## Binary to Decimal

A binary number (base-2) is converted to an equivalent decimal number using the binary-to-decimal conversion formula (base-10). In mathematics, numbers are expressed using a number system. It is a way to display numerical data. The four various number systems are as follows:

- System of Binary Numbers (Base-2)
- System of octal numbers (Base-8)
- System of Decimal Numbers (Base-10)
- System of Hexadecimal Numbers (Base-16)

The number system is crucial to computer systems. The number system is used by virtually all electronic devices, not just computer architecture. Therefore, learning how to convert between different number systems is crucial. Before we talk about converting from binary to decimal, let’s first grasp both number systems.

## Binary to Decimal- Binary Number System

A number that is employed in binary systems is referred to as a binary number system. The base 2 number system is another name for it. It uses two separate symbols to represent the numerical values, essentially 1 (one) and 0 (zero).

## Binary to Decimal- Decimal Number System

The base-10 numeral system is another name for the decimal number system. Ten digits, from 0 to 9, are used. The continuous locations to the left of the decimal point in the decimal number system stand in for units, tens, hundreds, thousands, and so forth. Consequently, 10 serves as the decimal number system’s base.

## Binary to Decimal Conversion

When converting a binary number (base-2) to a decimal number, this process is known as binary to decimal conversion (base-10). Understanding binary-to-decimal conversion is crucial for applications using computer programming. Only two digits, 0 and 1, are used to represent the binary number system, whereas all ten digits, from 0 to 9, are used to represent the decimal number system. In the subsequent section, we must learn how to translate a binary number into its matching decimal number.

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## How Binary to Decimal Converter Works?

We use the division method to multiply the binary integer to get its decimal equivalent. In this conversion method, each digit of the input number is multiplied from the Most Significant Bit (MSB) to the Least Significant Bit (LSB), lowering the base’s power, if a number with base n is to be turned into a number with base 10.

## Binary to Decimal Conversion Steps

- Write the specified binary number first, then count the powers of two going left to right (powers starting from 0)
- Now, from right to left, write each binary digit with the matching powers of 2, so that the first binary digit (MSB) will be multiplied by the largest power of 2.
- Add each of the items from the previous step.
- The required decimal number will be the final response.

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## Binary to Decimal Converter working Formula

We will learn the basic formula, let’s study how to convert a binary number to a decimal number. The formula to convert binary to decimal is provided as, where dn represents the digits of a binary number with ‘n’ digits.

The formula for conversing binary to decimal is given below

Where,

N is decimal equivalent,

b is the digit,

q is the base value

## Binary to Decimal Conversion Table

Binary Number |
Decimal Number |
Hex Number |
---|---|---|

0 | 0 | 0 |

1 | 1 | 1 |

10 | 2 | 2 |

11 | 3 | 3 |

100 | 4 | 4 |

101 | 5 | 5 |

110 | 6 | 6 |

111 | 7 | 7 |

1000 | 8 | 8 |

1001 | 9 | 9 |

1010 | 10 | A |

1011 | 11 | B |

1100 | 12 | C |

1101 | 13 | D |

1110 | 14 | E |

1111 | 15 | F |

10000 | 16 | 10 |

10001 | 17 | 11 |

10010 | 18 | 12 |

10011 | 19 | 13 |

10100 | 20 | 14 |

10101 | 21 | 15 |

10110 | 22 | 16 |

10111 | 23 | 17 |

11000 | 24 | 18 |

11001 | 25 | 19 |

11010 | 26 | 1A |

11011 | 27 | 1B |

11100 | 28 | 1C |

11101 | 29 | 1D |

11110 | 30 | 1E |

11111 | 31 | 1F |

100000 | 32 | 20 |

1000000 | 64 | 40 |

10000000 | 128 | 80 |

100000000 | 256 | 100 |

## Binary to Decimal Table

The conversion chart for Binary to Decimal conversion is given below

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 |

## Binary to Decimal Conversion Examples

**Example:** **Convert 1110 _{2}, from binary to decimal using the binary-to-decimal formula.**

**Solution:** We start doing the conversion from the rightmost digit, which is ‘0’ here.

(Decimal Number)_{10} = (d_{0 }× 2^{0}) + (d_{1} × 2^{1 })+ (d_{2} × 2^{2 })+ ….. (d_{n-1} × 2^{n-1}),

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

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

= 0 + 2 + 4 + 8

= 14

Therefore, 1110_{2} = 14_{10}

**Example**: **Convert the binary number 1001 to a decimal number.**

**Solution**: Given, binary number = 1001_{2}

Hence, using the binary-to-decimal conversion formula, we have:

1001_{2} = (1 × 2³) + (0 × 2²) + (0 × 2¹) + (1 × 2⁰)

= 8 + 0 + 0 + 1

= (9)_{₁₀}

**Example**: **Convert 1101001 _{2} into an equivalent decimal number.**

**Solution**: Using binary to decimal conversion method, we get;

(1101001)₂ = (1 × 2⁶) + (1 × 2⁵) + (0 × 2⁴) + (1 × 2³) + (0 × 2²) + (0 × 2¹) + (1 × 2⁰)

= 64 + 32 + 0 + 8 + 0 + 0 + 1

= (105)_{₁₀}

## 1111 Binary to Decimal

1111 is in binary to decimal conversion is equal to 15.

1111 |
15 |

## 1001 Binary to Decimal Conversion

In The binary conversion of 1001 means the 1 and the 8 bits are activated, but not the 2 or 4 bits. Thus, the binary-to-decimal conversion for 1001 is **8 + 1 = 9**.