Transcript Chapter 1 1 Number Systems

Chapter 1 1
Number Systems
Objectives
Understand why computers use binary (Base-2)
numbering.
Understand how to convert Base-2 numbers to Base10 or Base-8.
Understand how to convert Base-8 numbers to Base10 or Base 2.
Understand how to convert Base-16 numbers to Base10, Base 2 or Base-8.
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Why Binary System?
• Computers are made of a series of
switches
• Each switch has two states: ON or OFF
• Each state can be represented by a number
– 1 for “ON” and 0 for “OFF”
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Converting Base-2 to Base-10
1 1)
ON
2
ON
ON
OFF
0
OFF
(1 0
Exponent:
16 0 0 2 1
(19)10
Calculation:
+
+
4
+
+
=
• Number systems include decimal, binary,
octal and hexadecimal
• Each system have four number base
Number System Base
Symbol
Binary
Base 2
B
Octal
Base 8
O
Decimal
Base 10
D
Hexadecimal
Base 16
H
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1.1 Decimal Number System
• The Decimal Number System uses base 10. It
includes the digits {0, 1,2,…, 9}. The weighted
values for each position are:
Base
10^4
10^3 10^2
10000 1000 100
10^1 10^0 10^-1 10^-2 10^-3
10
left of the decimal point
1
0.1
0.01
0.001
Right of decimal point
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• Each digit appearing to the left of the decimal
point represents a value between zero and nine
times power of ten represented by its position in
the number.
• Digits appearing to the right of the decimal point
represent a value between zero and nine times an
increasing negative power of ten.
• Example: the value 725.194 is represented in
expansion form as follows:
• 7 * 10^2 + 2 * 10^1 + 5 * 10^0 + 1 * 10^-1 + 9 *
10^-2 + 4 * 10^-3
• =7 * 100 + 2 * 10 + 5 * 1 + 1 * 0.1 + 9 * 0.01 + 4 *
0.001
• =700 + 20 + 5 + 0.1 + 0.09 + 0.004
• =725.194
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•
1.2 The Binary Number Base
Systems
Most modern computer system using binary logic. The
computer represents values(0,1) using two voltage levels
(usually 0V for logic 0 and either +3.3 V or +5V for logic
1).
• The Binary Number System uses base 2 includes only the
digits 0 and 1
• The weighted values for each position are :
Base
2^5
2^4
2^3
2^2
2^1
2^0
2^-1 2^-2
32
16
8
4
2
1
0.5
0.25
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1.3 Number Base Conversion
• Binary to Decimal: multiply each digit by its
weighted position, and add each of the weighted
values together or use expansion formdirectly.
• Example the binary value 1100 1010 represents :
• 1*2^7 + 1*2^6 + 0*2^5 + 0*2^4 + 1*2^3 + 0*2^2 +
1*2^1 + 0*2^0 =
• 1 * 128 + 1 * 64 + 0 * 32 + 0 * 16 + 1 * 8 + 0 * 4 + 1 *
2+0*1=
• 128 + 64 + 0 + 0 + 8 + 0 + 2 + 0 =202
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• Decimal to Binary
There are two methods, that may be used to convert
from integer number in decimal form to binaryform:
1-Repeated Division By 2
•
•
•
•
For this method, divide the decimal number by 2,
If the remainder is 0, on the right side write down a 0.
If the remainder is 1, write down a 1.
When performing the division, the remainders which
will represent the binary equivalent of the decimal
number are written beginning at the least significant
digit (right) and each new digit is written to more
significant digit (the left) of the previous digit.
10
• Example: convert the number 333 to binary.
Division
333/2
166/2
83/2
41/2
20/2
Quotient
166
83
41
20
10
Remainder Binary
1
1
0
01
1
101
1
1101
0
01101
10/2
5/2
2/2
1/2
5
2
1
0
0
1
0
1
001101
1001101
01001101
101001101
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Octal System
Computer scientists are often looking for
shortcuts to do things
 One of the ways in which we can represent
binary numbers is to use their octal
equivalents instead
 This is especially helpful when we have to do
fairly complicated tasks using numbers

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• The octal numbering system includes
eight base digits (0-7)
• After 7, the next placeholder to the right
begins with a “1”
• 0, 1, 2, 3, 4, 5, 6, 7, 10, 11, 12, 13 ...
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Octal Placeholders
2
1
“Ones
”
4
“Eights”
“SixtyFours”
Number:
Value:
64*2
8*4
1*1
Exponential
Expression:
82*2
81*4
Placeholder
Name:
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80*1
Transform (44978)10 to Octal
Division
• .
Quotient
44978 / 8
5622
5622 / 8
702
87
10
1
0
702/8
87/8
10/8
1/8
Remainder Binary
2
6
6
7
2
1
2
62
662
7662
27662
127662
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