Binary Addition
The following table is useful when adding two binary numbers

A	B	SUM(A+B)	Carry
0	0	0	-
0	1	1	-
1	0	1	-
1	1	0	1

In 1+ 1= 10,is considered as sum 0and the 1as carry bit. This carry bit is added with the previous position of the bit pattern 
Example Add: 1011₂ + 1001₂
 

Example Perform Binary addition for the
following: 23₁₀ + 12₁₀
Step 1: Convert 23 and 12 into binary form

2310
2's Power	16	8	4	2	1
Binary Number	1	0	1	1	1
2310 =000101112

1210
2's Power	8	4	2	1
Binary Number	1	1	0	0
1210 =000011002

Step 2: Binary Addition of 23 and 12 :

Carry Bit \(\rightarrow \)			1	1
2310=0	0	0	1	0	1	1	1
1210=0	0	0	0	1	1	0	0
3510	0	1	0	0	0	1	1

Binary Subtraction
The table for Binary Subtraction is as follows:

A	B	Differnce (A-B)	Borrow
0	0	0	0
1	0	1	0
1	1	0	0
0	1	1	1

When subtracting 1 from 0, borrow 1 from the next Most Significant Bit, when borrowing from the next Most Significant Bit, if it is 1, replace it with O. If the next Mos.t Significant Bit is 0, you must borrow from a more significant bit that contains 1 and replace it with 0 and Os upto that point become 1s.

Example Perform binary addition for the following: (-21)₁₀ + (5)₁₀
Step 1: Change -21 and 5 into binary form

2110
2's Power	16	8	4	2	1
Binary Number	1	0	1	1	1
2110 =000101012

510
2's Power	4	2	1
Binary Number	1	1	0
510 =00001012

2110	0	0	0	1	0	1	0	1
1's Compliment	1	1	1	0	1	0	1	0
2's Compliment	1	1	1	0	1	0	1	1

Step 3:
Binary Addition of -21 and 5:

Carry Bit				1	1	1	1
-2110	1	1	1	0	1	0	1	1
510	0	0	0	0	0	1	0	1
-1610	1	1	1	1	0	0	0	0

A numbering system is a way of representing numbers. The most commonly used numbering system in real life is Decimal number system.· Other number systems are Binary, Octal, Hexadecimal number system. Each number system 'is uniquely identified by its base value or radix. Radix or base is the count of number of digits in each number system. Radix or base is the general idea behind positional numbering system.
Decimal Number System
It consists of 0,1,2,3,4,5,6,7,8,9(10 digits). It is the oldest and most popular number system used in our day to day life, In the positional number system, each decimal digit is weighted relative to its position in the number. This means that each digit in the number is multiplied by 10 raised to a power corresponding to that digit's position.

Binary Number System
There are only two digits in the Binary system, namely, 0 and 1. The numbers in the binary system are represented to the base 2 and the positional multipliers are the powers of2. The left most bit in the binary number is called as the Most Significant Bit (MSB) and it has the largest positional weight. The right most bit is the Least Significant Bit (LSB) and has the smallest positional weight.

Octal Number System
Octal number system uses digits 0,1,2,3,4,5,6 and 7-(8 digits). Each octal digit.has its own positional value or weight as a power of 8.
Example
The Octal sequence (547)₈ has the decimal equivalent:

Hexadecimal Number System:
A hexadecimal number is represented using base 16. Hexadecimal or Hex numbers are used as a shorthand form of binary sequence. This system is used to represent data in a more compact manner. Since 16 symbols are used, 0 to F, the notation is called hexadecimal. The first 10 symbols are the same as in the decimal system, 0 to 9 and the remaining 6 symbols are taken from the first 6 letters of the alphabet sequence, A to F, where A represents 10, B is 11, C is 12, D is 13, E is 14 and F is 15.

101110101
Decimal Equivalent;
= 1 x 2⁸ x 0x2⁷ x 1 x2⁶ + 1 x2⁵ +1 x 2⁴ +0x2⁵ + 1 x 2² + 0x2¹ + 1 x 2⁰ 
= 256 + 64 + 32 + 16 + 4 + 1 = 373₁₀
Octal Equivalent;
= \(\overline { \overset { 101 }{ \overset { \downarrow }{ 5 } } } \) \(\overline { \overset { 110 }{ \overset { \downarrow }{ 6 } } } \) \(\overline { \overset { 101 }{ \overset { \downarrow }{ 5 } } } \)
=565₈
Hexadecimal Equivalent;
= \(\overline { \overset { 10 }{ \overset { \downarrow }{ 1 } } } \) \(\overline { \overset { 1110 }{ \overset { \downarrow }{ 7 } } } \)\(\overline { \overset { 101 }{ \overset { \downarrow }{ 5 } } } \)
=175₁₆
= 10110101₂ = 373₁₀ =565₈ =175₁₆

6247₈
\(\overset { 6 }{ \underset { 110 }{ \downarrow } } \) \(\overset { 2 }{ \underset { 010 }{ \downarrow } } \) \(\overset { 4 }{ \underset { 100 }{ \downarrow } } \) \(\overset { 7 }{ \underset { 111 }{ \downarrow } } \)
6247₈ = 110010100111₂

145₈
= \(\overset { 1 }{ \underset { 1 }{ \downarrow } } \) \(\overset { 4 }{ \underset { 100 }{ \downarrow } } \) \(\overset { 5 }{ \underset { 101 }{ \downarrow } } \)
= 145₈ = 1100101₂