For Monday, Aug. 23

1
For Monday, Aug. 23

• Review Homework Problems
• A structured way to convert fractions
• Base 16 / 8 / 2hexadecimal, octal, binary
• Addition, Subtraction
• 2s Complement

2
Homework for Monday, August 23

• 1) (15 points)
• What decimal number is represented by the
  following numbers?A) 10110.1012 ?10
• B) 100110. 1002 ?10
• C) 341.15 ?10
• 2) (15 points)
• What is the base-n representation of the
  following decimal numbers?
• A) 54.7510 ?2
• B) 123.62510 ?2
• C) 123 ?5

3
1) (15 points) What decimal number ?

• A) 10110.1012 ?10
• 01 12 14 08 116 1(1/2) 0(1/4)
  1(1/8) 22.625
• B) 100111. 1002 ?10
• 11 12 14 08 016 132 1(1/2)
  39.5
• C) 341.15 ?10
• 325 45 11 1(1/5) 96.210

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2) (15 points)What is the base-n representation?

• Heuristically
• A) 54.7510 ?2
• 54 32 16 4 2 0.75 1/2 1/4 110110.112
• B) 123.62510 ?2
• 123 64 32 16 8 2 1 0.625 1/2 1/8
• 1111011.1012
• C) 123 ?5
• 123 425 45 3 4435

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2) (15 points)What is the base-n representation?

• A) 54.7510 ?2 B) 123.62510 ?2 C)123 ?5
• 54/2 27 0 123/261 1 123/524 3
• 27/2 13 1 61/2 30 1 24/5 4 4
• 13/2 6 1 30/2 15 0 4/5 0 4
• 6/2 3 0 15/2 7 1 12310 4435
• 3/2 1 1 7/2 3 1
• 1/2 0 1 3/2 1 1
• 5410 1101102 12310 11110112

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Check your work

• By converting back
• By using another method

1111011.1012 11 12 04 18 116
132 164 1(1/2) 0(1/4) 1(1/8)
123.62510
54 272 027 132 113 62 1 6
32 0 3 12 1 1 02 1 5410
1101102
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A Structured Way to ConvertDecimal to Binary

• Represent a decimal fraction as binary
  digits0.y10 0.abcde2 (a,b,c,d,e,f) ?(0,1)
• 0.y10 a2-1 b2-2 c2-3 d2-4 e2-5
• Multiply by two

0.y2 a b2-1 c2-2 d2-3 e2-4
an integer a fraction
The first integer after multiplying by 2 is the
most significant bit of the binary representation
of a fraction
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Continued multiplication

• 0.y2 a b2-1 c2-2 d2-3 e2-4
  an integer a fraction (call it z)

• z2 b c2-1 d2-2 e2-3

The second integer after multiplying by 2 is the
coefficient of 2-2 in the binary representation
? Continue multiplying and saving integers
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An Example

• What is the binary representation of 0.0625?
• 0.0625 2 0 .125
• 0.125 2 0 .250
• 0.250 2 0 .500
• 0.500 2 1 .000

Read integers from first to last .062510 .00012
(1/16)
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The result is sometimes a continued fraction

• What is the binary representation of 0.6510?
• 0.65 2 1 .30
• 0.30 2 0 .60
• 0.60 2 1 .20
• 0.20 2 0 .40
• 0.40 2 0 .80
• 0.80 2 1 .60
• 0.60 2 STOP! This would continue forever.
• 0.65 0.101001 (the overbar indicates the
  digits to be repeated)

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Octal

• Octal digits relate to powers of 8367.28 3
  82 6 81 7 80 2 8-1 3 64 6
  8 7 1 2 1/8 192 48 7 2/8
  247.25
• To convert from decimal to octal
• Hueristic combine powers of 8
• Structured
• Integer Repeatedly divide by 8 and record the
  remainders. The last remainder is the most
  significant digit
• Fraction Repeatedly multiply by 8 and record the
  integers. The first integer is the most
  significant digit

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Hexadecimal

• Hexadecimal digits relate to powers of 16367.216
  3 162 6 161 7 160 2 16-1 3
  256 6 16 7 1 2 1/16 768 96
  7 2/16 871.125
• To convert from decimal to Hexadecimal
• Hueristic combine powers of 16
• Structured
• Integer Repeatedly divide by 16 and record the
  remainders. The last remainder is the most
  significant digit
• Fraction Repeatedly multiply by 16 and record the
  integers. The first integer is the most
  significant digit

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Counting

• Decimal Octal Hexadecimal
• 0 0 0
• 1 1 1
• 2 2 2
• 3 3 3
• 4 4 4
• 5 5 5
• 6 6 6
• 7 7 7
• 8 10 8
• 9 11 9
• 10 12 ?

Decimal Octal Hexadecimal 10 12 A 11 13 B 12 14 C
13 15 D 14 16 E 15 17 F 16 20 10 17 21 11 25 31
19 26 32 1A
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Binary ? Octal Binary ? Hexadecimal

• Each octal digit exactly 3 binary digits
  (bits)Each hexadecimal digit exactly 4 bits
• Start from the decimal point and work
  outward123.62510 1111011.1012 1 111 011 .
  101 173.58123.62510 1111011.1012
  111 1011 . 1010 7B.A16
• These can be shortcuts to converting decimal ?
  binary1111011.1012 173.58 164 78 3
  5(1/8) 123.625
  123.62510 7B.A16 716 11 10(1/16)
  123.625
• To convert from octal ? hexadecimal, use binary

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Addition

• x y CarryIn Sum CarryOut
• 0 0 0 0 0
• 0 0 1 1 0
• 0 1 0 1 0
• 0 1 1 0 1
• 1 0 0 1 0
• 1 0 1 0 1
• 1 1 0 0 1
• 1 1 1 1 1

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Check your work 45 29 74
Carry
1 0 1 1 0 1 0 1 1 1 0 1
1 0
0 1
1 0
1 1
1 0
1 0
1
Sum
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Homework for Wednesday, Aug. 25

• 1. (15 points) Use the structured method to
  convert the following decimal number to
• a) binary, b) hexadecimal, c) octal
• 114.125
• 2. (10 points) Check your work for problem 1 by
  converting the binary equivalent you found in
  part a) to hexadecimal and to octal. (Note use
  groups of 3 and 4)
• 3. Perform the following addition in base 2.
  Check your work by converting each addend and the
  sum to decimal.
• 101011
• 111011