Number System

HOME- OUR MAIN WEBPAGE
ABOUT- Information about the following webpage
CONATCT US- Email for contact

Topics covered
1.Number System and it's use
2.Types of Number System
3.Binary Number System
4.Decimal Number System
5.Octal Number System
6.Hexadecimal Number System
7.Conversions
8.Conversion of Binary Number to Decimal Number
9.Conversion of Decimal Number to Binary Number
10.Conversion of Decimal Number to Octal Number
11.Conversion of Decimal Number to Hexadecimal Number
12.Conversion of Binary Number to Octal Number
13.Conversion of Binary Number to Hexadecimal Number
14.Binary Arithmetic
15.Binary Addition
16.Binary Subtraction
17.Binary Multiplication
18.Binary Division
19.Summary

What is number system and what are its use?

Number system is the technique to work with numbers. Arayabhatta is the father of the number system. Number system is also the way to represent numbers in the computer system. Every value that you are getting into/from computer memory has a defined number system. A numeral system which is also called as system of numeration is a process to express numbers. In day-to-day life we use the decimal number system. A computer can understand these combinations of numbers and responds accordingly. It is an organised way to represent numbers.

What are the types of number system?

There are four types of number system.
  1. Binary Number System
  2. Decimal Number System
  3. Octal Number System
  4. Hexadecimal Number System

    5. Binary Number System

It is the number system which has two states ON state and OFF state. The programs of instructions written in this number system can be directly understood by the computer without any language translatoes. ON state is represented by 1 and OFF state is represented by O. It is very easy to design, simple and accurate electronic circuit that operates with only two voltage levels. For this reason, almost every decimal number system uses the binary number system as the basic number system for its operations. Computer understands only the Binary numbers and uses this number system to represent data. Binary Number System was invented by a German mathematician, Gottfried Leibniz.

Decimal Number System

In the decimal number system, the numbers are represented with base 10. The way of denoting the decimal numbers with base 10 is also termed as decimal notation. This number system is widely used in computer applications. It is also called the base-10 number system which consists of 10 digits, such as, 0,1,2,3,4,5,6,7,8,9. Each digit in the decimal system has a position and every digit is ten times more significant than the previous digit. A number system which uses digits from 0 to 9 to represent a number with base 10 is the decimal system number. The number is expressed in base-10 where each value is denoted by 0 or first nine positive integers. Each value in this number system has the place value of power 10. It means the digit at the tens place is ten times greater than the digit at the unit place.

Octal Number System

Octal Number System has a base of eight and uses the numbers from 0 to 7. The octal numbers, in the number system, are usually represented by binary numbers when they are grouped in pairs of three. For example, an octal number 128 is expressed as 0010102 in the binary system, where 1 is equivalent to 001 and 2 is equivalent to 010. A number system which has its base as eight is called an Octal number system. It uses numbers from 0 to 7. Let us take an example, to understand the concept. As we said, any number with base 8 is an octal number like 248, 1098, 558, etc.

Hexadecimal Number System

The hexadecimal number system is a type of number system, that has a base value equal to 16. It is also pronounced sometimes as hex. Hexadecimal numbers are represented by only 16 symbols. These symbols or values are 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E and F. Each digit represents a decimal value. For example, D is equal to base-10 13. Hexadecimal number systems can be converted to other number systems such as binary number (base-2), octal number (base-8) and decimal number systems (base-10).

Coversions

Conversions in number system means to convert one number system to another number system, example to convert binary to decimal.
Here are some conversions and the way to convert them-

Binary Number to Decimal Number

We know that a positional value is given to every digit in the binary number. The positional value is in terms of powers of 2. The digits are multiplied with their positional values and the products are added to get the decimal number.
Binary Digit Positional Value
1 2 4
1 2 3
0 22
0 21
1 20
The bits in the given table are multiplied with the positional values and their products are added. =1x24+1x23+0x22+0x21+1x20
=16+8+0+0+1
=(25)10

Decimal Number to Binary Number

To convert a decimal number to a binary number, you need to divide the decimal number by 2 repeatedly and write the remainder on the right-hand side. This process should be continued till you get 0 quotient. To form binary number, you need to read the remainder from bottom to top.
Example:-
Convert(146)10 to binary number.
2 146
2 73 Reminder 0
2 36 Reminder 1
2 18 Reminder 0
2 9 Reminder 1
2 4 Reminder 0
2 2 Reminder 0
2 1 Reminder 0
0 Reminder 1
(146)10 = (10001010)2

Decimal Number to Octal Number

Converting a decimal number to octal involves dividing the decimal number by 8 repeatedly and noting down the remainders. Here's a step-by-step guide:
1.Divide the decimal number by 8.
2.Note down the remainder. This will be the least significant digit of the octal number.
3.Divide the quotient from the previous step by 8.
4.Note down the remainder again. This will be the next digit of the octal number, moving from right to left.
5.Repeat steps 3 and 4 until the quotient becomes zero.
6.Write down the remainders in reverse order. The last remainder you obtained will be the most significant digit of the octal number.
Example:-
 Convert (75)10 to octal number
8 75
8 9 3
8 1 1
0 1
(75)10 = (113)8

Decimal Number to Hexadecimal Number

You need to follow the given steps to convert decimal to hexadecimal number system.
1.First divide the number by 16.
2.Find the integer quotient for the number.
3.Get the remainder for the hexa digit
4.Repeat the division until the quotient is equal to 0.
5.Write the number in reverse order.
Example:-
 Convert (3562)10 to hexadecimal number
Decimal number Hexadecimal number
16 3562
16 222 10 A
16 13 14 E
16 0 13 D
(3562)10 = (DEA)16

Binary number to Octal Number

Converting a binary number to an octal number in a computer involves grouping the binary digits into sets of three, starting from the rightmost digit, and then converting each group into its octal equivalent.
Here's a step-by-step guide:-

1.Group the binary digits: Starting from the rightmost digit of the binary number, group the digits into sets of three. If the leftmost group has fewer than three digits, add leading zeros to make it a complete group.
2.Convert each group to octal: Now, convert each group of three binary digits into its octal equivalent.
Here is the table for reference:-
Binary numbers Octal numbers
000 0
001 1
010 2
011 3
100 4
101 5
110 6
111 7
3.Concatenate the octal digits: Once you have the octal equivalent for each group, simply concatenate them to get the final octal number.
Let's go through an example:-

Convert (101101101)2 to octal number
1.Grouping:-
>Group 1: 101 (no leading zeros needed)
>Group 2: 101 (no leading zeros needed)
>Group 3: 101 (no leading zeros needed)

2.Conversion:-
>Group 1: 101 (binary) = 5 (octal)
>Group 2: 101 (binary) = 5 (octal)
>Group 3: 101 (binary) = 5 (octal)

Concatenation:-
The octal digits are 555.
(101101101)2 = (555)8

Binary Number to Hexadecimal Number

Converting a binary number to a hexadecimal number in a computer follows a similar principle to converting to octal, but with groups of four binary digits instead of three.
Here's a step-by-step guide:-

1.Group the binary digits: Starting from the rightmost digit of the binary number, group the digits into sets of four. If the leftmost group has fewer than four digits, add leading zeros to make it a complete group.
2.Convert each group to hexadecimal: Now, convert each group of four binary digits into its hexadecimal equivalent.
Here is the table for reference:-
Binary numbers Hexadecimal numbers
0000 0
0001 1
0010 2
0011 3
0100 4
0101 5
0111 7
1000 8
1000 8
1001 9
1010 A
1011 B
1100 C
1101 D
1110 E
1111 F
3.Concatenate the hexadecimal digits: Once you have the hexadecimal equivalent for each group, simply concatenate them to get the final hexadecimal number.
Let's go through an example:-

Convert (1011010101)2 to hexadecimal number
1.Grouping the binary digits into sets of four: 10 1101 0101.
2.Converting each group to hexadecimal: 2 D 5.
3.Combine the hexadecimal digits: 2D5.
(1011010101)2 = (2D5)16

Binary Arithmetic

Binary Arithmetics operations are used to perform calculation on digital computers. There are four basic binary operations that can be performed using Binary Arithmetic, they are:-
  1. Binary Addition
  2. Binary Subtraction
  3. Binary Multiplication
  4. Binary Division
Let us learn about them one by one.

Binary Addition

This operation is used to add two binary numbers. You need to follow the four rules for the addition of binary digits.
Num1 Num2 Num1+Num2 Carry
0 0 0 0
0 1 1 0
1 0 1 0
1 1 0 1
In binary addition sum of 1+1=10 i.e. 0 is written in the sum column and a carry of 1 over to the carry column. And note that is there is number which has 1+1+1 than the answer will be 1 and 1 carry.
 Let's go through an example:-
(0011010)2+(0001100)2
0011010
+0001100
0100110
Answer-(0100110)2

Binary Subtraction

This operation is used to Subtract the two binary numbers. The rules which you need to follow for binary subtraction are as follows:-
Num1 Num2 Num1+Num2 Borrow
0 0 0 0
0 1 1 1
1 0 1 0
1 1 0 0
Borrow 1 is used when you have to Subtract 1 from 0.
Let's go through an example:-
0 10 -borrow
1100
-1010
0010
Answer-(0010)2

Binary Multiplication

In binary multiplication, you find out the product of two binary digits. The rules for binary multiplication are as follows:-
Num1 Num2 Num1 x Num2
0 0 0
0 1 0
1 0 0
1 1 1
Multiplication is always 0, when out of two inputs, one input is 0.
Let's go through an example:-
110010
x 100
----------------
x x 0 0 0 0 0
x 0 0 0 0 0 0
1 1 0 1 0 0 0
----------------
1 1 0 1 0 0 0
Answer-(1101000)2

Binary Division

The binary division helps to divide two binary digits. It has four parts divisor, dividend, quotient and remainder. The rules for binary division are as follows:-
Num1 Num2 Num1/Num2
0 0 Not defined
0 1 0
1 0 Not defined
1 1 1

Summary

Here is the Summary:- THANKS
FOR
READING