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
2 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, AddisonWesley as an imprint of Pearson, 2012
Donald D. Givone,
Digital Principles and
Design
, McGrawHill, 2003
John L. Hennessy and David A. Patterson,
Computer Organization and Design, The
Hardware/Software Interface
, 3
rd
Edition,
Morgan Kaufmann Publishers, Inc., 2005