1. Department of Computer Science Georgia State University
Data Manipulation
Department of Computer Science
Georgia State University

2. What will be covered? A review on binary notation Fractions in binary
Binary addition
Storing integers
Storing fractions

3. A review on binary notation
How to convert a binary into a decimal (base 10)?
( )2= ( ? )10

4. ( )2= ( ? )10 23 1
Position 4 3 2 1
Position’s
quantity 24 23 22 21 20 16 8 4 2 1
⨯ ⨯ ⨯ ⨯ ⨯ 1 =
= 23

5. Fractions in Binary

6. Fractions in Binary
Decimal Point

7. 1 0 1 . 0 1 1 Fractions in Binary Radix Point Integer Part
Integer Part
Fractional Part

8. ( )2= ( ? )10 1 .
Position 2 1 -1 -2 -3
Position’s
quantity 22 21 20
2-1
2-2
2-3 4 2 1 ⅛
⨯ ⨯ ⨯ ⨯ ½ ⨯ ¼ ⨯ ⅛
= ¼ + ⅛
= 5 ⅜

9. Practice 1 ( 1 0 . 0 1 1 )2= ( ? )10 ½ ¼ ⅛ Position 7 6 5 4 3 2 1 . -1
( )2= ( ? )10
Position 7 6 5 4 3 2 1 . -1 -2 -3
Position's quantity 27 26 25 24 23 22 21 20
2-1
2-2
2-3
128 64 32 16 8 ⅛

10. Practice 1
( )2= ( ? )10
Position's
quantity
128 64 32 16 8 4 2 1 . ⅛ + + ⅛ 2
= 2⅝
( )2= ( ½ + ⅛ )10=( 2⅝ ) 10

11. Binary Addition
Base 2
Base 10 ?
5 8 ?

12. Binary Addition
Base 10 1
5 8 5 8 2 7 8 5

13. Binary Addition
Base 2

14. Binary Addition
+ 0
+ 1 1
+ 0 1
+ 1 ⨯ 1 1 1 2

15. 1 1 1 0 1 0 + 1 1 0 1 1 1 1 1 1 Binary Addition Integers Base 2 1 1 1
Integers 1 1 1 1

16. Binary Addition
Base 2
Fractions 1 1 1 . 1

17. Storing Integers
In the “real world” of mathematics, computers must represent both positive and negative binary numbers for positive and negative integers.
For example, even when dealing with positive arguments, mathematical operations may produce a negative result.
Example : = -5
Thus needs to be a consistent method of representing negative numbers in computers.

18. Storing Integers Sign-Magnitude The simplest conceptual format
Two’s complement
The most popular system for representing integers within today’s computers.
Excess notation
Representing a fixed length binary number in computer circuits.

19. Sign-Magnitude Sign bit : leftmost bit
0 represents positive, 1 represents negative
Magnitude : the remaining bits
The same as binary notation of absolute value
E.g. 4-bit sign-magnitude format
But…....
Sign
Magnitude
0000
1000 +0 -0
E.g. Using 4-bit sign-magnitude format
Calculate 110+(-1)10
+1 is stored as 0001
-1 is stored as 1001
0011 represents 3
1011 represents -3
0001
1010
= -210

20. Two’s complement notation
Definition: “The two’s complement of a binary integer is the 1’s complement of the number plus 1.”
For each bit, change 0 to 1, and 1 to 0

21. Two’s complement notation
How to encode an integer in two’s complement notation?
For positive integers,
keep the binary representation
For negative integers,
Start with N-bit representation of their absolute values (positive integer)
Complement each bit (change 0 to 1, and 1 to 0)
Add one (Binary addition algorithm)
“Complement and Add One”

22. Two’s complement notation
How to encode an integer in two’s complement notation?
For positive integers, keep the binary representation.
For negative integers,
Start with N-bit representation of their absolute values (positive integer)
Complement each bit (change 0 to 1, and 1 to 0)
Add one (Binary addition algorithm)
E.g. Represent in Two’s complement
The positive integer:
610
Complement each bit:
Add one:
-610

23. Two’s complement notation
Another simple way to get the two’s complement notation for a negative integer
“Copy until one and Complement” 6
Copy the bits from right to left until a 1 has been copied
Complement the remaining bits
0  1
1  0
- 6

24. Two’s complement notation
Another simple way to get the two’s complement notation for a negative integer
“Copy until one and Complement”
The positive integer:
Copy until 1
1 0
Complement remaining:
-610
610
610
-610
The positive integer:
Complement each bit:
Add one:
“Complement and Add one”

25. Two’s complement notation
E.g. Represent in Two’s complement using 4-bit pattern
The positive integer:
410
Copy until 1:
Complement remaining:
-410
E.g. Represent in Two’s complement using 4-bit pattern
The positive integer:
810
Copy until 1:
Complement remaining:
-810

26. Figure 1.19 Two’s complement notation systems
-4 ~ +3
-8 ~ +7

27. Two’s complement notation
Range of integers with Two’s complement
3-bit : ~ +3
4-bit : ~ +7
5-bit : ~ +15
8-bit : ~ +127
N-bit : ~ ~ ~ ~
-2N-1 ~ +2N-1-1
So In order to store value 8 in two’ complement notation, how many bits should be used at least?

28. Two’s complement notation
How to decode a Two’s complement notation?
Check “Sign bit”
Sign bit = 0 ( positive number)
Convert the binary number to a decimal directly
Sign bit = 1 ( negative number)
Do the “ Copy until 1 and Complement ” process
Convert binary into decimal
E.g. Decode Two’s complement notation 1010
Two’s complement:
Negative
Copy until one:
1 0
Complement remaining:
The positive integer:
610
Result:
-610

29. Practice 2
Decode the following two’s complement (8-bit)

30. Practice 2
Decode the following two’s complement (8-bit)
Complement
Copy
1 0
= 1 *16 + 1*2
= 18
Final answer: - 18

31. Addition in Two’s Complement

32. Overflow problem 9 is not in range -8 ~ 7, thus error happened.
Problem in
base ten
Problem in
4-bit Two’s complement 5
+ 4
Answer in
base ten 9 -7
9 is not in range -8 ~ 7, thus error happened.
We call this phenomenon as overflow.

33. Excess Notation
Excess four conversion table 7 7 6 5 4 3 2 1

34. Storing Fractions
Besides storing the binary representation, we need to store the position of radix.
Radix Point
One popular way to store fractions in computer is called floating-point notation.

35. Floating-point Notation
Scientific notation
±M x BE (M < 1)
M is the mantissa, B is the exponent base (usually 2), and E is the exponent
Normalized form
E.g * 102 , * 22 , - 0.1* 2-3
8 bits
Sign bit
Exponent
( 3 bits)
Mantissa
( 4 bits)

36. Error can be reduced by using a longer mantissa field
Truncation errors
Encoding the value 2 ⅝ 2½
Error can be reduced by using a longer mantissa field

37. What you have learned ? Binary System Binary notation
Fractions in binary
Binary addition
How to store integers and fractions in computer?
Two’s complement notation
Overflow problem
Floating point notation (scientific notation)
Truncation error

38. Thank you !

39. Floating-point Notation
Decode a floating-point notation 1
8 bits
Sign bit
Exponent
Mantissa
Scientific notation : ± M x B E - 4 x x x 2¾
10.11

40. Practice 3 ( 5⅝)10= ( ? )2 0 1 0 1 . 1 0 1 ½ ¼ ⅛ 5 = 4+1 ⅝ = ½ + ⅛
( 5⅝)10= ( ? )2
Position's
quantity
128 64 32 16 8 4 2 1 . ⅛
5 = 4+1
⅝ = ½ + ⅛

41. Two’s complement notation
How to encode an integer in two’s complement notation?
For positive integers, keep the binary representation.
For negative integers,
Start with N-bit representation of absolute value (positive integer)
Complement each bit (change 0 to 1, and 1 to 0)
Add one (Binary addition algorithm)
E.g. Represent in Two’s complement
The positive integer:
410
Complement each bit:
Add one:
-410

42. Answer
( )2= ( 1*2 + 1*1 + 0*0.5+1*0.25 )10
( )2= ( 0*1 + 1* * * )10
( 4.5 )10= ( )2
( 5.625)10= ( )2
Position 7 6 5 4 3 2 1 . -1 -2 -3
Position's quantity 27 26 25 24 23 22 21 20
2-1
2-2
2-3
128 64 32 16 8
1/2
1/4
1/8

43. Patterns that Add Up to Zero
How to represent -1 ?
Find a particular bit pattern results that represents zero when added to one.
0001
0000 1 -1 1
How to represent -3 ?
Find a particular bit pattern results that represents zero when added to three.
0011
0000 3 -3 1

44. Floating-point Notation
Decode a floating-point notation 1
8 bits
Sign bit
Exponent
Mantissa
Scientific notation : ± M x B E - 4
* 2-1
* 21
* 22 x x x
10.11 2¾

45. Exercises 1 ( 5⅝)10= ( ? )2 5 = 4 +1 ⅝ = ½ + ⅛ 1 0 1 . 1 0 1 Position
( 5⅝)10= ( ? )2
5 = 4 +1
⅝ = ½ + ⅛
Position 7 6 5 4 3 2 1 . -1 -2 -3
Position's quantity 27 26 25 24 23 22 21 20
2-1
2-2
2-3
128 64 32 16 8
1/2
1/4
1/8

46. ( )2= ( ? )10 23 1 16 8 4 2 1 16 4 2 1
= 23