Binary number system

1. NADEEM UDDIN
ASSOCIATE PROFESSOR
OF STATISTICS
https://www.slideshare.net/NadeemUddin17
https://nadeemstats.wordpress.com/listofbooks/

2. Introduction to Decimal Number System:
A positional system of numeration that uses decimal digits and a base
ten. The number system we use every day, based on 10 digits
(0,1,2,3,4,5,6,7,8,9) position is important.The decimal numeral
system (also called base ten or occasionally denary) has ten as its
base. It is the numeral base most widely used by modern civilizations.
Binary Number System:
A method of representing number that has 2 as its base and uses only
the digits 0 and 1.Number system that uses only two values (0,1) to
represent codes and data. Since zeros and ones can be easily
represented by two voltages. The binary system is the foundation on
which digital technology is built. Every digital computer whether a
pocket calculator or a mainframe uses the same binary notation.

3. 2 123
2 61 1
2 30 1
2 15 0
2 7 1
2 3 1
1 1
Example -1 Decimal to Binary
(123)10 = (1111011)2

4. Example-2 Decimal Fraction to Binary Fraction
0
.
0.3125
x 2
0
6250
x 2
1
2500
x 2
0
5000
x 2
1 0000
(0.3125)10 = (0.0101)2

5. Example -3 Binary to Decimal
1001001 = 1 x 26 + 0 x 25 + 0 x 24 + 1 x 23 + 0 x 22 + 0 x 21 + 1x20
1001001 = 1 x 64 + 0 x 32 + 0 x 16 + 1 x 8 + 0 x 4 + 0 x 2 + 1 x 1
1001001 = 64 + 0 + 0 + 8 + 0 + 0 + 1
(1001001)2 = (73)10

6. Example -4 Binary Fraction to Decimal Fraction
0.0011 = 0 x 2-1 + 0 x 2-2 + 1 x 2-3 + 1 x 2-4
1 1 1 1
0 0 1 1
2 3 42 2 2 2
1 1 1 1
0 0 1 1
2 4 8 16
1 1
0 0
8 16
1 1
8 16
2 1
16
3
16
      
      
  


= 0.1875
(0.0011)2 = (0.1875)10

7. Addition Facts:
0 + 0 = 0
0 + 1 = 1
1 + 0 = 1
1 + 1 = 10 with a carry of 1
1+1+1 = 11
Example – 5 110101 + 11011
Solution:
1 1 0 1 0 1
+ 1 1 0 1 1
1 0 1 0 0 0 0

8. Example-6 1 0 1 1 1 1 + 1 1 0 1 0 1 + 1 0 1 1 0 1
Solution:
1 0 1 1 1 1
+ 1 1 0 1 0 1
11 0 0 1 0 0
+ 1 0 1 1 0 1
10 0 1 0 0 0 1

9. Subtraction Facts:
0 - 0 = 0
1 - 0 = 1
1 -1 = 0
Example-7
1 1 0 – 1 0
Solution:
1 1 0
- 1 0
1 0 0

10. Example-8 1 1 1 0 0 1 1 1 – 1 1 1 1 1 1 1
Solution:
1 1 1 0 0 1 1 1
- 1 1 1 1 1 1 1
1 1 0 1 0 0 0
Example-9 1 0 1 0 1 0 0 1 – 1 0 1 1 0 1
Solution:
1 0 1 0 1 0 0 1
- 1 0 1 1 0 1
1 1 1 1 1 0 0

11. Multiplication Facts:
0 x 0 = 0
0 x 1 = 0
1 x 0 = 0
1 x 1 = 1
Example-10: 1 1 0 1 x 1 0 0
Solution:
1 1 0 1
x 1 0 0
0 0 0 0
0 0 0 0 x
1 1 0 1 x x
1 1 0 1 0 0

12. Example-11: 1 1 0 0 1 x 1 0 0 1
Solution:
1 1 0 0 1
x 1 0 0 1
1 1 0 0 1
0 0 0 0 0 x
0 0 0 0 0 x x
1 1 0 0 1 x x x
1 1 1 0 0 0 0 1

13. Division Facts:
0 ÷ 1 = 0
1 ÷ 1 = 1
Example-12:
Solution:
1 0 1 0 1 0 ÷ 1 1 0
111
110 101010
11 0
10 0 1
1 1 0
1 1 0
1 1 0
x x x
1 0 1 0 1 0 = 1 1 1
1 1 0

14. Example-13: 1 0 0 0 0 1 1 1 ÷ 1 0 1
11011
101 10000111
1 0 1
0 0 1 1 0
1 0 1
0 0 1 1 1
1 0 1
1 0 1
1 0 1
x
1 0 0 0 0 1 1 1 = 1 1 0 1 1
1 0 1
Solution:

15. Example-14: 1 0 0 0 0 1 ÷ 1 1
Solution:
1011
11 100001
1 1
1 0 0
1 1
1 1
1 1
x
1 0 0 0 0 1 = 1011
1 1

16. Exercise-1.1
1. Convert the following decimal numbers into equivalent binary
numbers.
i) 32 ii) 57 iii) 67
iv) 89 v) 185 vi) 369
vii) 412 viii) 567 ix) 1853
x) 3922
2. Convert the following decimal fraction into equivalent binary
fractions.
i) 0.125 ii) 0.625 iii) 0.5625
iv) 0.859375 v) 0.078125
3. Convert the following binary numbers into equivalent decimal
numbers.
i) 101 ii) 1000 iii) 1001
iv) 1011 v) 1100 vi) 100001
vii) 1000001 viii) 1010001 ix) 10011101
x) 111000011

17. 4. Convert the following binary fraction into decimal fractions.
i) 0.001 ii) 0.101 iii) 0.1001
iv) 0.110111 v) 0.000101
5. Convert the following decimal into binary fractions.
i) 25.125 ii) 412.5625
6. Convert the following binary into decimal fractions.
i) 111001.101 ii) 10111001.000101

18. 7. Add the following binary numbers.
i) 10101 + 1101 vi) 111 +101 +10
ii) 10110 +11011 vii) 1011101 +111011 + 10111
iii) 10011 + 10101 viii) 110110 +101101+ 10101
iv) 10101 + 1101 ix) 1010101 + 110110 + 10010
v) 10111 +10101 x) 1110111 + 100101 + 111110
8. Subtract the following binary numbers.
i) 1110 – 101 iv) 110101 – 10011
ii) 100101 – 11010 v) 1100001 – 11110
iii) 11011 – 1010

19. 9. Perform the following multiplication.
i) 101 x 11 iii) 10110 x 1101
ii) 1100 x 1110 iv) 1011 x 110
v) 11010 x 10110
10. Perform the following division.
i) 11001 ÷ 101 iv) 1111110 ÷ 111
ii) 11110 ÷ 1010 v) 110010 ÷ 101
iii) 11011 ÷10

20. ANSWERS EXERCISE-1.1
1.
i)100000 vi)101110001
ii)111001 vii)110011100
iii)1000011 viii)1000110111
iv)1011001 ix)11100111101
v)10111001 x)111101010010
2.
i)0.001 iii)0.1001
ii)0.101 iv)0.110111
v)0.000101
3.
i)5 vi)33
ii)8 vii)65
iii)9 viii)81
iv)11 ix)157
v)12 x)451

21. 4.
i)0.125 iv)0.859375
ii) o.625 v)0.078125
iii)o.5625
5.
i)11001.001
ii)110011100.1001 6.i)57.625 ii)185.078125
7.
i)100010 vi)1110
ii)110001 vii)10101111
iii)101000 viii)1111000
iv)100010 ix)10011101
v)101100 x)11011010
8.
i)1001 iii)10001
ii)1011 iv)100010
v)1000011

22. 9.
i)1111 iii)100011110
ii)10101000 iv)1000010
v)1000111100
10.
i)101 iii)1101.1
ii)11 iv)10010
v)1010