Binary Representation
Major: All Engineering Majors
Authors: Autar Kaw, Matthew Emmons
http://numericalmethods.eng.usf.edu
Transforming Numerical Methods Education for STEM
Undergraduates
7/12/2016
http://numericalmethods.eng.usf.edu
1
Binary Representation
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How a Decimal Number is
Represented
257.76  2 10 2  5 101  7 100  7 10 1  6 10 2
3
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Base 2
 (1 23  0  2 2  1 21  1 20 )


(1011.0011) 2  
1
2
3
4 

(
0

2

0

2

1

2

1

2
) 10

 11.1875
4
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Convert Base 10 Integer to
binary representation
Table 1 Converting a base-10 integer to binary representation.
Quotient
11/2
5
5/2
2
2/2
1
1/2
0
Remainder
1  a0
1  a1
0  a2
1  a3
Hence
(11)10  (a3 a 2 a1a0 ) 2
 (1011) 2
5
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Start
Input (N)10
Integer N to be
converted to binary
format
i=0
Divide N by 2 to get
quotient Q & remainder R
i=i+1,N=Q
ai = R
No
Is Q = 0?
Yes
n=i
(N)10 = (an. . .a0)2
STOP
6
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Fractional Decimal Number
to Binary
Table 2. Converting a base-10 fraction to binary representation.
0.1875 2
0.375 2
0.75  2
0.5  2
Number
Number after
decimal
0.375
0.75
1.5
1.0
0.375
0.75
0.5
0.0
Number before
decimal
0  a1
0  a2
1  a 3
1  a4
Hence
(0.1875)10  (a1a 2 a 3a 4 ) 2
 (0.0011) 2
7
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Start
Fraction
F
to
be
converted to binary
format
Input (F)10
i  1
Multiply F by 2 to get
number before decimal,
S and after decimal, T
i  i  1, F  T
ai = R
No
Is T =0?
Yes
n=i
(F)10 = (a-1. . .a-n)2
STOP
8
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Decimal Number to Binary
11.187510  
?.?
2
Since
(11)10  (1011) 2
and
(0.1875)10  (0.0011) 2
we have
(11.1875)10  (1011.0011) 2
9
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All Fractional Decimal Numbers
Cannot be Represented Exactly
Table 3. Converting a base-10 fraction to approximate binary representation.
Number
0.3 2
0.6  2
0.2  2
0.4  2
0.8  2
0.6
1.2
0.4
0.8
1.6
Number
after
decimal
0.6
0.2
0.4
0.8
0.6
Number
before
Decimal
0  a1
1  a2
0  a3
0  a4
1  a 5
(0.3)10  (a1a2 a3a4 a5 ) 2  (0.01001) 2  0.28125
10
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Another Way to Look at
Conversion
Convert 11.187510 to base 2
1110  23  3
 23  21  1
2 2 2
3
1
0
 1 2  0  2  1 2  1 2
 10112
3
11
2
1
0
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0.187510  23  0.0625
 2 3  2  4
1
2
3
 0  2  0  2  1 2  1 2
4
 .00112
11.187510  1011.00112
12
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Additional Resources
For all resources on this topic such as digital audiovisual
lectures, primers, textbook chapters, multiple-choice
tests, worksheets in MATLAB, MATHEMATICA, MathCad
and MAPLE, blogs, related physical problems, please
visit
http://numericalmethods.eng.usf.edu/topics/binary_repr
esentation.html
THE END
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