Converting numbers between different numerical systems is a fundamental skill in computer science and digital electronics. Among these conversions, the transition from binary to decimal is particularly essential, as binary is the language of computers and decimal is the most common numerical system used by humans. This article aims to elucidate the process of converting binary numbers to their decimal equivalents, providing detailed explanations and examples.

## Understanding Binary and Decimal Systems

### Binary System

The binary system is a base-2 numerical system that uses only two digits: 0 and 1. Each digit in a binary number is referred to as a “bit”. The position of each bit represents a power of 2, starting from 2^0 at the rightmost bit.

For example, the binary number `1011`

can be broken down as follows:

- 1 × 2^3 = 8
- 0 × 2^2 = 0
- 1 × 2^1 = 2
- 1 × 2^0 = 1

### Decimal System

The decimal system, or base-10 system, uses ten digits: 0 through 9. Each position in a decimal number represents a power of 10, starting from 10^0 at the rightmost digit.

## Binary to Decimal Conversion Process

To convert a binary number to its decimal equivalent, follow these steps:

**Write down the binary number.****List the powers of 2 from right to left, starting with 2^0.****Multiply each bit by the corresponding power of 2.****Sum all the products to get the decimal equivalent.**

### Example 1: Converting Binary 1101 to Decimal

Let’s convert the binary number `1101`

to its decimal form.

**Write down the binary number:**yamlCopy code`<span class="hljs-number">1101</span>`

**List the powers of 2 from right to left:**Copy code`1 1 0 1 2^3 2^2 2^1 2^0`

**Multiply each bit by the corresponding power of 2:**Copy code`1 × 2^3 = 8 1 × 2^2 = 4 0 × 2^1 = 0 1 × 2^0 = 1`

**Sum all the products:**Copy code`8 + 4 + 0 + 1 = 13`

Thus, the binary number `1101`

converts to the decimal number `13`

.

### Example 2: Converting Binary 100101 to Decimal

Now, let’s convert the binary number `100101`

to its decimal form.

**Write down the binary number:**Copy code`100101`

**List the powers of 2 from right to left:**Copy code`1 0 0 1 0 1 2^5 2^4 2^3 2^2 2^1 2^0`

**Multiply each bit by the corresponding power of 2:**Copy code`1 × 2^5 = 32 0 × 2^4 = 0 0 × 2^3 = 0 1 × 2^2 = 4 0 × 2^1 = 0 1 × 2^0 = 1`

**Sum all the products:**Copy code`32 + 0 + 0 + 4 + 0 + 1 = 37`

Thus, the binary number `100101`

converts to the decimal number `37`

.

## Practice Problem

Convert the binary number `1010101`

to its decimal form.

**Solution:**

**Write down the binary number:**Copy code`1010101`

**List the powers of 2 from right to left:**Copy code`1 0 1 0 1 0 1 2^6 2^5 2^4 2^3 2^2 2^1 2^0`

**Multiply each bit by the corresponding power of 2:**Copy code`1 × 2^6 = 64 0 × 2^5 = 0 1 × 2^4 = 16 0 × 2^3 = 0 1 × 2^2 = 4 0 × 2^1 = 0 1 × 2^0 = 1`

**Sum all the products:**Copy code`64 + 0 + 16 + 0 + 4 + 0 + 1 = 85`

Thus, the binary number `1010101`

converts to the decimal number `85`

.

## Conclusion

Converting binary numbers to decimal is a straightforward process that involves understanding the positional values of each bit in the binary system and performing simple multiplications and additions. Mastering this conversion is crucial for anyone working with digital systems, computer science, or electronics. By practicing with various examples, you can become proficient in translating binary numbers into their decimal equivalents and deepen your understanding of numerical systems.