Number System

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:

Number system representation

dn -n-th digit.

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:

  1. Binary Number System
  2. Decimal Number System
  3. Hexadecimal Number System
  4. 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.

Number System Conversion

Binary to Decimal Conversions

Steps

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

Binary to Decimal

Binary to Octal

Steps

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

 Binary to Octal

Binary to Hexadecimal

Steps

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

 Binary to Hexadecimal

Octal to Binary

Steps

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

 Octal to Binary

Hexadecimal to Binary

Steps

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

 Hexadecimal to Binary

Decimal to Other Base Systems (Binary, Octal, Hexadecimal)

  • For Natural Numbers

Steps

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

  decimal to binary

So, Binary Number= (1000100)2

  • For decimal numbers

Steps

                                                           decimal number

  1. In a decimal number we know, it has WHOLE NUMBER PART and DECIMAL PART.
  2. The whole number part is converted to binary by division method. While the decimal part is converted to binary by using multiplication method.
  3. 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

  1. Multiply the DECIMAL PART with base. Base value for binary, octal, hexadecimal is 2,8,16.
  2. The RESULT from step 4 will be another decimal number. Its whole number part will be the MOST SIGNIFICANT DIGIT in StepA.
  3. Now again take the result of step 4.
  4. Multiply result excluding the whole number part with the base.
  5. Get the result of step 7. Its whole number part will be the next digit.
  6. Now take step7 result and perform Step7 and 8.till you reach 0.00.
  7. 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

  1. Divide the WHOLE NUMBER PART with base. Base value for binary, octal, hexadecimal is 2,8,16.
  2. Get the remainder of the step10. It is the LEAST SIGNIFICANT DIGIT in StepB.
  3. Now take the quotient of the step10.
  4. Divide quotient with the base.
  5. Record the remainder. This is the next digit.
  6. 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.

 decimal part

  • 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.

  whole number part

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

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

Octal to Decimal Conversions

Steps

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

  octal to decimal

Hexadecimal to Decimal Conversions

Steps

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

?Hexa decimal to decimal