Transcript Number Systems - Computer Science

Numbering Systems
Outline
 What
is a Numbering System
 Review of decimal numbering system
 Binary representation range
 Hexadecimal numbering system
 Converting decimal to binary
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What is a Numbering System
Can you count?
 What do you use to count if you are not allowed
to use a calculator?
 What are the unique digits that you use?
 How many are they?
 Humans use a decimal (base 10) numbering
system.
 Do you think the computer could count?
 What are the unique digits that a computer use?
 Computers use a binary (base 2) numbering
system.
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Decimal
(base 10)
Octal
(base 8)
0
0
0
0
1
1
1
1
2
2
10
2
3
3
11
3
4
4
100
4
5
5
101
5
6
6
110
6
7
7
111
7
8
10
1000
8
9
11
1001
9
10
12
1010
A
11
13
1011
B
12
14
1100
C
13
15
1101
D
14
16
1110
E
15
17
1111
F
16
20
10000
10
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Binary
(base 2)
Hexadecimal
(base 16)
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Review of Decimal Numbering
System
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Most of us are so familiar with the decimal numbering system,
that we normally do not think about the issues inherent in the
representation.
The decimal representation is a positional numbering system.
The decimal representation of any number specifies the value as
a sum of individual digits times powers of ten (which is the
base/radix of the decimal system).
The decimal number 432110 is actually:
1 x 100 =
1 plus
2 x 101 = 20 plus
3 x 102 = 300 plus
4 x 103 = 4000
4321
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Review of Decimal Numbering
System (cont’d)
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The positions are usually (informally) named according to the
numbers that they represent: thousands, hundreds, tens and ones
(units).
We can also name the positions after the corresponding power
of 10 that each represents: position 3 (thousands), position 2
(hundreds), position 1 (tens), and position 0 (units).
In mathematics and computer science positions start from 0
rather than 1.
The powers increase from right to left.
The number 10210 is actually:
2 x 100 = 2 plus
0 x 101 = 0 plus
1 x 102 = 100
102
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Exercises
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What is binary 10112 in decimal?
1 x 20 = 1 plus
1 x 21 = 2 plus
0 x 22 = 0 plus
1 x 23 = 8
1110
What is octal 2038 in decimal?
3 x 80 = 3 plus
0 x 81 = 0 plus
2 x 82 = 128
13110
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Hexadecimal Numbering System
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The binary numbering system is very cumbersome in use, as it requires so
many digits to represent even the relatively small values.
Hexadecimal (or hex) numbering system is of particular importance, as it
overcomes the above problem, by providing excellent abbreviation/concise
representation.
A binary number can be easily converted to hexadecimal by grouping the
binary digits into blocks of four digits, to make a single hexadecimal digit,
each representing a power of 16.
The hexadecimal number 1216 is:
1 x 161 plus 2 x 160 =
1 x 24 plus 2 x 20 =
1 x 24 plus 1 x 21 =
000100102
The binary number 1010010100112 is composed of 3 groups of 4 binary
digits:
1010 0101 0011
A
5
3
A5316
It could be seen how conversion is straight forward.
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More Exercises
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What is binary 011011102 in decimal?
0 x 20 = 0 plus
1 x 21 = 2 plus
1 x 22 = 4 plus
1 x 23 = 8 plus
0 x 24 = 0 plus
1 x 25 = 32 plus
1 x 26 = 64 plus
0 x 27 = 0 plus
11010
What is it in octal?
1568
What is it in hexadecimal?
6E16
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Binary Representation Range
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With a single bit you can represent two distinct numbers
(0 and 1).
By grouping bits together, you can represent more than
two unique patterns/values.
With two bits you can represent four distinct
patterns/values 00, 01, 10 and 11.
Therefore with m bits you can represent 2m distinct
patterns/values.
The distinct values that could be represented in m bits are
0, 1, 2, …, 2m- 1. (0 <= i <= 2m- 1 or 0 <= i < 2m )
16 bits (m=16) allow for representing 216 (65,536)
different patterns/values, ranging from 0 … 65,535.
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Converting Decimal to Binary I
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Since humans use decimal numbers and computers use binary,
it is also useful to know how to convert decimal numbers into
binary numbers.
One method of converting a decimal number to a binary one
involves repeatedly dividing the decimal number by 2. Then the
remainders are written from right to left in the order they are
generated.
Converting the decimal number 2910 to binary:
29/2 =14 rem 1
14/2 = 7 rem 0
7/2 = 3 rem 1
3/2 = 1 rem 1
1/2 = 0 rem 1
Ans:
111012  000111012 (in 8 bits)
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Exercise
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Convert the decimal number 11010 to binary:
110/2 = 55 rem 0
55/2 = 27 rem 1
27/2 = 13 rem 1
13/2 = 6 rem 1
6/2 = 3 rem 0
3/2 = 1 rem 1
1/2 = 0 rem 1
Ans:
11011102
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Converting Decimal to Binary II
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Another method for converting a decimal number to a binary
one involves finding those powers of two which, when added
together, produce the decimal result. You should work from the
largest power of two that fits in the number down to two to
power 0.
Convert the decimal number 2910 to binary:
128 64 32 16 8 4 2 1
27 26 25 24 23 22 21 20
0 0 0 1 1 1 0 1
29 - 16 = 13 – 8 = 5 – 4 = 1 – 1 = 0
Convert the decimal number 11010 to binary:
128 64 32 16 8 4 2 1
27 26 25 24 23 22 21 20
0 1 1 0 1 1 1 0
110 – 64 = 46 – 32 = 14 – 8 = 6 – 4 = 2 – 2 = 0
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Next lecture we will continue
Computer Representation of Information
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