How To Convert Binary To Decimal

Oriana Fall

2 min read

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21-01-2025

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Binary and decimal are two different ways of representing numbers. Binary, used in computers, uses only two digits (0 and 1), while decimal, the system we use every day, uses ten digits (0-9). Knowing how to convert between these systems is a fundamental skill in computer science and related fields. This guide will show you exactly how to convert binary numbers to their decimal equivalents.

Understanding the Place Value System

Both binary and decimal systems rely on a place value system. Each digit's position determines its value. In the decimal system, we're familiar with ones, tens, hundreds, thousands, and so on. These represent powers of 10 (10⁰, 10¹, 10², 10³, etc.).

Binary works similarly, but instead of powers of 10, it uses powers of 2. The rightmost digit represents 2⁰ (which is 1), the next digit to the left represents 2¹ (which is 2), then 2² (4), 2³ (8), and so on.

Step-by-Step Guide to Binary to Decimal Conversion

Let's illustrate the conversion process with an example: Convert the binary number 101101₂ to decimal. (The subscript "2" indicates it's a binary number.)

Here's how to do it:

1. Write down the binary number: 101101₂

2. Assign place values: Write the powers of 2 under each digit, starting from the rightmost digit (least significant bit) with 2⁰.

   1   0   1   1   0   1<sub>2</sub>
   2<sup>5</sup> 2<sup>4</sup> 2<sup>3</sup> 2<sup>2</sup> 2<sup>1</sup> 2<sup>0</sup>
   

3. Multiply each digit by its place value:

   • 1 × 2⁵ = 32
   • 0 × 2⁴ = 0
   • 1 × 2³ = 8
   • 1 × 2² = 4
   • 0 × 2¹ = 0
   • 1 × 2⁰ = 1

4. Add the results: 32 + 0 + 8 + 4 + 0 + 1 = 45

Therefore, 101101₂ = 45₁₀ (the subscript "10" indicates it's a decimal number).

Another Example: Converting a Larger Binary Number

Let's try a slightly more complex example: 11011101₂

1. Write down the binary number: 11011101₂

2. Assign place values:

   1   1   0   1   1   1   0   1<sub>2</sub>
   2<sup>7</sup> 2<sup>6</sup> 2<sup>5</sup> 2<sup>4</sup> 2<sup>3</sup> 2<sup>2</sup> 2<sup>1</sup> 2<sup>0</sup>
   

3. Multiply and add:

   • 1 × 2⁷ = 128
   • 1 × 2⁶ = 64
   • 0 × 2⁵ = 0
   • 1 × 2⁴ = 16
   • 1 × 2³ = 8
   • 1 × 2² = 4
   • 0 × 2¹ = 0
   • 1 × 2⁰ = 1

4. Sum: 128 + 64 + 0 + 16 + 8 + 4 + 0 + 1 = 221₁₀

Therefore, 11011101₂ = 221₁₀

Tips and Tricks for Easier Conversion

• Start from the right: Always begin with the rightmost digit (2⁰) when assigning place values.
• Use a calculator: For larger binary numbers, a calculator can greatly simplify the addition process.
• Practice: The more you practice, the easier and faster you'll become at converting binary to decimal.

This comprehensive guide should enable you to confidently convert any binary number to its decimal equivalent. Remember to break down the process step-by-step and focus on understanding the underlying principles of place value. Happy converting!