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Inorder to represent a set of quantities, a value set is required. This set of values is known as **number system**. In every number system, a number can be represented by:

*d _{n}*

*b* – Base..

*n* - Can start from negative number if the number has a fraction part.

*N*+1 - number of digits.

There are different number system, they are:

- Binary Number System
- Decimal Number System
- Hexadecimal Number System
- Octal Number System

*Binary Number System*

0 and 1 are the two digits in a binary number system. This number system comes with a base of 2.

*Decimal Number System*

Digits from 0-9 come under the decimal number system. These are the numbers that we commonly use. They have the base of 10.

*Hexadecimal Number System*

Hexadecimal number system uses digits 0-9 and A-F. This number system comes with a base of 16.

*Octal Number System*

This number system uses digits from 0-7 and has a base of 8.

*Steps*

- Find the positional value of each digit of the binary number.
- Start finding the positional value from the least significant digit to the most significant digit.
- Multiply the positional value with the respective digits.
- Now take the sum.
- Final obtained value is the Decimal Number.

*Steps*

- Divide the binary number into groups containing three digits. Start the grouping from right.
- Convert each group into a octal digit.

*Steps*

- Divide the binary number into groups containing four digits. Start the grouping from right.
- Convert each group into a hexadecimal digit.

*Steps*

- Convert each octal digit into a three digit binary.
- Combine them, the binary equivalent is obtained.

*Steps*

- Convert each hexadecimal digit into a four digit binary.
- Combine them, the binary equivalent is obtained.

**For Natural Numbers**

*Steps*

- Divide the decimal number with the base. Base value for binary, octal, hexadecimal is 2,8,16.
- Get the remainder of the step1. It is the LEAST SIGNIFICANT DIGIT.
- Now take the quotient of the step1.
- Divide quotient with the base.
- Record the remainder. This is the next digit.
- Repeat steps 4 and 5 until the quotient becomes zero. The last remainder obtained will be the MOST SIGNIFICANT DIGIT.

So, Binary Number= (1000100)_{2}

**For decimal numbers**

*Steps*

* *

- In a decimal number we know, it has WHOLE NUMBER PART and DECIMAL PART.
- The whole number part is converted to binary by division method. While the decimal part is converted to binary by using multiplication method.
- As given in the figure; if the actual number is 5.9. Then its DECIMAL PART is 0.9. WHOLE NUMBER PART is 5.

*StepA: First take the DECIMAL PART*

- Multiply the DECIMAL PART with base. Base value for binary, octal, hexadecimal is 2,8,16.
- The RESULT from step 4 will be another decimal number. Its whole number part will be the MOST SIGNIFICANT DIGIT in StepA.
- Now again take the result of step 4.
- Multiply result excluding the whole number part with the base.
- Get the result of step 7. Its whole number part will be the next digit.
- Now take step7 result and perform Step7 and 8.till you reach 0.00.
- You can continue the above process till you get 0.00. But 4 to 5 decimal places are taken here.

*StepB: Now take the WHOLE NUMBER PART*

- Divide the WHOLE NUMBER PART with base. Base value for binary, octal, hexadecimal is 2,8,16.
- Get the remainder of the step10. It is the LEAST SIGNIFICANT DIGIT in StepB.
- Now take the quotient of the step10.
- Divide quotient with the base.
- Record the remainder. This is the next digit.
- Repeat steps 4 and 5 until the quotient becomes zero. The last remainder obtained will be the MOST SIGNIFICANT DIGIT IN StepB.

At last, the binary number must be written as; (binary) _{WHOLE NUMBER PART}, DECIMAL POINT, (binary)_{ DECIMAL} _{PART}.

Consider the number (24.68)_{10}

DECIMAL PART= 0.68. Follow steps below.

- You can continue the above process till you get 0.00. But 4 to 5 decimal places are taken here.
- So the binary part is 0.1010111…

WHOLE NUMBER PART= 24. Follow steps below.

So the binary of the whole number part is = (11000)_{2}

Finally, the binary of (24.68)_{10 }= 11000. 1010111…

*Steps*

- Find the positional value of each digit.
- Start finding the positional value from the least significant digit to the most significant digit.
- Multiply the positional value with the respective digits.
- Now take the sum.
- Final obtained value is the Decimal Number

*Steps*

- Find the positional value of each digit.
- Start finding the positional value from the least significant digit to the most significant digit.
- Multiply the positional value with the respective digits.
- Now take the sum.
- Final obtained value is the Decimal Number.

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