Computer Systems Architecture Copyright © Genetic Computer School 2008 CSA 1- 0 Lesson 1 Number System

CSA 1- 1

Computer Systems Architecture

Copyright © Genetic Computer School 2008

Lesson 1

Number System

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LESSON OVERVIEW Different types of number systems Common bases Place values Conversion of bases Computer calculation Arithmetic of the computer Subtracting using twos complement Coding systems Binary coded decimal Floating-point numbers Numbers in standard form Integers and floating-point arithmetic

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NUMBER SYSTEMS

A number system is the set of symbols used to express quantities as the basis for counting, determining order, comparing amounts, performing calculations, and representing value.

It is the set of characters and mathematical rules that are used to represent a number.

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DIFFERENT TYPES OFNUMBER SYSTEMS

Decimal

Binary

Octal

Hexadecimal

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DECIMAL NUMBER SYSTEM

The decimal or denary number system, base 10, has a radix of 10.

Decimal uses different combinations of 10 symbols to represent any valy (i.e., 0,1,2,3,4,5,6,7,8 and 9)

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BINARY NUMBER SYSTEM

Binary is known as machine language.

Data is stored and manipulated inside the computer in binary.

The binary number system is based on two digits, 0 and 1.

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OCTAL NUMBER SYSTEM

The Octal number system has eight as its base; it uses the symbols 0, 1, 2, 3,4,5,6 and 7 only.

For the values eight and above, need to use two digits.

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HEXADECIMAL NUMBER SYSTEM

The Hexadecimal number has sixteen as its base; using 0,1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E and F.

A, B, C, D, E and F stand for the “digits” ten, eleven, twelve, thirteen, fourteen and fifteen.

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PLACE VALUE

Place value, positional value depends on the base used.

Example:

The third place from the right

in base 10 has the place value 100



In base 16 has the place value 256

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DECIMAL TO OTHER BASES

Divide the base into the quotient and keep repeating the process until there is a zero quotient. Reading off the remainder

in the reverse order of how you wrote them down gives the answer.

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EXAMPLE (1)

2 ) 132 ) 6 , remainder 12 ) 3 , remainder 02 ) 1 , remainder 1 0 , remainder 1

1310 = 11012

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EXAMPLE (2)

8 ) 2368 ) 29 remainder 48 ) 3 remainder 5

0 remainder 3

23610 = 3548

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EXAMPLE (3)

16 ) 47316 ) 29 remainder 916 ) 1 remainder D

0 remainder 1

47310 = 1D916

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Some decimal fractions cannot be represented exactly as binary fractions.

To reduce errors of this type, computers need to store such converted values to a large number of binary places.

The process involves repeatedly multiplying by 2 that part of the decimal fraction to the right of the decimal point, and writing down the whole number part of the product at each stage ( but not involving it in subsequent multiplication ). Reading the whole number parts down from the top gives the binary fraction to as many places as is necessary.

CHANGING DECIMAL FRACTION TO BINARY

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EXAMPLE (4)

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(e.g. 13.746)

Work separately on the whole and fraction parts. Then link the two answers together with a point.

TO CONVERT A MIXED DECIMAL NUMBER

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Multiple each digit with its place value

and then added together.

FROM OTHER BASES TO DECIMAL(Whole Number)

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EXAMPLE (5)

11012 = (1x8)+(1x4)+(0x2)+(1x1)

= 1310

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EXAMPLE (6)

11028 = (1x512) +(1x64) +(0x8) +(2x1)

= 57810

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EXAMPLE (7)

17F16 =(1x256) +(7x16) +(15x1)

= 38310

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1s compliment and 2s compliment used to represent positive and negative number.

Example

1s COMPLEMENT AND 2s COMPLEMENT

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Adding Binary Numbers

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Subtracting Binary NumbersUsing Twos Compliment

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CODING SYSTEMS

Three of the most popular coding systems are: ASCII (American Standard Code for Information Interchange) EBCDIC (Extended Binary Coded Decimal Interchange Code) BCD (Binary Coded Decimal)

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FLOATING POINT NUMBERS

Floating-point numbers allow a far greater range of values - integer, fractional or mixed numbers, - in a single word. Calculations in floating-point arithmetic are slower than those in fixed-length working.

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STANDARD FORM

The number 57429 in standard form is:

5.7429 X 104

where 5.7429 is the mantissa and

4 is the exponent.

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FLOATING POINT ADDITION (1)

(0.1011 x 25) + (0.1001 x 25)

= (0.1011 + 0.1001) x 25

= 1.0100 x 25

= 0.1010 x 26

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FLOATING POINT ADDITION (2)

(0.1001 x 23) + (0.1110 x 25)

= (0.001001 x 25) + 0.1001) x 25

= 1.000001 x 25

= 0.1000 x 26 (after truncation)

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FLOATING POINT MULTIPLICATION

(0.1101 x 26) x (0.1010 x 24)

= 0.1000001 x 210

= 0.1000 x 210 (after truncation)

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FLOATING POINT DIVISION

(0.1011 x 27) ÷ (0.1101 x 24)

= 0.1101 x 23

Documents

Computer Systems Architecture Copyright © Genetic Computer School 2008 CSA 1- 0 Lesson 1 Number System