The number system is a form of expressing the numbers. In **number system conversion**, we will study to convert a number of one base, to a number of another base. There are a variety of different number systems such as binary numbers, decimal numbers, hexadecimal numbers, octal numbers, which can be exercised. The general representation of number systems are:

Decimal Number – Base 10 – N_{10}

Binary Number – Base 2 – N_{2}

Octal Number – Base 8 – N_{8}

Hexadecimal Number – Base 16 – N_{16}

Number system conversions deal with the operations to change the base of the numbers. For example, to change a decimal number with base 10 to binary number with base 2. We can also perform the arithmetic operations like addition, subtraction, multiplication on the number system.

Here, we will learn the methods to convert the number of one base to the number of another base starting with decimal number systems. The representation of number system base conversion in general form for any base number is;

(Number)_{b} = d_{n-1 }d_{n-2}—–_{.}d_{1} d_{0 . }d_{-1 }d_{-2 }—- d_{-m}

In the above expression, d_{n-1 }d_{n-2}—–_{.}d_{1} d_{0 }represents the value of integer part and d_{-1 }d_{-2 }—- d_{-m} represents the fractional part.

Also, d_{n-1 } is the Most significant bit (MSB) and d_{-m } is the Least significant bit (LSB).

Now let us learn, conversion from one base to another.

Converting decimal number to other base numbers is easy. We have to divide the decimal number by the converted value of the new base. Suppose if we have to convert a decimal number to binary number, then divide the decimal number by 2.

**Example**

**1. **Convert (25)_{10 }to binary number.

**Solution: **Let us create a table based on this question.

Operation | Output | Remainder |

25 ÷ 2 | 12 | 1(MSB) |

12 ÷ 2` | 6 | 0 |

6 ÷ 2 | 3 | 0 |

3 ÷ 2 | 1 | 1 |

1 ÷ 2 | 0 | 1(LSB) |

Therefore, from the above table, we can write,

(25)_{10 }= (11001)_{2}

**In the same way, we can convert decimal numbers to other base numbers such as the octal number and hexadecimal number by dividing the decimal number from 8 and 16 respectively.**

In this conversion, binary number to a decimal number, we use multiplication method, in such a way that, if a number with base n has to be converted into a number with base 10, then each digit of the given number is multiplied from MSB to LSB with reducing the power of the base. Let us understand this conversion with the help of an example.

**Example**

**2.** Convert (1101)_{2 } into decimal number.

**Solution:** Given a binary number (1101)_{2}.

Now, multiplying each digit from MSB to LSB with reducing the power of the base number 2.

1 × 2^{3} + 1 × 2^{2 }+ 0 × 2^{1} + 1 × 2^{0}

= 8 + 4 + 0 + 1

= 13

Therefore, (1101)_{2 } = (13)_{10}

**In the same way, you can convert the octal number and hexadecimal number to decimal number system by multiplying each digit with decreasing power of the base number of 8 and 16 respectively.**

To convert hexadecimal numbers to binary and vice versa is easy, you just have to memorize the table given below.

Hexadecimal Number |
Binary |

0 | 0000 |

1 | 0001 |

2 | 0010 |

3 | 0011 |

4 | 0100 |

5 | 0101 |

6 | 0110 |

7 | 0111 |

8 | 1000 |

9 | 1001 |

A | 1010 |

B | 1011 |

C | 1100 |

D | 1101 |

E | 1110 |

F | 1111 |

You can easily solve the problems based on hexadecimal and binary conversions with the help of this table. Let us take an example.

**Example**

**3.** Convert (89)_{16 } into a binary number.

**Solution:** From the table, we can get the binary value of 8 and 9, hexadecimal base numbers.

8 = 1000 and 9 = 1001

Therefore, (89)_{16 }= (10001001)_{2}

Just like having a table for hexadecimal and its equivalent binary, in the same way, we have a table for octal and its equivalent binary number.

Octal Number |
Binary |

0 | 000 |

1 | 001 |

2 | 010 |

3 | 011 |

4 | 100 |

5 | 101 |

6 | 110 |

7 | 111 |

**Example**

**4.** Convert (214)_{8 }into a binary number.

Solution: From the table, we know,

2 → 010

1 → 001

4 → 100

Therefore,(214)_{8} = (010001100)_{2}