Decimal and Binary Number System

**Category : **JEE Main & Advanced

(1) **Decimal number system :** In a decimal number system, we have ten digits i.e. 0, 1, 2, 3, 4, 5, 6, 7, 8, 9.

A decimal number system has a base of ten (10)

e.g. \[=1000+900+70+1\] \[=1\times {{10}^{3}}+9\times {{10}^{2}}+7\times {{10}^{1}}+1\times {{10}^{0}}\]

LSD = Least significant digit

MSD = Most significant digit

(2) **Binary number system :** A number system which has only two digits i.e. 0 (Low) and 1 (High) is known as binary system. The base of binary number system is 2.

(i) Each digit in binary system is known as a bit and a group of bits is known as a byte.

(ii) The electrical circuit which operates only in these two state i.e. 1 (On or High) and 0 (i.e. Off or Low) are known as digital circuits.

**Different names for the digital signals**

State Code |
1 | 0 |

Name for the State | On | Off |

Up | Down | |

Close | Open | |

Excited | Unexcited | |

True | False | |

Pulse | No pulse | |

High | Low | |

Yes | No |

(3) **Decimal to binary conversion **

(i) Divide the given decimal number by 2 and the successive quotients by 2 till the quotient becomes zero.

(ii) The sequence of remainders obtained during divisions gives the binary equivalent of decimal number.

(iii) the most significant digit (or bit) of the binary number so obtained is the last remainder and the least significant digit (or bit) is the first remainder obtained during the division.

**For Example :** Binary equivalence of 61

2 | 61 | Remainder |

2 | 30 | 1 LSD |

2 | 15 | 0 |

2 | 7 | 1 |

2 | 3 | 1 |

2 | 1 | 1 |

0 | 1 MSD |

\[\Rightarrow \] \[{{(61)}_{10}}={{(111101)}_{2}}\]

(4) **Binary to decimal conversion :** The least significant digit in the binary number is the coefficient of 2 with power zero. As we move towards the left side of LSD, the power of 2 goes on increasing.

**For Example :**

\[{{(11111100101)}_{2}}=~1\times {{2}^{10}}+1\times {{2}^{9}}+1\times {{2}^{8}}+1\times {{2}^{7}}+1\times {{2}^{6}}+1\times {{2}^{5}}+0\times {{2}^{4}}+0\times {{2}^{3}}+1\times {{2}^{2}}+0\times {{2}^{1}}+1\times {{2}^{0}}=2021\]

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