Title: Homework Hints
1Homework Hints
2Converting 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
3Converting 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
4Example0110010010010012 to base 16
- 011 0010 0100 1001
- 3 2 4 9
- 3249
5Converting 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
6Example8EF16 to base 2
- 8 1000
- E 1110
- F 1111
- 100011101111
7Converting 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
8ExampleCAB16 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
9Converting 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
10Example1100101010112 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
11Converting 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
12Example327 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
13Converting 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
14Example327 to base 16
- 327 / 16 20 r 7 answer 716
- 20 / 16 1 r 4 answer 4716
- 1 / 16 0 r 1 answer 14716
15Twos 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
16Convert 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
17Example 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
18Convert 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
19Example -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
20Convert 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
21Example 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
22Convert 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
23Example -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
24Twos Complement Operations
- Binary addition
- Binary subtraction
- Binary comparison
- Hexadecimal addition
- Hexadecimal subtraction
- Hexadecimal comparison
25Binary 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
26Example
- 00000000101110102
- 00000000101010112
- 00000001011001012
27Binary 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
28Example
- 00000000101110102
- - 00000000101010112
- negate 00000000101010112 -gt
- 1111111101010100
- 1
- 1111111101010101
- 00000000101110102
- 11111111010101012
- 00000000000011112
29Binary 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
30Example
- 00000000101110102
- - 00000000101010112
- 00000000000011112
- First number is greater than the second
31Hexadecimal 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
32Example BA16 AB16
- BA16 AB16
- 1011 1010 1010 1011
- 0000000010111010 0000000010101011
- 0000000010111010
- 0000000010101011
- 0000000101100101
- 0000 0001 0110 0101
- 016516
33Alternative 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
34Example BA16 AB16
- (1)
- BA16
- AB16
- ------------
- 516
- BA16
- AB16
- ------------
- 16516
35Hexadecimal 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
36Example 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
37Alternative 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
38Example BA16 - AB16
- A
- B 1A16
- - A B16
- ------------
- F16