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Last modified on January 29th, 2024

A binary-to-decimal conversion is done to convert a binary number (base 2) to its equivalent decimal number (base 10) for simplicity in mathematical calculations.

The general formula for converting a binary into a decimal number in positional notation form can be written as:

(a_{n-1}….a_{3}a_{2}a_{1}a_{0})_{2} = (a_{n-1} × 2^{n-1}) + … + (a_{2} × 2^{2}) + (a_{1} × 2^{1}) + (a_{0} × 2^{0}), here a_{0}, a_{1}, a_{2}, …, a_{n-1} are digits of a binary number.

In the positional notation method, the value of a digit in a number is determined by the weight of its position.

Now, let us convert a binary number (11001)_{2} to its corresponding decimal step by step.

**Step 1:** While converting, we consider each digit in the binary number (also known as a bit) from right to left. Here, (11001)_{2} has 1 as its rightmost digit, then continues with 0, 0, 1, and 1 as we move left.

**Step 2: **Then, we multiply each binary digit with increasing powers of 2 from right to left, such that the rightmost digit is multiplied by the greatest power of 2.

⇒ 1 × 2^{4}, 1 × 2^{3}, 0 × 2^{2}, 0 × 2^{1}, and 1 × 2^{0}.

**Step 3:** On simplifying, 1 × 2^{4} + 1 × 2^{3} + 0 × 2^{2} + 0 × 2^{1} + 1 × 2^{0}, we get,

⇒ 16 + 8 + 0 + 0 + 1 = 25

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

**For Fractional Binary Numbers**

To convert a decimal binary number to its corresponding decimal, we follow the same steps for its integral part, while the fractional part is multiplied by negative powers of 2.

For example, on converting (0.0001)_{2} into its equivalent decimal, we get

⇒ 0 × 2^{0} + 0 × 2^{-1} + 0 × 2^{-2} + 0 × 2^{-3} + 1 × 2^{-4} = 0 + 0.0625 = 0.0625

**Find the decimal values of (1001) _{2} and (1111)_{2} using the positional notation method.**

Solution:

(1001)_{2} = 1 × 2^{3} + 0 × 2^{2} + 0 × 2^{1} + 1 × 2^{0} = 8 + 1 = 9

(1111)_{2} = 1 × 2^{3} + 1 × 2^{2} + 1 × 2^{1} + 1 × 2^{0} = 8 + 4 + 2 + 1 = 15

Thus, (1001)_{2} = (9)_{10} and (1111)_{2} = (15)_{10}.

**Find the decimal value of (11.01) _{2}.**

Solution:

(11.01)_{2} = 1 × 2^{1} + 1 × 2^{0} + 0 × 2^{-1} + 1 × 2^{-2} = 2 + 1 + 0 + 0.25 = 3.25

Thus, (11.01)_{2} = 3.25

Some binary numbers and their equivalent decimal numbers are in the following table:

Binary | Decimal |
---|---|

0 | 0 |

1 | 1 |

10 | 2 |

11 | 3 |

100 | 4 |

101 | 5 |

110 | 6 |

111 | 7 |

1000 | 8 |

1001 | 9 |

1010 | 10 |

1011 | 11 |

1100 | 12 |

1101 | 13 |

1110 | 14 |

1111 | 15 |

10000 | 16 |

10001 | 17 |

10010 | 18 |

10011 | 19 |

10100 | 20 |

Using this method, let us consider the previous binary number (11001)_{2} and convert it into a decimal.

**Step 1: **Unlike the first method, we consider each digit of the binary number from left to right. Here, (11001)_{2} has 1 at the leftmost.

**Step 2:** Now, we multiply 0 by 2 and add the leftmost digit of the binary number.

0 × 2 + 1 = 1

**Step 3:** Using the above result and multiplying it by 2, we get 1 × 2 = 2

On adding 2 and the second leftmost digit, we get 2 + 1 = 3

**Step 4: **Again, we repeat the above step until no digit is left.

On multiplying 3 by 2 and adding the third leftmost digit of (11001)_{2}, we get

3 × 2 + 0 = 6

On multiplying 6 by 2 and adding the next digit of (11001)_{2}, we get

6 × 2 + 0 = 12

On multiplying 12 by 2 and adding the next digit of (11001)_{2}, we get

12 × 2 + 1 = 25

Now, no digit is left, and we have a result of 25, the converted decimal number. Thus, (11001)_{2} = (25)_{10}.

**Convert the binary number (101) _{2} into decimal using the doubling method.**

Solution:

(101)_{2} has 1 at the leftmost, then 0, and 1.

Here, 0 × 2 + 1 = 1

1 × 2 + 0 = 2

2 × 2 + 1 = 5

Now, the final result is 5 for the binary number 101. Thus, (101)_{2} = (5)_{10}.