This document discusses different number systems used in computing, including binary, decimal, octal, and hexadecimal. It provides the base and symbols for each system, examples of 4-bit numbers converted to decimal, and techniques for converting between binary, decimal, octal, and hexadecimal. Key bits in a binary number are defined as the most significant bit (MSB) and least significant bit (LSB). Conversion methods include dividing decimal numbers by powers of two for binary, and representing octal and hexadecimal digits as groups of bits.

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Number Systems Number systems are the techniques to represent
numbers in the computer system architecture,
every value that we are saving or getting into/from
computer memory has a defined number system.
Types of Number Systems
1. Non-Positional Number Systems. Ex- I, II, II, IV…… etc.
2. Positional Number Systems.
a) Binary
b) Decimal
c) Octal
d) Hexadecimal
Base and Symbols of
Positional Number Systems
Number Systems Base Symbols
Binary 2 0, 1
Decimal 10 0, 1, 2, 3, 4, 5, 6, 7, 8, 9
Octal 8 0, 1, 2, 3, 4, 5, 6, 7
Hexa-
decimal
16
0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D,
E, F

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4-bit Numbers with their
Decimal Values
Binary Decimal Equivalent
0000 0
0001 1
0010 2
0011 3
0100 4
0101 5
0110 6
0111 7
4- bit Numbers with their
Decimal Values (cont.)
Binary Decimal Equivalent
1000 8
1001 9
1010 10
1011 11
1100 12
1101 13
1110 14
1111 15
Bit is the smallest unit of data in computing. It is
represented by a 0 or a 1.
The byte is a unit of digital information that most
commonly consists of eight bits, representing a
binary number.

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In any binary number, the right most digit is
called least significant bit (LSB) and leftmost digit
is called most significant bit (MSB).
1 0 1 1 0
(MSB) (LSB)
Conversion Among Bases
• The possibilities:
Hexadecimal
Decimal Octal
Binary
Binary to Decimal Conversion
Technique
Multiply each bit by 2n, where n is the “weight”
of the bit
The weight is the position of the bit, starting
from 0 on the right
Add the results
Example
1010112 => 1 x 25 = 32
0 x 24 = 0
1 x 23 = 8
0 x 22 = 0
1 x 21 = 2
1 x 20 = 1
4310
Bit “0”

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Octal to Decimal Conversion
Technique
Multiply each bit by 8n, where n is the “weight”
of the bit
The weight is the position of the bit, starting
from 0 on the right
Add the results
Example
7248 => 7 x 82 = 448
2 x 81 = 16
4 x 80 = 4
46810
Hexadecimal to Decimal
Conversion
Technique
Multiply each bit by 16n, where n is the
“weight” of the bit
The weight is the position of the bit, starting
from 0 on the right
Add the results
Example
ABC16 => A x 162 = 10 x 256 = 2560
B x 161 = 11 x 16 = 176
C x 160 = 12 x 1 = 12
274810

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Decimal to Binary Conversion
Technique
Divide by two, keep track of the remainder
First remainder is bit 0 (LSB, least-significant
bit)
Second remainder is bit 1
Example
12510 = ?2
2 125
62 12
31 02
15 12
7 12
3 12
1 12
0 1
12510 = 11111012
Reverse Order
Octal to Binary Conversion
Technique
Convert each octal digit to a 3-bit equivalent
binary representation
Example
7058 = ?2
7 0 5
111 000 101
7058 = 1110001012

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Hexadecimal to Binary
Conversion
Technique
Convert each hexadecimal digit to a 4-bit
equivalent binary representation
Example
10AF16 = ?2
1 0 A F
0001 0000 1010 1111
10AF16 = 00010000101011112