This document discusses different numbering systems, including binary, decimal, octal, and hexadecimal. It explains how each system works using different bases and digits. The document also covers how to convert between these numbering systems, and how to perform basic arithmetic operations like addition and subtraction in binary.

2. Outcome of this chapter
 Understand the base value for different
numbering system.
 Different numbering system basic difference.
 How the each of the four numbering system
works?
 How to convert one numbering system to
other.
 Understand the binary arithmetic operations.
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3. Understanding the base
 A value of each digit in a number can be determined
using
 A digit
 A position of the digit in the number
 The base of the number system
 Total number digit available in the number system
 For instance, let’s assume 278 is a base 10 number then
how do we get the value 278?
27810 = (2 x 102 )+(7 x 101 )+(8 x 100 )
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4. Number system
 A mathematical notation for representing numbers of
a given set, using digits or other symbols
 There are four types of number system
1. Binary number
 Base 2 and digit 0 and 1
2. Decimal number
 Base 10, digit (0,1,2,3,4,5,6,7,8,9)
3. Octal number
 Base 8, digit (0,1,2,3,4,5,6,7)
4. Hexadecimal number
 Base 16, digit (0,1,2,3,4,5,6,7,8,9 and A to F where A,B,C,D,E,F)
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5. Decimal number
 The base-10 number system is known as the decimal
system
 Range: 0 to 9
 A decimal point to separate whole numbers from
decimal fraction
 For instance, 4210, and 34510.
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6. Binary number
 A base-2 number system
 Consists either 0 and 1
 For instance, 101102 and 11112
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7. Octal number
 Base eight numbers meaning eight digits are used to
represent all the values
Octal 0 1 2 3 4 5 6 7
Binary 000 001 010 011 100 101 110 111
Decimal 0 1 2 3 4 5 6 7
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8. Hexadecimal Number
 Numbering system uses sixteen digits to represent
numbers
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9. Conversion
1. Decimal to other base system
i. Decimal to Binary
ii. Decimal to Octal
iii. Decimal to Hexadecimal
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10. Conversion (2)
2. Other base system to decimal
a. Binary to decimal
b. Octal to decimal
c. Hexadecimal to decimal
3. Binary to Octal
4. Octal to binary
5. Binary to hexadecimal
6. Hexadecimal to binary
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11. Decimal to any number and vice versa
1. Divide the decimal number by the base of the
number you want to convert and take it’s remainder
2. The process is repeated until it produces the result of
0
3. The resulting number is obtained by taking the
remainder from the bottom to the top
For doing vice versa, following formula to be considered:
N = anbn + an-1bn-1 + ….. + a1b1 + a0b0
Where: N: decimal equivalent, b: base, a: i-th digit
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12. Binary to Octal and Octal to Binary
1. At first make group of three digits starting from
right.
2. If in last the digits can’t be rounded up to three add 0
the left.
3. Replace each group accordingly with the help of the
table.
4. Sequence the numbers as the groups are created.
101112 = 278
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13. Octal to Binary
1. Replace each digit with it’s equivalent three digits
2. After joining the numbers if any 0s are left at the
leftmost erase them
3. For instance, 2 is replaced by 010 and 7 is replaced by
111
 Thus, the number becomes 010111 and removing the
leftmost zero as it bears no value the answer become
101112
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14. Binary to Hexadecimal and
Hexadecimal to Binary
1. Replace each digit with it’s equivalent four digits
2. After joining the numbers if any 0s are left at the
leftmost erase them
For instance, 10011112 = 4F16 and 4F16 = 010011112
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15. Binary arithmetic
 Binary addition
 As like decimal addition
 Add two numbers according to the column and forward
the carry to the next column
 Binary subtraction
- Borrow 1 from the next more significant bit when 0-1=1
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16. Binary addition
 Basic rules for adding two binary numbers
 0+0 = 0
 0+1 = 1
 1+0 = 1
 1+1 = 0, and carry 1 to the next more significant bit
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17. Binary subtraction
 Basic rules for binary subtraction as follows
 0 - 0 = 0
 0-1 = 1, and borrow 1 from next more significant bit
 1-0 = 1
 1-1 = 0
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