CPS120:%20Introduction%20to%20Computer%20Science

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CPS120 Introduction to Computer Science

• Computer Math Converting to Decimal

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Binary
Decimal is base 10 and has 10 digits
0,1,2,3,4,5,6,7,8,9 Binary is base 2 and has 2
digits 0,1
For a number to exist in a given number system,
the number system must include those digits. For
example The number 284 only exists in base 9 and
higher.
9
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Codes

• Given any positive integer base (RADIX) N, there
  are N different individual symbols that can be
  used to write numbers in the system. The value of
  these symbols range from 0 to N-1
• All systems we use in computing are positional
  systems
• 495 400 90 5

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Number Systems

• We use the DECIMAL (10 system
• Computers use BINARY (2 or some shorthand for it
  like OCTAL (8 or HEXADECIMAL (16

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Power of 2 Number System

Binary Octal Decimal
000 0 0
001 1 1
010 2 2
011 3 3
100 4 4
101 5 5
110 6 6
111 7 7
100 10 8
1001 11 9
1010 12 10
16
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Bases Higher than 10
How are digits in bases higher than 10
represented?
Base 16 0,1,2,3,4,5,6,7,8,9,A,B,C,D,E, and F
10
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Conversions
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Decimal Equivalents

• Assuming the bits are unsigned, the decimal value
  represented by the bits of a byte can be
  calculated as follows
• Number the bits beginning on the right using
  superscripts beginning with 0 and increasing as
  you move left
• Note 20, by definition is 1
• Use each superscript as an exponent of a power of
  2
• Multiply the value of each bit by its
  corresponding power of 2
• Add the products obtained

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Converting Octal to Decimal
What is the decimal equivalent of the octal
number 642?
6 x 8² 6 x 64 384 4 x 8¹
4 x 8 32 2 x 8º 2 x 1
2 418 in base 10
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Converting Hexadecimal to Decimal
What is the decimal equivalent of the hexadecimal
number DEF?
D x 16² 13 x 256 3328 E x 16¹
14 x 16 224 F x 16º 15 x 1
15 3567 in base 10
Remember, base 16 is 0,1,2,3,4,5,6,7,8,9,A,B,C,D,E
,F
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Converting Binary to Decimal
What is the decimal equivalent of the binary
number 010110?
1 x 26 1 x 64 64 1 x 25 1
x 32 32 0 x 24 0 x 16 0 1
x 23 1 x 8 8 1 x 22 1 x 4
4 1 x 21 1 x 2 2 0 x 2º
0 x 1 0 112 in base 10
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Horners Method

• Another procedure to calculate the decimal
  equivalent of a binary number
• Note This method works with any base
• Horners Method
• Step 1 Start with the first digit on the left
• Step 2 Multiply it by the base
• Step 3 Add the next digit
• Step 4 Multiply the sum by the base
• Step 5 Continue the process until you add the
  last digit