CSE 111
Representing Numeric Data in a Computer
Slides adapted from Dr. Kris Schindler
Unsigned Binary Numbers
Range: 02n-1where n is the number of bits
Positional Notation
Example: 101100two
11
22
33
22
11
00 222222
nn
nn bbbbbb
11
223210 228421
nn
nn bbbbbb
44328432116081412010
n -1
012345n -2n -1
n -2n -1
n -2 5 4
4
3 2
2
1
1
bbbbbbbb
0
2 2 3 2 1 6 8W e ig h tB it
P o s itio n
Unsigned Binary Numbers
How do we convert from a decimal number to a binary number?
Continue until q=0
012
22221
11110
0000
...
Re
......
,2/
,2/
,2/
bbbbNumberBinary
mainderr
Quotientq
kBitBinaryb
NumberDecimali
where
rbrqq
rbrqq
rbrqi
n
x
k
Unsigned Binary Numbers
How do we convert from a decimal number to a binary number?Example: 39ten
twoten
b
b
b
b
b
b
10011139
15
04
0
1
1
1
1,02/1
0,12/2
0,22/4
1,42/9
1,92/19
1,192/39
3
2
1
0
Bit Positions
MSBMost Significant BitLeftmost Bit Position
LSB Least Significant BitRightmost Bit Position
Signed Binary Numbers
The most significant bit (leftmost) represents the signNegative (-): 1Positive (+): 0
Signed Binary Numbers
Computers represent signed numbers using two’s complement notation
Signed Binary Numbers
Two’s ComplementRepresentation of a negative binary number
Consider an n-bit number, x The two’s complement of the number is 2n - xThis process is called taking the two’s complement of a numberTaking the two’s complement of a number negates it
Signed Binary Numbers
Two’s ComplementShortcut 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 1Keep that 1Invert every bit (01,1 0) after the first 1 as you continue to
move left
Signed Binary Numbers
Two’s ComplementExamples:
-4 Take the two’s complement of 4 (00000100) 11111100 = -4
-9 Take the two’s complement of 9 (00001001) 11110111 = -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 ComplementExamples:
+6 Since the number is positive, you don’t need to take the two’s
complement 000000110 = +6
+18 Since the number is positive, you don’t need to take the two’s
complement 000010010 = +18
Signed Binary Numbers
Two’s ComplementSince 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 00001100Taking the two’s complement results in -12 (11110100)Taking the two's complement of -12 results in +12 (00001100)
Floating Point
Very large/small numbersFractions
Example8.5 x 223
100.12 x 223
Normalized1.0012 x 227
ExponentBias = 127127+26 = 153 = 100110012
Significand: 00100000000000000000000Sign: 0Number: 01001000100100000000000000000000
S1B its
E x p o n e n t (E )
(-1) X 1.F X 2S (E-bias)
8S ig n if ic an d (F )
2 3
References
J. Glenn Brookshear, Computer Science - An Overview, 11th 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, 3rd Edition, Morgan Kaufmann Publishers, Inc., 2005