Binary ⇔ Decimal Converter

Binary to Decimal Conversion Made Simple

Every digital device around us — from your laptop and smartphone to modern appliances — runs on a special language called binary, made up of only two digits: 0 and 1. While machines rely on this base-2 system, humans are more comfortable with the decimal system, which uses digits from 0 to 9. Converting between these two systems is a key concept in computer science, programming, and electronics. Our Binary to Decimal Converter gives you instant results while also helping you understand how the calculation works step by step.

Where Number Systems Come From

The decimal system (base-10) has been used for centuries. It most likely became popular because humans have ten fingers, making counting natural and straightforward. With the addition of zero and positional notation, the decimal system became the foundation of trade, mathematics, and modern education worldwide.

The binary system (base-2), on the other hand, might look like a modern invention, but it has a rich history. The ancient scholar Pingala described a binary-like method as far back as the 3rd century BC. Later, in the 17th century, mathematician Gottfried Wilhelm Leibniz formalized binary mathematics. Today, thanks to the rise of electronics and computers, binary has become the universal language of machines.

How Positional Notation Works

The magic of number systems lies in positional notation. In decimal, each digit represents a power of 10. For example:

(3 × 10²) + (4 × 10¹) + (5 × 10⁰) = 300 + 40 + 5 = 345

In binary, the concept is the same, but each place value represents a power of 2. So, the binary number 1101 is calculated as:

(1 × 2³) + (1 × 2²) + (0 × 2¹) + (1 × 2⁰) = 8 + 4 + 0 + 1 = 13

That’s exactly how our Binary to Decimal Converter computes values instantly.

Manual Conversion: Binary to Decimal

Let’s convert the binary number 101101 by hand:

1. Write the binary digits with their positions, starting from the right:
   1 0 1 1 0 1
(Pos 5, Pos 4, Pos 3, Pos 2, Pos 1, Pos 0)
2. Find the power of 2 for each position:
   2⁵=32, 2⁴=16, 2³=8, 2²=4, 2¹=2, 2⁰=1
3. Multiply each digit by its value:
   (1 × 32) + (0 × 16) + (1 × 8) + (1 × 4) + (0 × 2) + (1 × 1)
4. Add them together:
   32 + 0 + 8 + 4 + 0 + 1 = 45

So, 101101 in binary is equal to 45 in decimal.

Manual Conversion: Decimal to Binary

To convert from decimal to binary, you divide the number by 2 repeatedly and record the remainders. Let’s convert 45 into binary:

1. 45 ÷ 2 = 22 remainder 1
2. 22 ÷ 2 = 11 remainder 0
3. 11 ÷ 2 = 5 remainder 1
4. 5 ÷ 2 = 2 remainder 1
5. 2 ÷ 2 = 1 remainder 0
6. 1 ÷ 2 = 0 remainder 1

Reading the remainders from bottom to top gives us 101101, confirming the result.

Other Useful Number Systems

Along with binary and decimal, two other systems are commonly used in programming: Octal and Hexadecimal.

Octal (Base-8)

Octal uses digits 0–7. Each octal digit represents three binary digits. For example, binary 101101 can be grouped as (101)(101), which equals 55 in octal.

Hexadecimal (Base-16)

Hexadecimal uses 16 symbols (0–9 and A–F). Each hex digit represents four binary digits. For instance, binary 11101011 is written as EB in hex, which is shorter and easier to read.

From Bits to Terabytes

Understanding binary isn’t just about numbers — it’s about how data is stored and measured:

• Bit: The smallest unit of data (0 or 1).
• Byte: 8 bits, usually one character of text.
• Kilobyte (KB): 1,024 bytes.
• Megabyte (MB): 1,024 KB.
• Gigabyte (GB): 1,024 MB.
• Terabyte (TB): 1,024 GB.

Frequently Asked Questions

1. Is this converter free to use?

Yes. The Binary to Decimal Converter on Converter For Free is completely free and runs directly in your browser. No downloads or sign-ups needed.

2. What’s the largest number I can convert?

Our tool supports very large values. With JavaScript precision, you can convert numbers up to 2^53 - 1, which is more than enough for most purposes.

3. What does base-2 and base-10 mean?

The "base" indicates how many digits a system uses. Binary (base-2) uses 0 and 1. Decimal (base-10) uses digits 0 through 9.

4. How do letters become binary?

Characters are encoded using standards like ASCII or Unicode. For example, the letter "A" is 65 in decimal, which equals 01000001 in binary.

Conclusion

Understanding how binary and decimal systems connect is essential for anyone learning computing. Our Binary to Decimal Converter makes the process simple, fast, and reliable. Use it as a quick utility or as a learning tool to strengthen your digital knowledge.