Info-Grade: Number System

Number System (Binary Calculation)

A number system is a system for expressing numbers that is mathematical notation used for manipulating other countable things. The numbers are represented in different based system in different situation. The decimal number system is used for human counting but not role in machines. Binary numbers are used in modern digital devices. Octal and Hexadecimal number system are just packages of 4 binary bits. In computer science we have to study four number systems. They are

Binary Number System (0, 1) / Base or radix = 2
Octal Number System (0 to 7) Base or radix = 8
Decimal Number System (0 to 9) Base or radix = 10
Hexadecimal Number System (0 to 9 and A to F) Base or radix = 16

The following table shows the number system equivalent to one another.

Binary	Octal	Decimal	Hexadecimal
0000	0	0	0
0001	1	1	1
0010	2	2	2
0011	3	3	3
0100	4	4	4
0101	5	5	5
0110	6	6	6
0111	7	7	7
1000	10	8	8
1001	11	9	9
1010	12	10	A
1011	13	11	B
1100	14	12	C
1101	15	13	D
1110	16	14	E
1111	17	15	F

Binary Number System : Any number having base or radix 2 and the number consists 0 and 1 only are called binary number system. E.g: 1110₂, 1001₂, 111001₂
Decimal Number System: Any number having base or radix 8 and the number consists 0, 1, 2, 3, 4, 5, 6 and 7 are called octal number system. E.g: 563₈, 1235₈, 5736₈
Decimal Number System: Any number having base or radix 10 and the numbers consists 0, 1, 2, ........ 9 are called decimal number system. E.g : 196₁₀, 258₁₀, 1236₁₀
Hexadecimal Number System : Any number having base or radix 16 and the number consists 0, 1, 2, 3, ...... 9 and character consist A, B, C, D, E and F are called hexadecimal number system. Where A=10, B=11, C=13, D=14, E=15 and F=15. E.g : 25A₁₆, 156B₁₆, 9D₁₆

Binary Calculation

The modern digital computer system performs every calculations in the form of binary format. They present 0 or 1, on or off, presence and absent. It represents the status.

Binary Addition:

Binary addition is performed in a similar way to decimal addition.

Binary Addition
X	Y	X+Y
0	0	0
0	1	0
1	0	0
1	1	10 (with a carry of 1)

Example: a) Add 11010₂+ 1001₂ b) 11110₂+ 11101₂

Binary Subtraction:

Binary subtraction, similar method is adopted as in decimal system.

Binary Subtraction
X	Y	X-Y
0	0	0
0	1	1 (with a borrow of 1)
1	0	1
1	1	0

Example : a) 10010 - 1011 b) 1101101 - 111101

Binary Multiplication

Binary multiplication is easier since there is no number as "carry" value. The following rule is adopted in binary multiplication. The values are added column wise as in binary addition.

Binary Multiplication
X	Y	X*Y
0	0	0
0	1	0
1	0	0
1	1	1

Example: a) 11101 x 111 b) 100011 x 110

Binary Division

Binary Division is similar to decimal division. If a number cannot be divided, put 0 to the quotient. If division is possible put 1 to the quotient. Multiplication and subtraction is as discussed in binary multiplication and binary subtraction.

Example: a) Divide 1011 by 11 b) 111 by 11