1. Tutorial: ITI1100 Dewan Tanvir Ahmed SITE, UofO

2. Binary Numbers Base (or radix) Number base conversion Complements2
example: 0110
Number base conversion
example: 41 =
Complements
1's complements ( 2n- 1 ) - N
2's complements 2n - N
Subtraction = addition with the 2's complement
Signed binary numbers
signed-magnitude,
signed 1's complement,
signed 2's complement

3. Binary Number System Base = 2 2 Digits: 0, 1 Examples:
= 8 + 1
= 9 
b = 1 * * * * =
= 173
Note: it is common to put binary digits in groups of 4 to make it easier to read them.

4. Ranges for Data Formats
No. of bits
Binary
BCD 1
0 – 1 2
0 – 3 3
0 – 7 4
0 – 15
0 – 9 7
0 – 127 8
0 – 255
0 – 99 16
0 - 65,535
0 – 9999 24
0 – 16,777,215
0 –

5. In General (binary)
No. of bits
Binary
Min
Max n
2n – 1

6. Signed Integers “unsigned integers” = positive values only
Must also have a mechanism to represent “signed integers” (positive and negative values!)
-1010 = ?2
Two common schemes:
sign-magnitude and
twos complement

7. Sign-Magnitude Extra bit on left to represent sign
0 = positive value
1 = negative value
6-bit sign-magnitude representation of +5 and –5:
+5:
+ve 5
-5:
-ve 5

8. Ranges (revisited) No. of bits Binary Unsigned Sign-magnitude Min Max1 2 3 -1 7 -3 4 15 -7 5 31
-15 6 63
-31

9. In General … No. of bits Binary Unsigned Sign-magnitude Min Max n
2n - 1
-(2n-1 - 1)
2n-1 - 1

10. Difficulties with Sign-Magnitude
Two representations of zero
Using 6-bit sign-magnitude… 0: 0:
Arithmetic is awkward!

11. Complementary Representations
1’s complement
2’s complement
9’s complement
10’s complement

12. Complementary Notations
What is the 3-digit 10’s complement of 207?
Answer:
What is the 4-digit 10’s complement of 15?
111 is a 10’s complement representation of what decimal value?

13. Exercises – Complementary Notations
What is the 3-digit 10’s complement of 207?
Answer:
What is the 4-digit 10’s complement of 15?
Answer:
111 is a 10’s complement representation of what decimal value?
Answer:

14. 2’s Complement Most common scheme of representing negative numbers
natural arithmetic - no special rules!
Rule to represent a negative number in 2’s C
Decide upon the number of bits (n)
Find the binary representation of the +ve value in n-bits
Flip all the bits
Add 1

15. 2’s Complement Example
Represent -5 in binary using 2’s complement notation
Decide on the number of bits
Find the binary representation of the +ve value in 6 bits
Flip all the bits
Add 1
6 (for example)
000101 +5
111010
111010
111011 -5

16. Sign Bit
In 2’s complement notation, the MSB is the sign bit (as with sign-magnitude notation)
0 = positive value
1 = negative value
+5:
+ve 5
-5:
-ve
2’s complement

17. “Complementary” Notation
Conversions between positive and negative numbers are easy
For binary (base 2)…

18. Example +5
2’s C -5
2’s C
+ve
-ve
2’s C

19. Range for 2’s Complement
For example, 6-bit 2’s complement notation
100000
100001
111111
000000
000001
011111
Negative, sign bit = 1
Zero or positive, sign bit = 0

20. Ranges No. of bits Binary Unsigned Sign-magnitude 2’s complement Min
Max 1 2 3 -1 -2 7 -3 -4 4 15 -7 -8 5 31
-15
-16 6 63
-31
-32

21. In General (revisited)
No. of bits
Binary
Unsigned
Sign-magnitude
2’s complement
Min
Max n
2n - 1
-(2n-1 - 1)
2n-1-1
-2n-1
2n-1 - 1

22. What is -6 plus +6? Zero, but let’s see
-6: :
Sign-magnitude
-6: :
2’s complement

23. 2’s Complement Subtraction
Easy, no special rules
Subtract??
Actually … addition!
A – B = A + (-B)
add
2’s complement of B

24. What is 10 subtract 3? 7, but… Let’s do it (we’ll use 6-bit values)
10 – 3 = 10 + (-3) = 7
+3:
-3:

25. What is 10 subtract -3? 13, but… Let’s do it (we’ll use 6-bit values)
10 – (-3) = 10 + (-(-3)) = 13
-3:
+3:

26. M - N M + 2’s complement of N If M  N If M < N
M + (2n - N) = M - N + 2n
If M  N
Produce an carry, which is discarded
If M < N
results in 2n - (N - M), which is the 2’s complement of (N-M)

27. Overflow Carry out of the leading digit
If we add two positive numbers and we get a carry into the sign bit we have a problem
If we add two negative numbers and we get a carry into the sign bit we have a problem
If we add a positive and a negative number we won't have a problem
Assume 4 bit numbers (+7 : -8)

28. N = 4 Number Represented Binary 0000 0001 0010 0011 0100 0101 0110
0111
1000
1001
1010
1011
1100
1101
1110
1111
Unsigned 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
Signed
Mag 1 2 3 4 5 6 7 -0 -1 -2 -3 -4 -5 -6 -7
1's
Comp 1 2 3 4 5 6 7 -7 -6 -5 -4 -3 -2 -1 -0
2's
Comp 1 2 3 4 5 6 7 -8 -7 -6 -5 -4 -3 -2 -1

29. Overflow
If we add two positive numbers and we get a carry into the sign bit we have a problem

30. Overflow
If we add two negative numbers and we get a carry into the sign bit we have a problem

31. Overflow
If we add a positive and a negative number we won't have a problem

32. Overflow
If we add two positive numbers and we get a carry into the sign bit we have a problem
carry in 0
carry out 0
carry in 1
carry out 0

33. Overflow
If we add two negative numbers and we get a carry into the sign bit we have a problem
carry in 1
carry out 1
carry in 0
carry out 1

34. Overflow
If we add a positive and a negative number we won't have a problem
carry in 1
carry out 1
carry in 1
carry out 1

35. Binary Codes n-bit binary code BCD – Binary Coded Decimal (4-bits)
2n distinct combinations
BCD – Binary Coded Decimal (4-bits)
… …
BCD addition
Get the binary sum
If the sum > 9, add 6 to the sum
Obtain the correct BCD digit sum and a carry

36. Binary Codes ASCII code Error-detection code
American Standard Code for Information Interchange
alphanumeric characters, printable characters (symbol), control characters
Error-detection code
one parity bit - an even numbered error is undetected
“A” 41:100| >
0100|0001 (even),
1100|0001 (odd)

37. Binary Logic Boolean algebra Binary variables: X, Y Logical operations
two discrete values (true or false)
Logical operations
AND, OR, NOT
Truth tables

38. Logic Gates Logic circuits Computations and controls Logic Gates
circuits = logical manipulation paths
Computations and controls
combinations of logic circuits
Logic Gates

39. Timing diagram

40. Thank You!