- A number system is a way of writing any number.
- It can be represented by using any symbols or digits.
- It provides distinctive representation for each number.
- We can calculate value of any digit with the help of its position and base value.

## Decimal Number System

- This is the most commonly used number system with base value of ten.
- It uses ten digits from 0 to 9.
- The leftmost position from decimal is units, tens, hundred and so on.
- The number 5312 has 2 in units place, 1 in tens place, 3 in hundreds place and 5 in thousands place.
- We can also express the number as

(5×1000) + (3×100) + (1×10) + (2×l)

=(5×10^{3}) + (3×10^{2}) + (1×10^{1}) + (2×l0^{0})

=5000 + 300 + 10 + 2

=5312

## Binary Number System

- This number system has only two digits 0 and 1.
- The base of the binary number system is 2.
- The position of each number is the power of the base.

Convert 10101_{2} to decimal number?

Step | Binary Number | Decimal Number |

Step 1 | 10101_{2} | ((1 × 2^{4}) + (0 × 2^{3}) + (1 × 2^{2}) + (0 × 2^{1}) + (1 × 2^{0}))_{10} |

Step 2 | 10101_{2} | (16 + 0 + 4 + 0 + 1)_{10} |

Step 3 | 10101_{2} | 21_{10} |

**Note:** We can write 10111_{2} as 10111.

## Octal Number System

- The base value of this number system is eight.
- It has total eight digits from 0 to 7.
- Therefore we call it the octal number systems.
- The position of each number is the power of the base.

Convert 12569_{8} to decimal number?

Step | Octal Number | Decimal Number |

Step 1 | 12569_{8} | ((1 × 8^{4}) + (2 × 8^{3}) + (5 × 8^{2}) + (6 × 8^{1}) + (9 × 8^{0}))_{10} |

Step 2 | 12569_{8} | (4096 + 1024 + 320 + 48 + 9)_{10} |

Step 3 | 12569_{8} | 5497_{10} |

**Note:** We can write 12575_{8} as 12575 in octal.

## Hexadecimal Number Systems

- This number system consists of ten digits and six letters.
- The digits are 0-9 and letters A-F.
- The value of these letters in the form of numbers is given as A = 10, B = 11, C = 12, D = 13, E = 14, F = 15.
- The base value of this number system is sixteen.
- The position of each number is equal to the power of the base.

Convert 19FBA_{16} to decimal number?

Step | Hexadecimal Number | Decimal Number |

Step 1 | 19FBA_{16} | ((1 × 16^{4}) + (9 × 16^{3}) + (F × 16^{2}) + (B × 16^{1}) + (A × 16^{0}))_{10} |

Step 2 | 19FBA_{16} | ((1 × 16^{4}) + (9 × 16^{3}) + (15 × 16^{2}) + (11 × 16^{1}) + (10 × 16^{0}))_{10} |

Step 3 | 19FBA_{16} | (65536 + 36864 + 3840 + 176 + 10)_{10} |

Step 4 | 19FDA_{16} | 106426_{10} |

**Note:** We can write 19FDA_{16} as 19FDA in hexadecimal.

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