CSA 1- 1
Computer Systems Architecture
Copyright © Genetic Computer School 2008
Lesson 1
Number System
CSA 1- 2
Computer Systems Architecture
Copyright © Genetic Computer School 2008
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
CSA 1- 3
Computer Systems Architecture
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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.
CSA 1- 4
Computer Systems Architecture
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DIFFERENT TYPES OFNUMBER SYSTEMS
Decimal
Binary
Octal
Hexadecimal
CSA 1- 5
Computer Systems Architecture
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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)
CSA 1- 6
Computer Systems Architecture
Copyright © Genetic Computer School 2008
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.
CSA 1- 7
Computer Systems Architecture
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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.
CSA 1- 8
Computer Systems Architecture
Copyright © Genetic Computer School 2008
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.
CSA 1- 9
Computer Systems Architecture
Copyright © Genetic Computer School 2008
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 2 has the place value 4
in base 8 has the place value 64
In base 16 has the place value 256
CSA 1- 10
Computer Systems Architecture
Copyright © Genetic Computer School 2008
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.
CSA 1- 11
Computer Systems Architecture
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EXAMPLE (1)
2 ) 132 ) 6 , remainder 12 ) 3 , remainder 02 ) 1 , remainder 1 0 , remainder 1
1310 = 11012
CSA 1- 12
Computer Systems Architecture
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EXAMPLE (2)
8 ) 2368 ) 29 remainder 48 ) 3 remainder 5
0 remainder 3
23610 = 3548
CSA 1- 13
Computer Systems Architecture
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EXAMPLE (3)
16 ) 47316 ) 29 remainder 916 ) 1 remainder D
0 remainder 1
47310 = 1D916
CSA 1- 14
Computer Systems Architecture
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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
CSA 1- 16
Computer Systems Architecture
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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
CSA 1- 17
Computer Systems Architecture
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Multiple each digit with its place value
and then added together.
FROM OTHER BASES TO DECIMAL(Whole Number)
CSA 1- 18
Computer Systems Architecture
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EXAMPLE (5)
11012 = (1x8)+(1x4)+(0x2)+(1x1)
= 1310
CSA 1- 19
Computer Systems Architecture
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EXAMPLE (6)
11028 = (1x512) +(1x64) +(0x8) +(2x1)
= 57810
CSA 1- 20
Computer Systems Architecture
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EXAMPLE (7)
17F16 =(1x256) +(7x16) +(15x1)
= 38310
CSA 1- 21
Computer Systems Architecture
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1s compliment and 2s compliment used to represent positive and negative number.
Example
1s COMPLEMENT AND 2s COMPLEMENT
CSA 1- 22
Computer Systems Architecture
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Adding Binary Numbers
CSA 1- 23
Computer Systems Architecture
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Subtracting Binary NumbersUsing Twos Compliment
CSA 1- 24
Computer Systems Architecture
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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)
CSA 1- 25
Computer Systems Architecture
Copyright © Genetic Computer School 2008
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.
CSA 1- 26
Computer Systems Architecture
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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.
CSA 1- 27
Computer Systems Architecture
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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
CSA 1- 28
Computer Systems Architecture
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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)
CSA 1- 29
Computer Systems Architecture
Copyright © Genetic Computer School 2008
FLOATING POINT MULTIPLICATION
(0.1101 x 26) x (0.1010 x 24)
= 0.1000001 x 210
= 0.1000 x 210 (after truncation)