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

3. 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 -> 0, > 1, > 2, >3
0100 -> 4, > 5, > 6, > 7
1000 -> 8, > 9, > A, > B
1100 -> C, > D, > E, > F

4. Example: to base 16
3249

5. Converting positive numbers: base 16 to base 2
Convert each digit into a group of 4 according to the following translation
0 -> 0000, 1 -> 0001, 2 -> 0010, 3 -> 0011
4 -> 0100, 5 -> 0101, 6 -> 0110, 7 -> 0111
8 -> 1000, 9 -> 1001, A -> 1010, B -> 1011
C -> 1100, D -> 1101, E -> 1110, F -> 1111

6. Example: 8EF16 to base 2
8 = 1000
E = 1110
F = 1111

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. Example: CAB16 to base 10 C x 162 + A x 161 + B x 160 =
12 x x x 1 = =
3243

9. 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. Example: to base 10
1x x x29 + 0x28 + 1x27 + 0x26 + 1x25 + 0x24 + 1x23 + 0x22 + 1x21 + 1x20 =
1x x x27 + 1x25 + 1x23 + 1x21 + 1x20 = = =
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. Example: 327 to base 2 327 / 2 = 163 r 1 answer 1

13. 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->A, 11->B, 12->C, 13->D, 14->E, 15->F
If the result is 0 then stop
Use the result as the next number and go to step 1

14. Example: 327 to base 16 327 / 16 = 20 r 7 answer 716

15. Two’s Complement
Convert a positive base 10 number to binary two’s complement
Convert a negative base 10 number to binary two’s complement
Convert a positive base 10 number to hexadecimal two’s complement
Convert a negative base 10 number to hexadecimal two’s complement

16. Convert a positive base 10 number to binary two’s complement
Convert the number to base 2 as described on slide 11
Fill in all leading spaces with 0’s

17. Example: 52 to binary two’s complement
Extend to 16-bit field:

18. Convert a negative base 10 number to binary two’s complement
Convert the positive value of number to binary as described on slide 11
Fill in all leading spaces with 0’s
Flip all digits: 0 -> 1, 1 -> 0
Add binary 1 to the number, remembering the following rules:
0+0=0, 0+1=1, 1+0=1, 1+1=10

19. Example: -52 to binary two’s complement
extend to 16 bits
Flip the bits and add 1

20. Convert a positive base 10 number to hexadecimal two’s complement
Convert the number to base 2 as described on slide 11
Fill in all leading spaces with 0’s
Put the digits into groups of 4 starting at the right
Convert each group of 4 according to the following translation
0000 -> 0, > 1, > 2, >3
0100 -> 4, > 5, > 6, > 7
1000 -> 8, > 9, > A, > B
1100 -> C, > D, > E, > F

21. Example: 52 to hexadecimal two’s complement
52/2 = 26 r 1 1
26/2 = 13 r 0 01
13/2 = 6 r 1 101
6/2 = 3 r
3/2 = 1 r
1/2 = 0 r
003516

22. Convert a negative base 10 number to binary two’s complement
Convert the positive value of number to binary as described on slide 11
Fill in all leading spaces with 0’s
Flip all digits: 0 -> 1, 1 -> 0
Add binary 1 to the number, remembering the following rules:
0+0=0, 0+1=1, 1+0=1, 1+1=10
Put the digits into groups of 4 starting at the right
Convert each group of 4 according to the following translation
0000 -> 0, > 1, > 2, >3
0100 -> 4, > 5, > 6, > 7
1000 -> 8, > 9, > A, > B
1100 -> C, > D, > E, > F

23. Example: -52 to binary two’s complement
FFCB16

24. Two’s 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

27. Binary subtraction Convert the second number into it’s negative by
Flipping each bit (0->1, 1->0)
Add binary 1 to the number, remembering the following rules:
0+0=0, 0+1=1, 1+0=1, 1+1=10
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
negate >

29. Binary comparison Subtract the two numbers as described in slide 27
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
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 -> 0, > 1, > 2, >3
0100 -> 4, > 5, > 6, > 7
1000 -> 8, > 9, > A, > B
1100 -> C, > D, > E, > F

32. Example: BA16 + AB16
BA16 AB16
016516

33. Alternative algorithm: hexadecimal addition table1 2 3 4 5 6 7 8 9 A B C D E F 10 11 12 13 14 15 16 17 18 19 1A 1B 1C 1D 1E

34. Example: BA16 + AB16
(1)
BA16
+AB16
516
16516

35. Hexadecimal subtraction
Convert both numbers into binary as described in slide 5
Convert the second number into it’s negative by
Flipping each bit (0->1, 1->0)
Add binary 1 to the number, remembering the following rules:
0+0=0, 0+1=1, 1+0=1, 1+1=10
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
negate >
000F16

37. Alternative algorithm: hexadecimal subtraction table borrow not needed borrow needed1 2 3 4 5 6 7 8 9 A B C D E F

38. Example: BA16 - AB16 A
B 1A16
- A B16
F16