1. Data Representation – Chapter 3
Sections 3-2, 3-3, 3-4

2. Programming Assignment
Questions/Requests/Comments/Concerns?

3. Number Range What is meant by “range”?
How do you compute the range of a given set of digits?

4. More Representations
Integer values are represented by a power series regardless of radix (base)
1410 = 1 x x 100
11102 = 1 x x x x 20
What about fractions?

5. Fraction Representations
Same principle applies
= 1 x x x x 10-2
“decimal point”
= 1 x x x x 2-2
“binary point”

6. Fraction Representations
To convert a decimal fraction to binary
Multiple decimal fractional part by 2
Output the integer part
Repeat until you’ve reached the desired accuracy =
.3125
x 2
0.6250
.6250
x 2
1.2500
.2500
x 2
0.5000
.5000
x 2
1.0000

7. Fraction Representations
More on binary floating point representations later

8. Addition
Decimal number addition
carry
121
+19 40

9. Addition Binary number addition 1 111 carries 10101 +10011 101000
It works the same way in binary as it does in decimal

10. Subtraction
Decimal number subtraction
borrow 21
+19 02

11. Subtraction Binary number subtraction 01 borrow 10101 -10011 00010
It works the same way in binary as it does in decimal

12. Subtraction Questions
What about when a subtraction results in a negative value?
How do you represent a negative value in binary?
How do you represent a negative value in decimal?

13. Negative Numbers Decimal - 27
Place a “-” in front of the decimal digits

14. Negative Numbers Binary -11011
Place a “-” in front of the binary digits
This is called signed-magnitude representation
In the hardware the sign is a designated bit where 0 means “+” and 1 mean “–”

15. Negative Numbers
Signed-magnitude is convenient for humans doing symbolic manipulations
It’s not convenient for computer computations
Why not?
Consider subtraction (which is really addition of a negative number)

16. Signed-Magnitude Subtraction
21 Minuend
-17 Subtrahend
4 Difference
17 Minuend
-21 Subtrahend
- 4 Difference

17. Signed-Magnitude Subtraction
if (M >= S)
compute M-S
else
compute S-M
place “-” on difference
To perform subtraction we must first perform a comparison!
Therein lies the inconvenience

18. Negative Numbers Revisited
Complement representation
“(r-1)’s” complement
Given Nr , an n-digit number of radix r
“(r-1)’s” complement is (rn – 1) - Nr

19. “(r-1)’s” complement Decimal 1234510 n=5, r=10, N=12345 (105-1)-12345
( )-12345
87654
“9’s” complement

20. “(r-1)’s” complement Binary 101102 n=5, r=2, N=10110 (25-1)-10110
( )-10110
01001
“1’s” complement
Note that the result is just an inversion of the bits of the original number (1/0 and 0/1)

21. “(r-1)’s” complement So, how does this help us do subtraction?
It doesn’t really!
Furthermore the representation of 0 is ambiguous
0000 “+0”
1111 “-0”
Try it!

22. Another Complement “r’s” complement
Given Nr , an n-digit number of radix r
“r’s” complement is “(r-1)’s” complement + 1
(rn – 1) – Nr+ 1 rn – Nr if (Nr != 0)
if (Nr == 0)

23. “r’s” complement
Let’s concentrate on the 2’s complement (binary number system)
“…leaving all least significant 0’s and the first 1 unchanged, and then replacing 1’s by 0’s and 0’s by 1’s in all other higher significant bits.”
Yeah, right!
Take the “1’s” complement and add 1

24. “2’s” Complement 101102 n=5, r=2, N=10110 (25-1)-10110
( )-10110
’s complement
’s complement

25. 2’s Complement “Subtraction”
Original Problem
2’s Complement Problem
0111
-0101
0111
+1011
10010
Carry Out
Answer
2’s Complement

26. 2’s Complement “Subtraction”
Original Problem
2’s Complement Problem
0101
-0111
0101
+1001
01110
Carry Out
Answer
2’s Complement

27. 2’s Complement “Subtraction”
2’s Complement Problem
2’s Complement Problem
0111
+1011
10010
0101
+1001
01110
Carry Out
Answer
2’s Complement
Carry Out
Answer
2’s Complement
How do we know the sign of the result?

28. 2’s Complement “Subtraction”
Recall that we’re doing “fixed bit-length” math
When the numbers are “signed” then the most significant bit (MSB) represents the sign
MSB=1 defined as negative
MSB=0 defined as positive

29. Sign of the Result?
2’s Complement Problem
2’s Complement Problem
0111
+1011
10010
0101
+1001
01110
Carry Out
Answer
2’s Complement
Carry Out
Answer
2’s Complement
Sign Bit
Sign Bit

30. Sign of the Result? The result is in 2’s complement form
If the sign bit is 1, take the 2’s complement of the result to read off the magnitude in “normal” binary notation
If the sign bit is 0, the result is in “normal” binary notation

31. Overflow What if the result is too big?
Remember, this is fixed bit-length math
When you add to n-bit numbers the result may be up to n+1 bits long

32. Overflow
When a signed positive number is added to a signed (2’s complement) negative number an overflow cannot occur
The positive number magnitude will only get smaller
When two signed negative numbers or two positive numbers are added an overflow may occur
Magnitude is going to get bigger
When two signed positive numbers are added, the sign bit is treated as part of the magnitude of the number and the end carry (overflow) is not treated as “overflow”

33. Overflow
So, when is that “extra bit” an overflow and when is it part of the result?
Check the sign bit and the overflow bit
If the two are not equal then an overflow has occurred
If the two are equal then there is no overflow

34. Overflow In an overflow condition
If the numbers are both positive, then the sign bit becomes part of the magnitude of the result
If the numbers are both negative, then the overflow bit is treated as the sign
As we’ll see next time (logic gates) there is a very simple way to detect overflow
Questions?

35. “2’s” Complement
Is this cheating on the representation of 0 since we handled it with an conditional statement?
No! Try it based on the formula only!
Don’t really need to specify it as two cases
I only did it because the book did

36. “2’s” Complement So, how does this help us do subtraction?
Addition of a 2’s complement subtrahend.
Take 2’s complement of the subtrahend (negate the subtrahend)
Add minuend to subtrahend
Difference is in 2’s complement notation
Note we’re using a fixed, pre-specified number of bits
If the result overflows, just ignore it for now

37. Other Binary Formats Binary Coded Decimal (BCD) Even parity/Odd parity
Convert each decimal digit to its binary representation and store
Requires 4 bits per digit – why?
Forget about BCD arithmetic
Even parity/Odd parity
For a binary number, count the number of 1’s
If the number is odd, attach an extra “1” for even parity or a “0” for odd parity
If the number is even, attach an extra “0” for even parity or a “1” for odd parity
What good is this?
Gray code
Count through a sequence of binary numbers in such a way that only 1 bit at a time changes
American Standard Code for Information Interchange (ASCII)
Standard set of binary patterns assigned to 256 characters
How many bits comprise the ASCII code?
Unicode
Up-to-date replacement for ASCII
Various schemes utilizing various bit lengths

38. Homework Textbook pages 89 – 91 Begin reading Chapter 1
3-1, 3-3, 3-4, 3-5, 3-10, 3-13, 3-15, 3-16, 3-17
Due beginning of next lecture
Begin reading Chapter 1