1. Representing Integer Data
ITEC 1000 “Introduction to Information Technology”
Lecture 3
Representing Integer Data

2. Lecture Template: Number Representation Unsigned Integer
Complementary Representation
9’s Decimal Representation (skip)
1’s Binary Representation (skip)
10’s Decimal Representation
2’s Binary Representation

3. Number Representation
An integer is a number which has no fractional part.
Numbers can be represented as a combination of
Sign (plus or minus)
Value or magnitude

4. Unsigned Integer (Natural Number)
8-bit storage location
28 different values between 0 and 255
16-bit storage location
216 different values between 0 and 65535
multiple storage locations
4 consecutive 1-byte storage locations
provide 32 bits of range
232, or 4,294,967,296 different values
difficult to calculate and manipulate

5. 32-bit multiple storage location

6. Signed-Integer Representation
No obvious direct way to represent the sign in binary notation
Options:
Sign-and-magnitude representation
1’s complement (skip – confusing)
2’s complement (most common)

7. Sign-and-Magnitude Use left-most bit for sign
0 = plus; 1 = minus
Total range of integers the same
Half of integers positive; half negative
Magnitude of largest integer half as large
Example using 8 bits:
Unsigned: = +255
Signed: = = -127
Note: 2 values for 0: +0 ( ) and -0 ( )

8. Calculation Algorithms
Sign-and-magnitude algorithms complex and difficult to implement in hardware
Must test for 2 values of 0
Useful with BCD (see slides at end)
Order of signed number and carry/borrow makes a difference
Example: Decimal addition algorithm
Addition: 2 Positive Numbers
Addition: 1 Signed Number

9. Ranges No. of bits Binary Unsigned Sign-magnitude Min Max 1 2 3 -1 72 3 -1 7 -3 4 15 -7 5 31
-15 6 63
-31
Etc.

10. Ranges: General Rule No. of bits Binary Unsigned Sign-magnitude Min
Max n
2n - 1
-(2n-1 - 1)
2n-1 - 1

11. Complementary Representation
Sign of the number does not have to be handled separately
Consistent for all different signed combinations of input numbers
Two methods
Radix: value used is the base number
Diminished radix: value used is the base number minus 1 (skip – confusing)
9’s complement: base 10 diminished radix
1’s complement: base 2 diminished radix

12. 10’s Decimal Complement
Complement representation: (1) positive number: remains itself; (2) negative number: subtracting its absolute value from a standard basis value
Decimal (base 10) system: radix complement
Radix = as the basis
3-digit example: base value = 1000
Range of possible values 0 to 999 arbitrarily split at 500
Numbers
Negative
Positive
Representation method
Complement
Number itself
Range of decimal numbers
-500 -1 +0
499
Calculation
1000 minus abs(number)
none
Representation example
500
999
– Increasing value +
1000 – 500

13. 10’s Decimal Complement
Necessary to specify number of digits or word size
10’s complement representation in “1000” base
Example: representation of 3-digit number
First digit = 0 through positive number
First digit = 5 through negative number

14. Examples with 3-Digit Numbers
10’s complement representation of 247
247 (positive number)
Example 2:
10’s complement representation of -247
1000 – 247 = 753 (negative number)
Example 3:
Find a value with 10’s complement representation 777
Negative number because first digit is 7
1000 – 777 = 223
Signed value = -223

15. 10’s Decimal Complement
Conversion to sign-and-magnitude value for 9’s complement representation(3 digits)
321 remains +321
521: take the difference: base minus complement (999 – 521) with negative sign: – 479

16. Addition: Counting Upwards
Counting upward on scale corresponds to addition
Example in 10’s complement: does not cross the modulus
+250
Representation
500
650
900
999
170
420
499
Number
represented
-500
-350
-100 -1 +0

17. Addition with Wraparound
Count to the right to add a negative number to the representation (since the complement 700 represents the value of -300)
Wraparound scale used to extend the range for the negative result
+700
Representation
500
999
200
499
900
Number
represented
-500 -1
-100
-300

18. Addition with End-around Carry
Count to the right crosses the modulus
+300
Representation
500
800
999
100
499
Number
represented
-500
-200 -1
800
300
1100
100

19. Overflow Fixed word size has a fixed range size
Overflow: combination of numbers that adds to result outside the range
End-around carry in modular arithmetic avoids problem
Complementary arithmetic: numbers out of range have the opposite sign
Test: If both inputs to an addition have the same sign and the output sign is different, an overflow occurred

20. 2’s Binary Complement Binary system complement
Binary (base 2) system radix complement
Radix = as the basis
Inversion: change 1’s to 0’s and 0’s to 1s (add 1)
Numbers beginning with 0 are positive
Numbers beginning with 1 are negative
Example with 8-bit binary numbers

21. 2’s Complement 2’s complement representation:
Positive value represents itself
Negative value: invert and add “1”
Numbers
Negative
Positive
Representation method
Complement
Number itself
Range of decimal numbers
-12810
-110
+010
12710
Calculation
Inversion
None
Representation example

22. Example: 2’s Complement
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
Invert all the bits:
Add 1
6 (for example)
000101 +5
111010
111010
111011 -5

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

24. Estimating Integer Value of 2’s Complement Representation
Positive numbers begin with 0
Small negative numbers (close to 0) begin with multiple 1’s
= -2 in 8-bit 2’s complements
= -128, largest negative 2’s complements
Invert all 1’s and 0’s, add “1” and approximate the value

25. Exercise: 2’s complement conversions
What is -20 expressed as an 8-bit binary number in 2’s complement representation?
Answer:
is a 7-bit binary number in 2’s complement representation. What is the decimal sign-and-magnitude value?
Skip answer
Answer

26. Exercise: 2’s complement conversions
Answer
What is -20 expressed as an 8-bit binary number in 2’s complement notation?
Answer:
is a 7-bit binary number in 2’s complement notation. What is the decimal value?
Answer:

27. Detail for >
-2010: Positive Value =
Invert:
Add 1:

28. Detail for >
2’s Complement Rep:
Invert:
Add One:
Converts to: =

29. Arithmetic in 2’s Complement
Add 2 positive 8-bit numbers
Add 2 8-bit numbers with different signs
Take the 1’s complement of 58 (i.e. invert, add 1) = 45 58
103 = 45
–58
–13
Invert to get magnitude = 13

30. Addition with Carry in 2’s Complement
8-bit number
Invert (add 1) (210)
Add
drop final carry out =
106 –2
104
(drop 1)

31. Subtraction 8-bit number Invert (add 1) 0101 1010 (9010) 1010 0110 Add
drop final carry out =
106 90 –
(drop 1) 16

32. Overflow 8-bit number Add
256 different numbers
Positive numbers: 0 to 127
Add
Test for overflow
2 positive inputs produced negative result overflow!
Wrong answer!
Programmers beware: some high-level languages, e.g., some versions of BASIC, do not check for overflow adequately = 64 65
-127
12710
Invert to get magnitude

33. Overflow and Carry Conditions
Carry flag: set when the result of an addition or subtraction exceeds fixed number of bits allocated
Overflow: result of addition or subtraction overflows into the sign bit

34. Overflow/Carry Examples
Correct result
No overflow, no carry
Example 2:
Incorrect result
Overflow, no carry
0100 =
(+ 4)
0010
+ (+ 2)
0110
(+ 6)
0100 =
(+ 4)
0110
+ (+ 6)
1010
(– 6)
0101
+ 1
0110
Invert, then
add 1 to get
magnitude

35. Overflow/Carry Examples
Result correct ignoring the carry
Carry but no overflow
Example 4:
Incorrect result
Overflow, carry ignored
1100 =
(– 4)
1110
+ (– 2)
11010
(– 6)
1100 =
(– 4)
1010
+ (– 6)
10110
(+ 3)

36. 2’s Complement Subtraction
Just add the opposite value!
A – B = A + (-B)
add
2’s complement rep. of -B

37. Not quite finished => BCD
Reading: Lecture slides and Chapter 5
Not quite finished => BCD

38. Unsigned Integers Binary equivalent BCD: Binary-Coded Decimal
conversion as discussed in Lecture 2
4 bits can hold 16 different values 0 – 15
8 bits can hold 28 different values between 0 and 255
BCD: Binary-Coded Decimal
digit-by-digit individual conversion to binary
4 bits per decimal digit, i.e. 10 different values 0 - 9
8-bit storage location can hold 2 BCD digits, i.e. 100 different values

39. Binary vs. BCD Decimal Binary BCD 68 = 0100 0100 = 0110 1000= =
= = = 68
= = 6 23 = 8 99
(largest 8-bit BCD) = =
= =
= = 99 = =
255 (largest 8-bit binary) = =
= 28 – 1 = 255
8 bits = =
12 bits

40. Value Range: Binary vs. BCD
No. of Bits
BCD Range
Binary Range 4
0-9
1 digit
0-15
1+ digit 8
0-99
2 digits
0-255
2+ digits 12
0-999
3 digits
0-4,095
3+ digits 16
0-9,999
4 digits
0-65,535
4+ digits 20
0-99,999
5 digits
0-1 million
6 digits 24
0-999,999
0-16 million
7+ digits 32
0-99,999,999
8 digits
0-4 billion
9+ digits 64
0-(1016-1)
16 digits
0-16 quintillion
19+ digits

41. Simple BCD Multiplication

42. Conventional Binary vs. BCD
Binary representation generally preferred
greater range of value for given number of bits
calculations easier
BCD is still used
in business applications to maintain decimal rounding and decimal precision
in applications with a lot of input and output, but limited calculations