This document discusses different number systems used in digital computers and their conversions. It explains that computers understand numbers in binary format and all other data needs to be converted to binary. It then describes the commonly used number systems - decimal, binary, hexadecimal and octal. Decimal uses 10 digits, binary uses two digits 0 and 1, hexadecimal uses 16 digits with 0-9 and A-F, and octal uses 8 digits from 0-7. The document provides examples and methods to convert between these number systems.

1. HOW DOES TECHNOLOGY
WORKS?

2. THE NUMBER SYSTEM

3. WHAT IS NUMBER SYSTEM?
• Computers understand machine language. Every letter, symbol etc.
that we write in the instructions given to computer, it gets converted into machine
language.
• This machine language comprises of numbers. In order to
understand the language used by computers and other digital
system it is crucial to have a better understanding of number
system.

4. • Commonly Used Number system
• Consist of 10 Digits (0-9).Base-10
• Example: 107,78,98,2786
Types of Number System
Decimal
Number
System
Binary Number
System
Hexadecimal
Number System
Octal Number
System
• Use in Digital Computers
• Consist of 2 Digits (0 and 1).Base-2
• Example: 10001,10100
• Consist of 16 Digits (0-9) (A-F)
• Base-16
• Example: D1CE,2E6
• Consist of 8 Digits (0-7)
• Base-8
• Example: 6675,0453,655

5. The Decimal System (Base) 10
• The word decimal is derived from the Latin root
decem(ten)
• Base b = 10
• Ten symbols: S = {0,1,2,3,4,5,6,7,8,9}
Example:45,452,501,4521
• The symbols in this system are often referred to
as decimal digits or just digits.

6. THE BINARY SYSTEM (BASE) 2
• The word binary is derived from the Latin root
• bini (double).
• Base b = 2.
• Ten symbols: S = {0, 1}
Binary is a base 2 number system invented by Gottfried Leibniz that is made up of only
two numbers: 0 and 1. This number system is the basis for all binary code, which is used to
write data such as the instructions that computer processors use, or the digital text you
read every day.

7. Computers Understand
ON OF
FTRU
E
FALSE
ONE ZER
O

8. DECIMAL TO BINARY
72
520
5078
=
=
=
1001000
1000001000
1001111010110
TEXT TO BINARY
HI
1001000 1101001
HELLO WORLD
1101000 1100101 1101100
1101100 1101111 0100000
1110111 1101111 1110010
1101100 1100100
TCCSTFI
1010100 1000011 1000011
1010011 1010100 1000110
1001001
48753 1011111001110001=

9. WORLD
01110111 01101111 01110010
01101100 01100100

10. EVERY GAME YOU PLAY
ARE MADE-UP WITH BINARY

11. EVERY SITE YOU VISIT
ARE MADE-UP WITH BINARY

12. CONVERT
DECIMAL
TO
BINARY

13. DECIMAL BINARYTO
2 1 2
10
=
212 / 2 RESULT 106 REMAINDER 0
106 /
53
26
13
6
3
1
2 RESULT 53 REMAINDER 0
/
/
/
/
/
/
2 RESULT 26 REMAINDER
02 RESULT 13 REMAINDER
2 RESULT 6 REMAINDER
1
1
2 RESULT 3 REMAINDER 0
2 RESULT 1 REMAINDER 1
2 RESULT 0 REMAINDER 1
1 1 0 1 0 1 0 0
2

14. DECIMAL BINARYTO
5 8 6 210
=
5862 / 2 RESULT 2931 REMAINDER 0
1 0 1 1 0 1 1 1
2
2931 / 2 RESULT 1465 REMAINDER 1
1465 / 2 RESULT 732 REMAINDER 1
732 / 2 RESULT 366 REMAINDER 0
366 / 2 RESULT 183 REMAINDER 0
183 / 2 RESULT 91 REMAINDER 1
91 / 2 RESULT 45 REMAINDER 1
45 / 2 RESULT 22 REMAINDER 1
22 / 2 RESULT 11 REMAINDER 0
11 / 2 RESULT 5 REMAINDER 1
5 / 2 RESULT 2 REMAINDER 1
2 / 2 RESULT 1 REMAINDER 0
1 / 2 RESULT 0 REMAINDER 1
0 0 1 1 0

15. CONVERT THE FOLLOWING
PROBLEM?
DECIMAL BINARYTO
301410
648910
125610
2245 10
=
=
=
=752 10
=
101111000110
2
1100101011001
10011101000
100011000101
1011110000
2
2
2
2
1.
2.
3.
4.
5.

16. CONVERT
BINARY
TO
DECIMAL

17. THE BINARY TABLE
128 64 32 16 8 4 2 1

18. DECIMALBINARY TO
1 0 1 0 1
2
2
0
2
1
2
2
2
3
2
4
1 X 10 X 21 X 40 X 81 X 16
01234
16 0 4 0 1+ + + + = 21
21
10
=
128 64 32 16 8 4 2 1
THE BINARY TABLE

19. DECIMALBINARY TO
1 0 1 0 1 0 0 1 1
2
2
0
2
1
2
2
2
3
2
4
1 X 11 X 20 X 40X 81 X 16
01234
339
10
=
128 64 32 16 8 4 2 1
THE BINARY TABLE
2
5
2
6
2
7
2
8
0 X 32
5
1 X 64
6
0X 128
7
1 X 256
8
+ + + + + + + +256 0 64 0 16 0 0 2 1 = 339

20. DECIMALBINARY TO
1 0 1 1 0 1 0 1 0
2 10
=
1248163264128256
256 64+ + 2+ =32 + 8
362
362

21. CONVERT THE FOLLOWING
PROBLEM?
DECIMALBINARY TO
10
=
=
=
=
=
1001000101001
21.
2.
3.
4.
5.
10111000111100
10011110011101
11000010101001
10101110100101
2
2
2
2
4649
11836
10141
12457
11173
10
10
10
10

23. CONVERT THE FOLLOWING PROBLEM?
10=
=
=
=
=
111101110101021.
2.
3.
4.
5.
100101011010
11001010111
11010111010100
1110000101010000
2
2
2
2
7914
2394
1623
13780
57680
10
10
10
10
6.
7.
8.
9.
10.
= 1011000010011111
2
45215
10
= 1100011010001102
25414
10
= 1010010111010011
2
42451
10
= 10010011010010100
2
75412
10
= 11010000100111110000
2
854512
10

24. THE HEXADECIMAL SYSTEM
(BASE 16)
• The word hexadecimal is derived from the Greek
root hex (six) and Latin root decem (ten).
• Base b = 16.
• Ten symbols: S = {0, 1,2,3,4,5,6,7, 8, 9, A, B, C, D,
E, F}
• Example: 2AF , 3DF4 , 5FAE3 , 8CF09
• The symbols in this system are often referred to
• as hexadecimal digits.

25. THE RELATIONSHIP BETWEEN
DECIMAL TO HEXADECIMAL
0
1
2
3
4
5
6
7
9
0
1
2
3
4
5
6
7
9
DECIMAL TO HEXADECIMAL
10
11
12
13
14
15
=
=
=
=
=
=
A
B
C
D
E
F
=
=
=
=
=
=
=
=
=

26. CONVERT
HEXADECIMAL
TO
DECIMAL

27. 2 3 E
HEXADECIMAL DECIMAL
=16
16
0
16
1
16
2
2 X 16
2
+ 3 X 16
1
+ 14 X 16
0
10
11
12
13
14
15
=
=
=
=
=
=
A
B
C
D
E
F
DH
2 X 256 + 3 X 16 + 14
512 + 48 + 14 574=
574 10
TO

28. HEXADECIMAL DECIMALTO
10
11
12
13
14
15
=
=
=
=
=
=
A
B
C
D
E
F
DH
A B 0 9
16
=
16
0
16
1
16
2
16
3
10 X 16
3
+ 11 X 16
2
+ 0 X 16
1
+ 9 X 16
0
10 X 4096 + 11 X 256 + 0 + 9
40960 + 2816 + 0 + 9 43785=
4378510
X 16

29. THE OCTAL SYSTEM (BASE 8)
• The word octal is derived from the Latin root octo
• (eight).
• Base b = 8.
{0, 1, 2, 3, 4, 5, 6, 7}
Example: {1524 , 576 , 342}

30. CONVERT
HEXADECIMAL
TO
OCTAL

31. HEXADECIMAL OCTALTO
A C
16
10
11
12
13
14
15
=
=
=
=
=
=
A
B
C
D
E
F
DH
10
CONVERT
TO DECIMAL
12
8 4 2 1 8 4 2 1
1 0 1 0 1 1 0 0
=
001101010
2 5 4 = 2 5 4
2 5 4
8
124 124 124

32. HEXADECIMAL OCTALTO
10
11
12
13
14
15
=
=
=
=
=
=
A
B
C
D
E
F
DH
1 E F
16
8 4 2 1
8 4 2 1
8 4 2 1
100 0 1 1 1 1
1 1 1 0
1 1 1 11 1 1 0100 0
124124124124
7 5 7 = 7 5 7
7 5 78
CONVERT
TO DECIMAL
1
14
15