103

1s multiplier

1




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4202 = 2x100 + 0x101 + 2x102 + 4x103

10s multiplier

10




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4202 = 2x100 + 0x101 + 2x102 + 4x103

100s multiplier

100




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4202 = 2x100 + 0x101 + 2x102 + 4x103

1000s multiplier

1000




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Binary (base 2) numbers are alsoexpressed in the positional notation

10011 = 1x20 +1x21 +0x22 +0x23 +1x24

The right-most is the least significant digit

The left-most is the most significant digit



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10011 = 1x20 +1x21 +0x22 +0x23 +1x24

1s multiplier

1




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10011=1x20 + 1x21 +0x22 +0x23 +1x24

2s multiplier

2




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10011=1x20 +1x21 + 0x22 +0x23 +1x24

4s multiplier

4




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10011=1x20 +1x21 +0x22 + 0x23 +1x24

8s multiplier

8




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10011 = 1x20 +1x21 +0x22 +0x23 + 1x24

16s multiplier

16




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Counting in

Decimal

0

1

23

4

5

6

7

8

9

10

11

1213

14

15

16

17

18

19

20

21

2223

24

25

26

27

28

29

30

31

3233

34

35

36

.

.

.

0

1

1011

100

101

110

111

1000

1001

1010

1011

11001101

1110

1111

10000

10001

10010

10011

10100

10101

1011010111

11000

11001

11010

11011

11100

11101

11110

11111

100000100001

100010

100011

100100

.

.

.

Counting

in Binary



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Number system

A number system uses a specific radix (base) e.g. Decimal base 10

Binary base 2

In digital electronics (computer) the only choice of

base in which to perform arithmetic is base-2, that isbinary arithmetic, using the only two digits available, 0and 1.



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Decimal number system

uses the common 0 thru 9 digits Single-digit numbers 0 to 9 make up the set numbers

called units or ones

When the count is higher than 9 the number is carried

over to what is called the tens position

If the number is over 99 the carried over number is in thehundreds position.

After the hundreds the position are thousands, ten

thousands, hundred thousand, millions, etc.

Each position, or place value, to the left increases by afactor of 10.

Thus, the decimal number system is the base 10 system



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Binary Number System

Also called the Base 2 systemThe binary number system is used to

model the series of electrical signals

computers use to represent information0 represents the no voltage or an off state

1 represents the presence of voltage or an

on state known as machine language



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Binary Digit (Bit)

A bit (binary digit) is the smallest unit of information

can have two values - 1 and 0.

Binary digits, or bits, can representnumbers, codes, or instructions.

Byte: a grouping of eight bits of

information



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Binary Number SystemThe names of the first few places in the binary notation are:

16s 8s 4s 2s 1s

(the places are powers of two):

2

x

2 2

x x

2 2 2

x x x

2 2 2 2 1

16s 8s 4s 2s 1s

Each place is double the last place from the right to left



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Octal Number System

Also known as the Base 8 SystemUses digits 0 - 7

Readily converts to binary

Groups of three (binary) digits can beused to represent each octal digit

Also uses multiplication and division

algorithms for conversion to and from base10



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Hexadecimal Number System

Base 16 system

Uses digits 0-9 &

letters A,B,C,D,E,F

Groups of four bits

represent eachbase 16 digit



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Integer Representation

Use 8, 16, 32 or 64 bit to represent number usingpositional notation

Unsigned -- 10101010=170

Cant represent negative numbers

Simple

8 bit unsigned integer from 0 to 255

Signed -- 10101010= -42

The most significant bit of a binary number is usuallyused as the sign bit.

Uses MSB for sign 1 = - , 0 = +

computers do not subtract very well

8 bit signed integer from 127 to 128



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(4bit)Signed numbers



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1s compliment & 2s complement used torepresent positive and negative number

Allows the computer to use addition to

deal with subtraction

Integer Representation

1s complement

5 00000101

-5 11111010

2s complement

5 00000101

-5 11111010 + 1 = 11111010CS 211 COMPUTER FUN DAMENTALSMEP_MIT-AQUINAS 1st term sy 2011-2012


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Fractional numbers (Floating point Representation)

single precision binary floating-pointformat ( binary32)

Sign bit: 1 bit (1 negative, 0 positive)

Exponent: width: 8 bitsSignificand precision(mantissa): 23

(IEEE 754 single precision binary floating-point format)



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Binary Code

A digital system requires that all its information be inbinary form. But the External world uses the alphabetic

characters, decimal digits, and special Characters (e.g.,

periods, commas, plus and minus signs) to represent

information.

A unique pattern of 0s and 1s is used to represent eachrequired character. This pattern is the code

corresponding to that character.

several codes that are commonly used in digital

systems. Binary Coded Decimal

Alphanumeric Codes



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Binary Coded Decimal

Each Decimal digit is coded into 4 bits

DECIMAL BCD

0 0000 5 0101

1 0001 6 0110

2 0010 7 01113 0011 8 1000

4 0100 9 1001

Only ten of the possible sixteen (24) patterns of bits are

used.e.g. 43210 would represented as:

0100 0011 0010



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Alphanumeric Codes

Extended BCD Interchange Code (EBCDIC)

American Standard Code for Information Interchange (ASCII).



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NUMBER SYSTEM

CONVERSION



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Decimal to Binary Conversion

The easiest way to convert a decimal number to itsbinary equivalent is to use the Division Algorithm

This method repeatedly divides a decimal number by 2

and records the quotient and remainder (stop whenquotient is zero)

The remainder digits (a sequence of zeros and ones)

form the binary equivalent.

The first remainder is the least significant digit(LSD),and

the last Remainder is the most significant digit(MSD).



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Decimal to Binary ConversionConvert decimal number 2810 into binary.

28/2 = 14 0 (LSD)

14/2 = 7 0

7/2 = 3 1

3/2 = 1 1

1/2 = 0 1 (MSD)

The Binary number is 111002

Convert decimal number 12210 into binary.

122/2 = 61 0 (LSD)

61/2 = 30 1

30/2 = 15 0

15/2 = 7 1

7/2 = 3 1

3/2 = 1 1

1/2 = 0 1 (MSD)

The Binary number is 11110102 CS 211 COMPUTER FUN DAMENTALSMEP_MIT-AQUINAS 1st term sy 2011-2012


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Binary to Decimal Conversion

The easiest method for converting abinary number to its decimal equivalent isto use the Multiplication Algorithm

Multiply the binary digits by increasingpowers of two, starting from the right

Then, to find the decimal number

equivalent, sum those products



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Convert binary number 111002 into decimal.

1 1 1 0 0

1 x 24 = 16 1 x 23 = 8 1 x 22 = 4 0 x 21 = 0 0 x 20 =0

16 + 8 + 4 = 2810

Binary to Decimal Conversion

Convert binary number 11110102 into decimal.

1 1 1 1 0 1 0

1 x 26 =64

1 x 25 =32

1 x 24 =16

1 x 23 =8

0 x 22 =0

1 x 21 =2

0 x 20

=0

64 + 32 + 16 + 8 + 2 = 12210



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To convert a decimal number into hexadecimal,divide the number repeatedly by 16 and takethe remainders. The first remainder is the LSD,and the last remainder is the MSD.

Decimal to Hexadecimal Conversion

Convert Decimal number 68410 into Hexadecimal.C (LSD)

A

2 (MSD)

The Hexadecimal number is 2AC16

Convert Decimal number 83010 into Hexadecimal.

E (LSD)3

3 (MSD)

The Hexadecimal number is 33E16



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To convert a decimal number into octal, divide thenumber repeatedly by 8 and take the remainders.

The first remainder is the LSD, and the last remainder is

the MSD.

Decimal to Octal Conversion

Convert Decimal number 15910 into Octal.

159/8 = 19 7 7 (LSD)

19/8 = 2 3 3

2/0 = 0 2 2 (MSD)

The Hexadecimal number is 2378

Convert Decimal number 46010 into Octal.

460/8 = 57 4 4 (LSD)

57/8 = 7 1 1

7/0 = 0 7 7 (MSD)

The Hexadecimal number is 7148



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Binary to Hexadecimal Conversion

To convert a binary number into hexadecimal, arrangethe number in groups of four and find the hexadecimal

equivalent of each group (use a substitution code).

If the number cannot be divided exactly into groups offour, insert zeros to the left of the number as needed so

the number of Digits are divisible by four.



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Each hex number converts to 4 binary digitsConvert binary number 100111112 into decimal.

1001 1111

1 0 0 1 1 1 1 1

1 x 23

= 8 0 x 22

= 0 0 x 21

= 0 1 x 20

= 1 1 x 23

= 8 1 x 22

= 4 1 x 21

= 2 1 x 20

= 1

8 + 1 = 910 = 9 8 + 4 + 2 + 1 = 1510 = F

100111112 = 9F16



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Convert 0101011010101110011010102 to hexusing the 4-bit substitution code :

1010


0101 0110 1010 1110 0110

5 6 A E 6 A

56AE6A16



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Substitution code can also be used to convertbinary to octal by using 3-bit groupings:

Convert binary 010101101010111001101010

255271528

Binary to Octal Conversion

010 101 101 010111 001

101 010

2 5 5 2 7 1 5 2



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To convert an octal number into binary,write the 3-bit binary equivalent of each

octal digit

Octal to Binary Conversion

convert 758 to binary7 5

111 101




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To convert an octal number into binary,write the 3-bit binary equivalent of each

octal digit

Octal to Binary Conversion

convert 758 to binary7 5

111 101


convert 6538 to binary6 5 3

110 101 011

The Binary number is 1101010112CS 211 COMPUTER FUN DAMENTALSMEP_MIT-AQUINAS 1st term sy 2011-2012


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Octal to Decimal Conversion

To convert an octal number into decimal, calculatethe sum of the powers of 8 of the number.

Convert 6538 to itsdecimal equivalent:

6 5 3

6 x 82 =384

5 x 81 =40

3 x 80 =3

384 + 40 + 3 = 427

6538 = 427

Convert 2378 to itsdecimal equivalent:

2 x 82 = 128

3 x 81 = 24

7 x 80 = 7

2378 = 159



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Two step Octal to Binary

Binary to Hexadecimal

Octal to Hexadecimal Conversion



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Decimal fraction to Binary fraction

Convert decimal number 0.87510into binary.

2 x0.875

= 1.750 1

2 x0.750

= 1.5 1

2 x 0.5 = 1.0 1

The Binary number is 0.1112

Convert decimal number0.12510 into binary.

2 x0.125

= 0.250 0

2 x0.250

= 0.5 0

2 x 0.5 = 1.0 1


repeatedly doubling the decimal fraction.Until a 0 remains to the left of the decimalpoint or until the desired precision



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Convert decimal number 0.34510into binary.

2 x 0.345 = 0.69 0

2 x 0.69 = 1.38 1

2 x 0.38 = 0.76 0

2 x 0.76 = 1.52 1

2 x 0.52 = 1.04 1

2 x 0.04 = 0.08 0

2 x 0.08 = 0.16 0

2 x 0.16 = 0.32 0

2 x 0.32 = 0.64 0

2 x 0.64 = 1.28 1


Decimal fraction to Binary fraction



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Binary fraction to Decimal fraction

Convert binary number 0.0012 into decimal.. 0 0 1

0 x 2-1 = 0 0 x 2-2 = 0 1 x 2-3 = 0.125

0.0012

= 0.12510

Convert binary number 0.1112 into decimal.

. 1 1 1

1 x 2-1 = 0.5 1 x 2-2 = 0.25 1 x 2-3 = 0.125

0.1112 = 0.5 + 0.25 + 0.125 =

0.87510



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Binary fraction to Decimal fraction

Convert binary number 0.01011000012 into decimal.. 0 1 0 1 1 0 0 0 0 1

0 x2-1

=0

1 x 2-2

=0.25

0 x2-3

=0

1 x 2-4

=0.0625

1 x 2-5

=0.03125

0 x2-6

=0

0 x2-7

=0

0 x2-8

=0

0 x2-9

=0

1 x 2-10

=0.0009765625

0.01011000012 = 0.34472710

CS 211 COMPUTER FUN DAMENTALSMEP_MIT-AQUINAS 1st

term sy 2011-2012


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Binary fraction to Decimal fractionConvert binary number 0.01011000012 into decimal.

. 0 1 0 1 1 0 0 0 0 1

0 x2-1

=0

1 x 2-2

=0.25

0 x2-3

=0

1 x 2-4

=0.0625

1 x 2-5

=0.03125

0 x2-6

=0

0 x2-7

=0

0 x2-8

=0

0 x2-9

=0

1 x 2-10

=0.0009765625

0.01011000012 = 0.34472710

Convert decimal number 0.34510 into binary.

2 x 0.345 = 0.69 0

2 x 0.69 = 1.38 1

2 x 0.38 = 0.76 0

2 x 0.76 = 1.52 1

2 x 0.52 = 1.04 1

2 x 0.04 = 0.08 0

2 x 0.08 = 0.16 0

2 x 0.16 = 0.32 0

2 x 0.32 = 0.64 0

2 x 0.64 = 1.28 1

The Binary number is 0.0101102 CS 211 COMPUTER FUN DAMENTALSMEP_MIT-AQUINAS 1st

term sy 2011-2012


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BINARY ARITHMETIC


term sy 2011-2012


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Binary Addition

similar to the addition of decimal numbers.Numbers in each column are added togetherwith a possible carry from a previous column.

carry bit

00

0

+ 10

1

+ 01

1

+ 11

0

+

1

1

11

1

+

1

1

+

1


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Binary Addition

115

61

1011101

1 0

11

000

1

+ +

150

53

1111100

1 1100

+ +2 01

0

11

CS 211 COMPUTER FUN DAMENTALSMEP_MIT-AQUINAS 1

st

term sy 2011-2012


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Binary Addition

11 1010 1

0101

257

09.

+ +3.5. 5

0

.1 1.

00 .

01 10

10 01000

5

2520.

++

9.

11. 57

.

1 1.11 .

1

11

11


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term sy 2011-2012


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Binary Subtraction

With aborrow

of 1

00

0

- 10

1

- 01

0

-11

1

-

e.g.

163

31

1 00011 0 1

1

11

011

- - 0

1

6.25.5

.751

1 0.00 1

1

1

1

1

1

- - 1

0

4 .

.CS 211 COMPUTER FUN DAMENTALS

MEP_MIT-AQUINAS 1st

term sy 2011-2012


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1s complement

Switch all 0s to 1s and 1s to 0s

Binary # 10110011

1s complement 01001100

2s complement

Step 1: Find 1s complement of the number

Binary # 11000110

1s complement 00111001 Step 2: Add 1 to the 1s complement

00111001

00000001

00111010

Binary Subtraction using complement

Binary subtraction is tricky due to the borrowing process and prone

to error and alternative and easy way is using number complement


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Binary Subtraction using 1s complement

16

331

1 000

111

11

011- -

01

1 000

000011

+

0

1 110

1

1

1011 When subtraction is performed in the

1s complement system, anyend-around carry is added to the leastsignificant bit

Involves forming the 1s complement of the subtrahend andthen adding this complement to the minuend

1s complement


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Binary Subtraction using 2s complement

16

331

1 000

111

11

011- -

01

1 000

011011

+

0

1 110

1

1

When subtraction is performed in the2s complement system, anyend-around carry is dropped

Involves forming the 2s complement of the subtrahend andthen adding this complement to the minuend

2s complement

001 11 +

10110


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011 0.0

0 1

1

1

1

1

1

- 1

0

.

.

111 0 .0

1 0

01

0

1

+ 1.

.

1

00

11

1 1111.00

111 0.0

1 1

11

0

1

+ 1.

.

0

00

1

1

using 2s complementusing 1s complement

Binary Subtraction


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Binary Multiplication and division

00

0

x10

0

x0

1

1

x1

1

0

x

multiplication rule

0 1 0

Division rule

=1 1 1 =


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Binary Multiplication

10

120

x 12 1 000

111

+ 11 10

0 0001 001

0 0001 0011 000

2.5

625

x1.25

250

1 011

11

+ .1 .0

1010 00

1 10

. 010

3.125

1


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25

5

5

11 0 11 00101 1

01 1

01 101 1

11 1 10 00101 0

01101 111

1

292.416

12

0.

0

0

0000

01 011

0000

01 0 00

110. 10101..

Binary Division

CS 211 COMPUTER FUN DAMENTALSst

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Number System Mep