UNIT-3 Number System

3.1.Decimal, Binary, Octal, Hexadecimal Number System & conversion

3.2.9's and 10's complements decimal subtraction

3.3.Calculation in Binary – addition, subtraction, One’s and Two’s Complement Methods of binary subtraction.

Number System

A way to representing a number with specific pattern and specific base is called number system. For example: (3450)₈, (110110)₂, (34FA)_H etc.

Types of Number system

1) Non-positional number system.

2) Positional number system.

Non-positional number system is such type of number system which is not used any types of arithmetic calculation. For example: a roman number system.

Positional number system is such type of number system which is used in any types of arithmetic calculation. For example: a Binary, Quainary, Octal, Decimal and Hexadecimal number system etc.

Types of Positional number system

i. Binary Number System: - The numbers having from 0 to 1 and base is 2 is called binary number system. It is used in mathematical and digital system. In computer, binary 0 means off/false/low and binary 1 means on/true/high. For example: (1100)₂, (00101)₂ etc.

ii. Quinary Number System: - The numbers having from 0 to 4 and base is 5 is called quinary number system. For example: (234)₅, (02303)₅ etc.

iii. Octal Number System: - The numbers having from 0 to7 and base is 8 is called octal number system. The word octal is taken from Latin octo or "eight". It is used in some programming language. For example: (2450)₈, (745602)₈ etc.

iv. Decimal Number System: - The numbers having from 0 to 9 and base is 10 is called decimal number system. It is used in all financial institutions and daily activities. For example: (2490)₁₀, (78902)₁₀ etc.

v. Hexadecimal Number System: - The numbers having from 0 to9 & A to F and base is 16 or H is called hexadecimal number system. It is used for representing addresses of memory locations in microprocessors and assembly language programming. For example: (24C9)_H, (745DF)₁₆ etc.

Some Basic terms:

i. Bit: - The bit stands for binary digit. Each digit in a binary number system is called a bit. For example 0 or 1.

ii. Nibble: - A group of 4-bits is called nibble or one nibble.

iii. Byte: - A group of 8-bits or 2-nibbles is called Byte.

iv. Word: -A group of 2-bytes is called word or one word.

v. Base: - A base is a type of digit which represents a type of number system. It is also called radix.

Binary Arithmetic

Binary arithmetic is essential part of all the digital computers and many other digital systems.

Binary Addition

It is a key for binary subtraction, multiplication, division. There are four rules of binary addition.

In fourth case, a binary addition is creating a sum of (1 + 1 = 10) i.e. 0 is written in the given column and a carry of 1 over to the next column.

Example − Addition

Binary Subtraction

Subtraction and Borrow, these two words will be used very frequently for the binary subtraction. There are four rules of binary subtraction.

Example − Subtraction

Binary Multiplication

Binary multiplication is similar to decimal multiplication. It is simpler than decimal multiplication because only 0s and 1s are involved. There are four rules of binary multiplication.

Example − Multiplication

Binary Division

Binary division is similar to decimal division. It is called as the long division procedure.

Example − Division

Conversion

Decimal Number Conversion

A repeated division and remainder algorithm can convert decimal to binary, octal, or hexadecimal.

Divide the decimal number by the desired target radix (2, 8, or 16).
Append the remainder as the next most significant digit.
Repeat until the decimal number has reached zero.

Decimal to Binary

Here is an example of using repeated division to convert 1792 decimal to binary:

Decimal Number	Operation	Quotient	Remainder	Binary Result
1792	÷ 2 =	896	0	0
896	÷ 2 =	448	0	00
448	÷ 2 =	224	0	000
224	÷ 2 =	112	0	0000
112	÷ 2 =	56	0	00000
56	÷ 2 =	28	0	000000
28	÷ 2 =	14	0	0000000
14	÷ 2 =	7	0	00000000
7	÷ 2 =	3	1	100000000
3	÷ 2 =	1	1	1100000000
1	÷ 2 =	0	1	11100000000
0	done.

Decimal to Octal

Here is an example of using repeated division to convert 1792 decimal to octal:

Decimal Number	Operation	Quotient	Remainder	Octal Result
1792	÷ 8 =	224	0	0
224	÷ 8 =	28	0	00
28	÷ 8 =	3	4	400
3	÷ 8 =	0	3	3400
0	done.

Decimal to Hexadecimal

Here is an example of using repeated division to convert 1792 decimal to hexadecimal:

Decimal Number	Operation	Quotient	Remainder	Hexadecimal Result
1792	÷ 16 =	112	0	0
112	÷ 16 =	7	0	00
7	÷ 16 =	0	7	700
0	done.

The only addition to the algorithm when converting from decimal to hexadecimal is that a table must be used to obtain the hexadecimal digit if the remainder is greater than decimal 9.

Decimal:	0	1	2	3	4	5	6	7
Hexadecimal:	0	1	2	3	4	5	6	7

Decimal:	8	9	10	11	12	13	14	15
Hexadecimal:	8	9	A	B	C	D	E	F

Hexadecimal to Binary

Converting from hexadecimal to binary is as easy as converting from binary to hexadecimal. Simply look up each hexadecimal digit to obtain the equivalent group of four binary digits.

Hexadecimal:	0	1	2	3	4	5	6	7
Binary:	0000	0001	0010	0011	0100	0101	0110	0111

Hexadecimal:	8	9	A	B	C	D	E	F
Binary:	1000	1001	1010	1011	1100	1101	1110	1111

Hexadecimal =	A	2	D	E
Binary =	1010	0010	1101	1110	= 1010001011011110 binary

Hexadecimal to Octal

When converting from hexadecimal to octal, it is often easier to first convert the hexadecimal number into binary and then from binary into octal. For example, to convert A2DE hex into octal:

(from the previous example)

Hexadecimal =	A	2	D	E
Binary =	1010	0010	1101	1110	= 1010001011011110 binary

Add leading zeros or remove leading zeros to group into sets of three binary digits.

Binary: 1010001011011110 = 001 010 001 011 011 110

Then, look up each group in a table:

Binary:	000	001	010	011	100	101	110	111
Octal:	0	1	2	3	4	5	6	7

Binary =	001	010	001	011	011	110
Octal =	1	2	1	3	3	6	= 121336 octal

Therefore, through a two-step conversion process, hexadecimal A2DE equals binary 1010001011011110 equals octal 121336.

Labels: CH-3

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Thursday, March 3, 2016