**Number systems** are the technique to represent numbers in the computer system architecture, every value that you are saving or getting into/from computer memory has a defined number system.

Computer architecture supports following number systems.

**Binary number system****Octal number system****Decimal number system****Hexadecimal (hex) number system**

A Binary number system has only two digits that are **0 and 1**. Every number (value) represents with 0 and 1 in this number system. The base of binary number system is 2, because it has only two digits.

Octal number system has only eight (8) digits from** 0 to 7**. Every number (value) represents with 0,1,2,3,4,5,6 and 7 in this number system. The base of octal number system is 8, because it has only 8 digits.

Decimal number system has only ten (10) digits from** 0 to 9**. Every number (value) represents with 0,1,2,3,4,5,6, 7,8 and 9 in this number system. The base of decimal number system is 10, because it has only 10 digits.

A Hexadecimal number system has sixteen (16) alphanumeric values from **0 to 9** and **A to F**. Every number (value) represents with 0,1,2,3,4,5,6, 7,8,9,A,B,C,D,E and F in this number system. The base of hexadecimal number system is 16, because it has 16 alphanumeric values. Here **A is 10**, **B is 11**, **C is 12**, **D is 13**, **E is 14** and **F is 15**.

**Table of the Numbers Systems with Base, Used Digits, Representation, C language representation:**

Number system | Base | Used digits | Example | C Language assignment |

Binary | 2 | 0,1 | (11110000)_{2} |
int val=0b11110000; |

Octal | 8 | 0,1,2,3,4,5,6,7 | (360)_{8} |
int val=0360; |

Decimal | 10 | 0,1,2,3,4,5,6,7,8,9 | (240)_{10} |
int val=240; |

Hexadecimal | 16 | 0,1,2,3,4,5,6,7,8,9, A,B,C,D,E,F |
(F0)_{16} |
int val=0xF0; |

**Decimal Number System to Other Base**

[for example: Decimal Number System to Binary Number System]**Other Base to Decimal Number System**

[for example: Binary Number System to Decimal Number System]**Other Base to Other Base**

[for example: Binary Number System to Hexadecimal Number System]

To convert Number system from **Decimal Number System** to **Any Other Base **is quite easy; you have to follow just two steps:

**A)** Divide the Number (Decimal Number) by the base of target base system (in which you want to convert the number: Binary (2), octal (8) and Hexadecimal (16)).

**B)** Write the remainder from step 1 as a Least Signification Bit (LSB) to Step last as a Most Significant Bit (MSB).

Decimal to Binary Conversion | Result |

Decimal Number is : (12345) _{10} |
Binary Number is(11000000111001)
_{2} |

Decimal to Octal Conversion | Result |

Decimal Number is : (12345) _{10} |
Octal Number is(30071)
_{8} |

Decimal to Hexadecimal Conversion | Result |

Example 1Decimal Number is : (12345) _{10} |
Hexadecimal Number is(3039)
_{16} |

Example 2Decimal Number is : (725) _{10} |
Hexadecimal Number is(2D5)
_{16}Convert
10, 11, 12, 13, 14, 15 to its equivalent... A, B, C, D, E, F |

To convert Number System from **Any Other Base System** to **Decimal Number System**, you have to follow just three steps:

**A)** Determine the base value of source Number System (that you want to convert), and also determine the position of digits from LSB (first digit’s position – 0, second digit’s position – 1 and so on).

**B)** Multiply each digit with its corresponding multiplication of position value and Base of Source Number System’s Base.

**C)** Add the resulted value in step-B.

*Explanation regarding examples:*

Below given exams contains the following rows:

**A)** __Row 1__ contains the **DIGITs** of number (that is going to be converted).

**B)** __Row 2__ contains the **POSITION** of each digit in the number system.

**C)** __Row 3__ contains the multiplication: **DIGIT* BASE^POSITION**.

**D)** __Row 4__ contains the calculated result of **step C**.

**E)** And then add each value of **step D**, resulted value is the Decimal Number.

Binary to Decimal Conversion | |

Binary Number is : (11000000111001) _{2} |

Octal to Decimal Conversion | Result |

Octal Number is : (30071) _{8} |
=12288+0+0+56+1=12345Decimal Number is: (12345)
_{10} |

Hexadecimal to Decimal Conversion | Result |

Hexadecimal Number is : (2D5) _{16} |
=512+208+5=725Decimal Number is: (725)
_{10} |

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