Chapter 1 1 Number Systems

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Chapter 1 1Number Systems
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Objectives

• Understand why computers use binary (Base-2)
  numbering.
• Understand how to convert Base-2 numbers to
  Base-10 or Base-8.
• Understand how to convert Base-8 numbers to
  Base-10 or Base 2.
• Understand how to convert Base-16 numbers to
  Base-10, 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
OFF
OFF
ON/OFF
ON
ON
ON
Exponent
21
22
23
24
20
0
0
16
2
1

Calculation




(19)10
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• Number systems include decimal, binary, octal
  and hexadecimal
• Each system have four number base

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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
Right of decimal point
left of the 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
• 7 102 2 101 5 100 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
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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
• 127 126 025 024 123 022
  121 020
• 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.

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• Example convert the number 333 to binary.

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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
Ones
Placeholder Name
Eights
Sixty-Fours
642
84
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Value
Exponential Expression
822
814
801
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Transform (44978)10 to Octal

• .