CSE 111
Representing Numeric Data in a
Computer
Slides adapted from Dr. Kris
Schindler

Unsigned Binary Numbers

Range: 0

2 n -1
 where n is the number of bits

Positional Notation
P n n 5 0
3 8 b b b
0

Example: 101100 two
0
2
0
1
2
1
2
2
2
3
2
3 b b
1
0 1
4
2
8
3
1 8 44

Unsigned Binary Numbers

How do we convert from a decimal number to a binary number?
q i q
0
1
/
/
/
2
2

 q
0
, q
1
, r
0 r
1
2
 q
2
, r
2
...
i where
 b k r q
Decimal

 x

Binary
Re
Number
Quotient
Bit mainder k b b b
1
0
2
...


 r
0 r
1 r
2
Binary Number
 b n
...
b
2 b
1 b
0

Continue until q =0

Unsigned Binary Numbers

How do we convert from a decimal number to a binary number?

Example: 39 ten
39 / 2

19
19
9 /
/ 2
2


9
4 , 1
, 1
, 1
1
4 / 2
2 / 2
/ 2



2 ,
1 , 0
0
0 , 1 b b b b b b
2
1
0
3

1

1


1
0
5
4


1
0
39 ten

100111 two

Bit Positions

MSB

Most Significant Bit

Leftmost Bit Position

LSB

Least Significant Bit

Rightmost Bit Position

Signed Binary Numbers

The most significant bit (leftmost) represents the sign

Negative (-): 1

Positive (+): 0

Signed Binary Numbers
 Computers represent signed numbers using two’s complement notation

Signed Binary Numbers
 Two’s Complement

Representation of a negative binary number
 Consider an n -bit number, x
 The two’s complement of the number is 2 n x
 This process is called taking the two’s complement of a number
 Taking the two’s complement of a number negates it

Signed Binary Numbers
 Two’s Complement
 Shortcut for taking the two’s complement of a number
 Start at the least significant (rightmost) bit and move left (toward the most significant bit)
 Keep every bit until you reach the first 1
 Keep that 1

Invert every bit (0

1,1

0) after the first 1 as you continue to move left

Signed Binary Numbers
 Two’s Complement

Examples:
 4
 9
 Take the two’s complement of 4 (00000100)

1 1111100 = 4
 Take the two’s complement of 9 (00001001)

1 1110111 = 9
 Since the above are negative, taking the two’s complement will allow you to determine the magnitude, which is the positive equivalent

Signed Binary Numbers
 Two’s Complement

Examples:
 + 6
 Since the number is positive, you don’t need to take the two’s complement

0 00000110 = + 6

+ 18
 Since the number is positive, you don’t need to take the two’s complement

0 00010010 = + 18

Signed Binary Numbers
 Two’s Complement
 Since taking the two’s complement of a number negates it, taking the two’s complement twice gives you the original number back

Example:

+12 is represented by 00001100
 Taking the two’s complement results in -12 (11110100)

Taking the two's complement of -12 results in +12 (00001100)

Floating Point

Very large/small numbers

Fractions

Example
1
E o t )
8

8.5 x 2 23
-
S

100.1
2 x 2 23

Normalized

1.
001
2 x 2 27

Exponent

Bias = 127
 127+26 = 153 = 10011001
2

Significand: 00100000000000000000000

Sign: 0
 Number: 0 10010001 00100000000000000000000 i n i n (

References
 J. Glenn Brookshear, Computer Science - An
Overview , 11 th edition, Addison-Wesley as an imprint of Pearson, 2012

Donald D. Givone, Digital Principles and
Design , McGraw-Hill, 2003

John L. Hennessy and David A. Patterson,
Computer Organization and Design, The
Hardware/Software Interface , 3 rd Edition,
Morgan Kaufmann Publishers, Inc., 2005