Decimal (base-10) to binary (base-2) conversion:

Step 1: Divide the decimal number by 2 (2 comes from the base value of binary number system).

Step 2: The remainder is the LSB of the binary number.

Step 3: Divide the quotient from*Step 1* by 2 again.

Step 4: The new remainder from*Step 3* is the next digit (from right to left) of the binary number.

Step 5: The process continues until we get 0 as the quotient. The remainder from this step is the MSB of the binary number.

Step 2: The remainder is the LSB of the binary number.

Step 3: Divide the quotient from

Step 4: The new remainder from

Step 5: The process continues until we get 0 as the quotient. The remainder from this step is the MSB of the binary number.

Decimal number: 45_{10}

Numerator | Denominator | Quotient | Remainder |
---|---|---|---|

45 | 2 | 22 | 1 (LSB) |

22 | 2 | 11 | 0 |

11 | 2 | 5 | 1 |

5 | 2 | 2 | 1 |

2 | 2 | 1 | 0 |

1 | 2 | 0 | 1 (MSB) |

So, the binary representation: 101101_{2}

Decimal fraction to binary (base-2) conversion:

Step 1: Multiply the fraction number by 2 (2 comes from the base value of binary number system).

Step 2: The result (without the fraction part if any) is the MSB of the binary number.

Step 3: Take only the fraction part from*Step 2* and multiply it by 2 again.

Step 4: The result (except the fraction part if any) from*Step 3* is the next digit (from left to right) of the binary number.

Step 5: The process continues until we get X.00 as the final result (here X = 0 or 1). From this step X is the LSB of the binary number.

Step 2: The result (without the fraction part if any) is the MSB of the binary number.

Step 3: Take only the fraction part from

Step 4: The result (except the fraction part if any) from

Step 5: The process continues until we get X.00 as the final result (here X = 0 or 1). From this step X is the LSB of the binary number.

Decimal number: 0.625_{10}

Steps | Multiplicand | Multiplier | Result (without fraction part) | Result (only the fraction part) |
---|---|---|---|---|

1 | .625 | 2 | 1 (MSB) | .25 |

2 | .25 | 2 | 0 | .5 |

3 | .5 | 2 | 1 (LSB) | .0 |

So, the binary representation: .101_{2}

Decimal fraction to infinite binary (base-2) number conversion:

We will follow the same procedures explained in *decimal fraction to binary (base-2) conversion*, but here we will continue the steps until we obtain a repetition of a sequence of digits. If there's no repetition of digits, then we will stop after calculating up to our required number of digits.

Decimal number: 0.975_{10}

Steps | Multiplicand | Multiplier | Result (without fraction part) | Result (only the fraction part) |
---|---|---|---|---|

1 | .975 | 2 | 1 (MSB) | .95 |

2 | .95 | 2 | 1 | .9 |

3 | .9 | 2 | 1 | .8 |

4 | .8 | 2 | 1 | .6 |

5 | .6 | 2 | 1 | .2 |

6 | .2 | 2 | 0 | .4 |

7 | .4 | 2 | 0 | .8 |

8 | .8 | 2 | 1 | .6 |

9 | .6 | 2 | 1 | .2 |

10 | .2 | 2 | 0 | .4 |

11 | .4 | 2 | 0 | .8 |

From the above table we can observe that steps 4-7 are repeated again and again.

So, the binary representation: .1111100110011001100....._{2}

Or, binary representation:*.111**1100*_{2}

So, the binary representation: .1111100110011001100.....

Or, binary representation:

Decimal: 45.625_{10}

Binary: 101101.101_{2}

A radix point [.] is used to separate the integer part (to the left of the radix point) of a number from its fractional part (to the right of the radix point).

Decimal: 45.975_{10}

Binary: 101101.1111100110011001100....._{2}

*Or, 101101.111**1100*_{2}

Binary (base-2) to decimal (base-10) conversion:

From right (LSB) to left (MSB) we will take each digit and multiply them by 2^{x} and later add the results together. Here x is the corresponding position of a digit and 2 comes from the base value of binary number system.

Binary number: 101101_{2}

Number: | 1 (MSB) | 0 | 1 | 1 | 0 | 1 (LSB) |
---|---|---|---|---|---|---|

Position: | 5 | 4 | 3 | 2 | 1 | 0 |

(1 x 2^{5}) + (0 x 2^{4}) + (1 x 2^{3}) + (1 x 2^{2}) + (0 x 2^{1}) + (1 x 2^{0})

= 32 + 0 + 8 + 4 + 0 + 1

= 45

So, the decimal representation: 45_{10}

= 32 + 0 + 8 + 4 + 0 + 1

= 45

So, the decimal representation: 45

Binary fraction to decimal fraction conversion:

From left (MSB) to right (LSB) we will take each digit and multiply them by 2^{x} and later add the results together. Here x is the corresponding position of a digit and 2 comes from the base value of binary number system.

Binary number: .101_{2}

Number: | 1 (MSB) | 0 | 1 (LSB) |
---|---|---|---|

Position: | -1 | -2 | -3 |

(1 x 2^{-1}) + (0 x 2^{-2}) + (1 x 2^{-3})

= 0.5 + 0 + 0.125

= 0.625

So, the decimal representation: 0.625_{10}

= 0.5 + 0 + 0.125

= 0.625

So, the decimal representation: 0.625

Binary: 101101.101_{2}

Decimal: 45.625_{10}