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Computer Systems Architecture
Lesson 1
Number System
Copyright © Genetic Computer School 2008
CSA 1- 0
Computer Systems Architecture
LESSON OVERVIEW
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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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Computer Systems Architecture
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 OF
NUMBER 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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Computer Systems Architecture
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 2 has the place value 4
in base 8 has the place value 64
In base 16 has the place value 256
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Computer Systems Architecture
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 ) 13
2 ) 6 , remainder 1
2 ) 3 , remainder 0
2 ) 1 , remainder 1
0 , remainder 1
1310 = 11012
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EXAMPLE (2)
8 ) 236
8 ) 29 remainder 4
8 ) 3 remainder 5
0 remainder 3
23610 = 3548
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EXAMPLE (3)
16 ) 473
16 ) 29 remainder 9
16 ) 1 remainder D
0 remainder 1
47310 = 1D916
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Computer Systems Architecture
CHANGING DECIMAL
FRACTION TO BINARY
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.
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Computer Systems Architecture
EXAMPLE (4)
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TO CONVERT A MIXED
DECIMAL NUMBER
(e.g. 13.746)
Work separately on the whole and
fraction parts. Then link the two
answers together with a point.
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FROM OTHER BASES
TO DECIMAL
(Whole Number)
Multiple each digit with its place value
and then added together.
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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 COMPLEMENT AND
2s COMPLEMENT
1s compliment and 2s compliment used to
represent positive and negative number.
Example
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Adding Binary Numbers
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Subtracting Binary Numbers
Using 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 floatingpoint 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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Computer Systems Architecture
FLOATING POINT
MULTIPLICATION
(0.1101 x 26) x (0.1010 x 24)
= 0.1000001 x 210
= 0.1000 x 210
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(after truncation)
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FLOATING POINT DIVISION
(0.1011 x 27) ÷ (0.1101 x 24)
= 0.1101 x 23
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