This Slide may help you to know that What is number System and its types, and conversion between different number systems.

1. Number System
Created By: DARPAN CHELANI
Roll No: 2K17-CSM-14

2. INTRODUCTION
A number system is a way to represent
numbers in the computer system
architecture, every value that you are
saving or getting into/from computer
memory has a defined number system.
Computer architecture supports
following number system:
 Decimal Number System
 Binary Number System
 Octal Number System
 Hexadecimal Number System

3. Decimal Number System
• It consist of ten digit i.e. 0, 1, 2, 3, 4, 5, 6, 7, 8,
9 with the base 10.
• Each number can be used individually or they
can be grouped to form a numeric value as
85,48,35,456 etc.

4. BINARY NUMBER SYSTEM
• The Binary Number System consist of only two
digits– 0 and 1.
• Since this system use two digits, it has the
base 2.
• All digital computer use this number system
and convert the data input from the decimal
format into its binary equivalent.

5. Why Binary?
Since the computer is made up of electronic
components; it can have only two states, either
• On(1)
• Off(0)
The data which is given to the computer is
converted into binary form because a computer
understand only binary language.
It further converts the binary results into their
decimal equivalents for output.

6. Octal Number System
 In the Octal Number System it consist of 8
digits i.e. 0, 1, 2, 3, 4, 5, 6, 7 with a base 8.
 The sequence of octal number goes as 0, 1, 2,
3, 4, 5, 6, 7, 10, 11, 12, 13, 14, 15, 16, 17, 20,
21, 22, …..as go on.
 See each successive number after 7 is a
combination of two or more unique symbols of
octal system.

7. Hexadecimal Number System
 The Hexadecimal system use base 16.
 It has 16 possible digit symbol.
 It use the digit 0 through 9 plus the letters A, B,
C, D, E, and F as the 16 digit symbols.

8. Relationship between Hexadecimal, Octal,
Decimal, and Binary
 Notice that each
hexadecimal digit
represent a
group of four
binary digit. It Is
important to
remember that
Hex(Abbreviation
for Hexadecimal)
digit A through F
are equivalent to
the decimal value
10 through 15.
Hexadeci
mal
Octal Decimal Binary
0 0 0 0000
1 1 1 0001
2 2 2 0010
3 3 3 0011
4 4 4 0100
5 5 5 0101
6 6 6 0110
7 7 7 0111
8 10 8 1000
9 11 9 1001
A 12 10 1010
B 13 11 1011
C 14 12 1100
D 15 13 1101
E 16 14 1110
F 17 15 1111

9. Decimal-to-Binary Conversion
The method of converting Decimal to binary is
repeated-division method. For conversion follow
the rules:
1. Divide the given decimal number with the
base 2.
2. Write down the remainder and divide the
quotient by 2.
3. Repeat step 2 till the quotient is zero.

10. Convert 20010 to Binary Number
Reading the
remainders from the
bottom to top, the result
is
20010 = 110010002
2 200 Remainders
2 100 0 LSB
2 50 0
2 25 0
Write
2 12 1 in
2 6 0 this
2 3 0
order
2 1 1
0 1 MSB

11. Binary-to-Decimal Conversion
To convert a binary number follow the steps:
1. Multiply each binary number with 2 having the
power 0 for last position, starting from the right
digit.
2. Increase the power one by one, with base as
2.
3. Sum up all the products to get decimal
number.

12. Convert 1100010012 to Decimal Number
1100010012= 1 X 28 + 1 𝑋 27 + 0 𝑋 26 +
0 𝑋 25 + 0 𝑋 24 + 1 𝑋 23 +
0 𝑋 22
+ 0 𝑋 21
1 𝑋 20
= 256 + 128 + 0 + 0 + 0 + 8 + 0 + 0 + 1
= 393
Thus, 1100010012 = 39310

13. Decimal-to-Octal
The method of converting Decimal to Octal is
repeated-division method. For conversion follow
the rules:
1. Divide the given decimal number with the
base 8,
2. Write down the remainder and divide the
quotient by 8,
3. Repeat step 2 till the quotient is zero.

14. Convert 50010 to Octal Number
Reading the
remainders from the
bottom to top, the
result is
26610 = 4128
8 266 Remainders
8 33 2 LSB
8 4 1
0 4
MSB

15. Octal-to-Decimal Conversion
To convert a octal number follow the steps:
1. Multiply each Octal number with 8 having the
power 0 for last position, starting from the right
digit.
2. Increase the power one by one, with base as
8.
3. Sum up all the products to get decimal
number.

16. Convert (372)8 to Decimal Number
3728 = 3 X 82 + 7 𝑋 81 + 2 𝑋 80
= 3 X 64 + 7 X 8 + 2 X 1
= 192 + 56 + 2
= 25010
Thus, 3728 = 25010
So, an octal number can be easily converted to
its decimal equivalent by multiplying each octal
digit by its position weight.

17. Octal-to-Binary Conversion
The conversion from octal to binary is performed
by converting each octal digit to its 3-bit binary
equivalent.
The eight possible digits are converted as
indicated below:
Using these conversions, any octal number is
converted to binary by individually converting
each digit.
Octal Digit 0 1 2 3 4 5 6 7
Binary
Equivalent
000 001 010 011 100 101 110 111

18. Convert 54318 to Binary Number
We convert 54318 to binary using 3 bits for each
octal digit as follows:
5 4 3 1
101 100 011 001
Thus, 54318 = 1011000110012

19. Binary-to-Octal Conversion
Converting from binary integers to octal integers
is simply the reverse of the foregoing process.
Firstly you have to do is:
1. Group the binary integer into 3-bits starting at
the Least Significant Bit(LSB).
2. If unable to form group then, add one or two
0s.
3. Each group Is converted to its octal
equivalent.
It illustrated below for binary number 11010110
0 1 1 0 1 0 1 1 0
3 2 6
Thus, 110101102 = 3268

20. Decimal-to-Hexadecimal Conversion
The method of converting Decimal to
Hexadecimal is repeated-division method. For
conversion follow the rules:
1. Divide the given decimal number with the
base 16.
2. Write down the remainder and divide the
quotient by 16.
3. Repeat step 2 till the quotient is zero.

21. Convert 42310 to Hexadecimal
Reading the
remainders from
the bottom to top,
the result is
42310 = 1𝐴716
Note: Any
remainder greater
than 9 are
represented by
letters A through F.
16 423 Remainders
16 26 7 LSB
16 1 A
0 1 MSB

22. Hexadecimal-to-Decimal Conversion
To convert a Hexadecimal number follow the
steps:
1. Multiply each hexadecimal number with 16
having the power 0 for last position, starting
from the right digit.
2. Increase the power one by one, with base as
16.
3. Sum up all the products to get decimal
number.

23. Convert 2𝐴𝐹16 to Decimal Number
2𝐴𝐹16 = 2 X 162 + 10 𝑋 161 + 15 𝑋 160
= 512 + 160 + 15
= 68710
Thus, 2𝐴𝐹16 = 68710

24. Binary-to-Hexadecimal Conversion
Converting from binary integers to hexadecimal
integers is simple. Firstly you have to do is:
1. Group the binary integer into 4-bits starting at the
Least Significant Bit(LSB).
2. If unable to form group then, add one or two 0s.
3. Each group Is converted to its Hexadecimal
equivalent.
It illustrated below for binary number 1010111010
0 0 1 0 1 0 1 1 1 0 1 0
2 B A
Thus, 10101110102 = 2𝐵𝐴16

25. Hexadecimal-to-Binary Conversion
The conversion from Hexadecimal to binary is
performed by converting each Hexadecimal digit
to its 4-bit binary equivalent.
This is illustrated below:
9𝐹216 = 9 F 2
1001 1111 0010
Thus, 9𝐹216 = 1001111100102

26. Binary addition:-
A B Sum Carry
0 0 0 0
0 1 1 0
1 0 1 0
1 1 0 1
Binary subtraction:-
A B Difference Borrow
0 0 0 0
0 1 1 1
1 0 1 0
1 1 0 0