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Title: 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

4
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

5
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

6
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
7
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
8
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
9
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)
10
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)

11
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

12
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

13
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
14
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

15
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

16
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
17
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
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