Homework Hints

1
Homework Hints

• Algorithms

2
Converting Positive Numbers

• Base 2 to base 16
• Base 16 to base 2
• Base 16 to base 10
• Base 2 to base 10
• Base 10 to base 2
• Base 10 to base 16

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Converting positive numbers base 2 to base 16

• Put the digits into groups of 4 starting at the
  right
• If the last group has last than 4 digits, extend
  it with leading 0s
• Convert each group of 4 according to the
  following translation
• 0000 -gt 0, 0001 -gt 1, 0010 -gt 2, 0011 -gt3
• 0100 -gt 4, 0101 -gt 5, 0110 -gt 6, 0111 -gt 7
• 1000 -gt 8, 1001 -gt 9, 1010 -gt A, 1011 -gt B
• 1100 -gt C, 1101 -gt D, 1110 -gt E, 1111 -gt F

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Example0110010010010012 to base 16

• 011 0010 0100 1001
• 3 2 4 9
• 3249

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Converting positive numbers base 16 to base 2

• Convert each digit into a group of 4 according to
  the following translation
• 0 -gt 0000, 1 -gt 0001, 2 -gt 0010, 3 -gt 0011
• 4 -gt 0100, 5 -gt 0101, 6 -gt 0110, 7 -gt 0111
• 8 -gt 1000, 9 -gt 1001, A -gt 1010, B -gt 1011
• C -gt 1100, D -gt 1101, E -gt 1110, F -gt 1111

6
Example8EF16 to base 2

• 8 1000
• E 1110
• F 1111
• 100011101111

7
Converting positive numbers base 16 to base 10

• Expand the number into its component values
  face-value x position value
• Position value
• rightmost digit, position value 160
• Moving right to left, the exponent in each
  position increases by 1 (161, 162, 163, )
• Face value
• 0 - 9 are their face values
• A - F are 10 15 respectively
• Convert each exponent to its decimal equivalent
• Multiply each face-value pair together
• Add the values together

8
ExampleCAB16 to base 10

• C x 162 A x 161 B x 160
• 12 x 256 10 x 16 11 x 1
• 3072 160 11
• 3243

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Converting positive numbers base 2 to base 10

• Expand the number into its component values
  face-value x position value
• Position value
• rightmost digit, position value 20
• Moving right to left, the exponent in each
  position increases by 1 (21, 22, 23, )
• Face value
• 0 0, 1 1
• Eliminate all the 0 terms
• Simplify each 1x2n to 2n
• Convert each exponent to its decimal equivalent
• Add the values together

10
Example1100101010112 to base 10

• 1x211 1x210 0x29 0x28 1x27 0x26 1x25
  0x24 1x23 0x22 1x21 1x20
• 1x211 1x210 1x27 1x25 1x23 1x21 1x20
  
• 211 210 27 25 23 21 20
• 2048 1024 128 32 8 2 1
• 3243

11
Converting positive numbers base 10 to base 2

• Divide the number by 2
• Use the remainder as the next number (beginning
  at the right)
• If the result is 0 then stop
• Use the result as the next number and go to step
  1

12
Example327 to base 2

• 327 / 2 163 r 1 answer 1
• 163 / 2 81 r 1 answer 11
• 81 /2 40 r 1 answer 111
• 40 / 2 20 r 0 answer 0111
• 20 / 2 10 r 0 answer 00111
• 10 / 2 5 r 0 answer 000111
• 5 / 2 2 r 1 answer 1000111
• 2 / 2 1 r 0 answer 01000111
• 1 / 2 0 r 1 answer 101000111

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Converting positive numbers base 10 to base 16

• Divide the number by 16
• Use the remainder as the next number (beginning
  at the right)
• If the remainder is less than 10, use the number
  directly, else convert it to a letter using
  10-gtA, 11-gtB, 12-gtC, 13-gtD, 14-gtE, 15-gtF
• If the result is 0 then stop
• Use the result as the next number and go to step
  1

14
Example327 to base 16

• 327 / 16 20 r 7 answer 716
• 20 / 16 1 r 4 answer 4716
• 1 / 16 0 r 1 answer 14716

15
Twos Complement

• Convert a positive base 10 number to binary twos
  complement
• Convert a negative base 10 number to binary twos
  complement
• Convert a positive base 10 number to hexadecimal
  twos complement
• Convert a negative base 10 number to hexadecimal
  twos complement

16
Convert a positive base 10 number to binary twos
complement

• Convert the number to base 2 as described on
  slide 11
• Fill in all leading spaces with 0s

17
Example 52 to binary twos complement

• 52/2 26 r 1 1
• 26/2 13 r 0 01
• 13/2 6 r 1 101
• 6/2 3 r 0 0101
• 3/2 1 r 1 10101
• 1/2 0 r 1 110101
• Extend to 16-bit field
• 0000000000110101

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Convert a negative base 10 number to binary twos
complement

• Convert the positive value of number to binary as
  described on slide 11
• Fill in all leading spaces with 0s
• Flip all digits 0 -gt 1, 1 -gt 0
• Add binary 1 to the number, remembering the
  following rules
• 000, 011, 101, 1110

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Example -52 to binary twos complement

• 52/2 26 r 1 1
• 26/2 13 r 0 01
• 13/2 6 r 1 101
• 6/2 3 r 0 0101
• 3/2 1 r 1 10101
• 1/2 0 r 1 110101
• extend to 16 bits
• 0000000000110101
• Flip the bits and add 1
• 1111111111001010
• 1
• 1111111111001011

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Convert a positive base 10 number to hexadecimal
twos complement

• Convert the number to base 2 as described on
  slide 11
• Fill in all leading spaces with 0s
• Put the digits into groups of 4 starting at the
  right
• Convert each group of 4 according to the
  following translation
• 0000 -gt 0, 0001 -gt 1, 0010 -gt 2, 0011 -gt3
• 0100 -gt 4, 0101 -gt 5, 0110 -gt 6, 0111 -gt 7
• 1000 -gt 8, 1001 -gt 9, 1010 -gt A, 1011 -gt B
• 1100 -gt C, 1101 -gt D, 1110 -gt E, 1111 -gt F

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Example 52 to hexadecimal twos complement

• 52/2 26 r 1 1
• 26/2 13 r 0 01
• 13/2 6 r 1 101
• 6/2 3 r 0 0101
• 3/2 1 r 1 10101
• 1/2 0 r 1 110101
• 0000000000110101
• 0000 0000 0011 0101
• 003516

22
Convert a negative base 10 number to binary twos
complement

• Convert the positive value of number to binary as
  described on slide 11
• Fill in all leading spaces with 0s
• Flip all digits 0 -gt 1, 1 -gt 0
• Add binary 1 to the number, remembering the
  following rules
• 000, 011, 101, 1110
• Put the digits into groups of 4 starting at the
  right
• Convert each group of 4 according to the
  following translation
• 0000 -gt 0, 0001 -gt 1, 0010 -gt 2, 0011 -gt3
• 0100 -gt 4, 0101 -gt 5, 0110 -gt 6, 0111 -gt 7
• 1000 -gt 8, 1001 -gt 9, 1010 -gt A, 1011 -gt B
• 1100 -gt C, 1101 -gt D, 1110 -gt E, 1111 -gt F

23
Example -52 to binary twos complement

• 52/2 26 r 1 1
• 26/2 13 r 0 01
• 13/2 6 r 1 101
• 6/2 3 r 0 0101
• 3/2 1 r 1 10101
• 1/2 0 r 1 110101
• 0000000000110101
• 1111111111001010
• 1
• 1111111111001011
• 1111 1111 1100 1011
• FFCB16

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Twos Complement Operations

• Binary addition
• Binary subtraction
• Binary comparison
• Hexadecimal addition
• Hexadecimal subtraction
• Hexadecimal comparison

25
Binary addition

• Add the two numbers
• If the carry into the leftmost digit is not the
  same as the carry out, define it as an overflow

26
Example

• 00000000101110102
• 00000000101010112
• 00000001011001012

27
Binary subtraction

• Convert the second number into its negative by
• Flipping each bit (0-gt1, 1-gt0)
• Add binary 1 to the number, remembering the
  following rules
• 000, 011, 101, 1110
• Add the two numbers
• If the carry into the leftmost digit is not the
  same as the carry out, define it as an overflow
  and stop

28
Example

• 00000000101110102
• - 00000000101010112
• negate 00000000101010112 -gt
• 1111111101010100
• 1
• 1111111101010101
• 00000000101110102
• 11111111010101012
• 00000000000011112

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Binary comparison

• Subtract the two numbers as described in slide 27
• Comparison
• If result is positive (first bit 0), first number
  is greater
• If result is negative (first bit 0 and not all
  others are 0), first number is smaller
• If result is zero, numbers are equal

30
Example

• 00000000101110102
• - 00000000101010112
• 00000000000011112
• First number is greater than the second

31
Hexadecimal addition

• Convert both numbers into binary as described in
  slide 5
• Add the two numbers
• If the carry into the leftmost digit is not the
  same as the carry out, define it as an overflow
  and stop
• Put the digits into groups of 4
• Convert each group of 4 according to the
  following translation
• 0000 -gt 0, 0001 -gt 1, 0010 -gt 2, 0011 -gt3
• 0100 -gt 4, 0101 -gt 5, 0110 -gt 6, 0111 -gt 7
• 1000 -gt 8, 1001 -gt 9, 1010 -gt A, 1011 -gt B
• 1100 -gt C, 1101 -gt D, 1110 -gt E, 1111 -gt F

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Example BA16 AB16

• BA16 AB16
• 1011 1010 1010 1011
• 0000000010111010 0000000010101011
• 0000000010111010
• 0000000010101011
• 0000000101100101
• 0000 0001 0110 0101
• 016516

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Alternative algorithmhexadecimal addition table
0 1 2 3 4 5 6 7 8 9 A B C D E F
0 0 1 2 3 4 5 6 7 8 9 A B C D E F
1 1 2 3 4 5 6 7 8 9 A B C D E F 10
2 2 3 4 5 6 7 8 9 A B C D E F 10 11
3 3 4 5 6 7 8 9 A B C D E F 10 11 12
4 4 5 6 7 8 9 A B C D E F 10 11 12 13
5 5 6 7 8 9 A B C D E F 10 11 12 13 14
6 6 7 8 9 A B C D E F 10 11 12 13 14 15
7 7 8 9 A B C D E F 10 11 12 13 14 15 16
8 8 9 A B C D E F 10 11 12 13 14 15 16 17
9 9 A B C D E F 10 11 12 13 14 15 16 17 18
A A B C D E F 10 11 12 13 14 15 16 17 18 19
B B C D E F 10 11 12 13 14 15 16 17 18 19 1A
C C D E F 10 11 12 13 14 15 16 17 18 19 1A 1B
D D E F 10 11 12 13 14 15 16 17 18 19 1A 1B 1C
E E F 10 11 12 13 14 15 16 17 18 19 1A 1B 1C 1D
F F 10 11 12 13 14 15 16 17 18 19 1A 1B 1C 1D 1E
34
Example BA16 AB16

• (1)
• BA16
• AB16
• ------------
• 516
• BA16
• AB16
• ------------
• 16516

35
Hexadecimal subtraction

• Convert both numbers into binary as described in
  slide 5
• Convert the second number into its negative by
• Flipping each bit (0-gt1, 1-gt0)
• Add binary 1 to the number, remembering the
  following rules
• 000, 011, 101, 1110
• Add the two numbers
• If the carry into the leftmost digit is not the
  same as the carry out, define it as an overflow

36
Example BA16 - AB16

• BA16 AB16
• 1011 1010 1010 1011
• 0000000010111010 0000000010101011
• 00000000101110102
• - 00000000101010112
• negate 00000000101010112 -gt
• 1111111101010100
• 1
• 1111111101010101
• 00000000101110102
• 11111111010101012
• 00000000000011112
• 0000 0000 0000 11112
• 000F16

37
Alternative algorithmhexadecimal subtraction
table borrow not needed borrow needed
0 1 2 3 4 5 6 7 8 9 A B C D E F
0 0 1 2 3 4 5 6 7 8 9 A B C D E F
1 F 0 1 2 3 4 5 6 7 8 9 A B C D E
2 E F 0 1 2 3 4 5 6 7 8 9 A B C D
3 D E F 0 1 2 3 4 5 6 7 8 9 A B C
4 C D E F 0 1 2 3 4 5 6 7 8 9 A B
5 B C D E F 0 1 2 3 4 5 6 7 8 9 A
6 A B C D E F 0 1 2 3 4 5 6 7 8 9
7 9 A B C D E F 0 1 2 3 4 5 6 7 8
8 8 9 A B C D E F 0 1 2 3 4 5 6 7
9 7 8 9 A B C D E F 0 1 2 3 4 5 6
A 6 7 8 9 A B C D E F 0 1 2 3 4 5
B 5 6 7 8 9 A B C D E F 0 1 2 3 4
C 4 5 6 7 8 9 A B C D E F 0 1 2 3
D 3 4 5 6 7 8 9 A B C D E F 0 1 2
E 2 3 4 5 6 7 8 9 A B C D E F 0 1
F 1 2 3 4 5 6 7 8 9 A B C D E F 0
38
Example BA16 - AB16

• A
• B 1A16
• - A B16
• ------------
• F16