Have you ever wondered how computers “understand” numbers? While we use the decimal system (base-10) in everyday life, computers rely entirely on the binary number system, a language of 0s and 1s. Whether you’re a student, a developer, or just curious about how digital systems work, knowing how to convert between binary and decimal is a fundamental skill in computer science.
In this blog, you will learn step-by-step everything you need to know about converting numbers between binary and decimal. From manual conversion methods to instant results with our binary to decimal and decimal to binary conversion calculator, we have got you covered. Plus, we’ll explain how these conversions power modern computing, electronics, and programming behind the scenes.
So, if you’re ready to understand how a number like 1101 becomes 13, or how 45 becomes 101101, this guide is for you.
What are Decimal Numbers?
Decimal numbers are the numbers we use every day, in counting, money, measurements, and calculations. Also known as the base-10 number system, decimal uses ten digits:
0, 1, 2, 3, 4, 5, 6, 7, 8, 9
Each position in a decimal number represents a power of 10, based on its place value from right to left. For example, in the number 653, we interpret it as:
(6×102)+(5×101)+(3×100)=600+50+3=653(6 × 10^2) + (5 × 10^1) + (3 × 10^0) = 600 + 50 + 3 = 653(6×102)+(5×101)+(3×100)=600+50+3=653
The decimal system is called Hindu-Arabic because it was developed in ancient India and introduced to Europe through Arabic scholars. It’s the most commonly used number system in the world today.
What are Binary Numbers?
Binary numbers are numbers expressed in the base-2 system, using only two digits:
0 and 1
Each digit in a binary number is called a bit (short for binary digit). Binary numbers are the language of computers, where:
- 0 represents OFF (no signal)
- 1 represents ON (signal present)
In binary, each digit’s place represents a power of 2. For example, the binary number 1101 is calculated as:
(1×23)+(1×22)+(0×21)+(1×20)=8+4+0+1=13(1 × 2^3) + (1 × 2^2) + (0 × 2^1) + (1 × 2^0) = 8 + 4 + 0 + 1 = 13(1×23)+(1×22)+(0×21)+(1×20)=8+4+0+1=13
Because of its simplicity and compatibility with digital electronics, the binary number system is used in everything from programming and networking to data storage and circuit design. You can learn more in detail from our other blog on the Binary Number System.
How to Convert Binary to Decimal:
Converting a binary number to decimal is a simple process once you understand how positional value and powers of 2 work. Each digit in the binary number is multiplied by 2 raised to the power of its position, starting from the rightmost bit (LSB).
Binary to Decimal Conversion Formula:
Decimal=(bn×2n)+(bn−1×2n−1)+…+(b1×21)+(b0×20)\text{Decimal} = (b_n × 2^n) + (b_{n-1} × 2^{n-1}) + … + (b_1 × 2^1) + (b_0 × 2^0)Decimal=(bn×2n)+(bn−1×2n−1)+…+(b1×21)+(b0×20)
Where:
- bnb_nbn is the binary digit (0 or 1)
- nnn is the position of the digit from the right (starting from 0)
- The final sum gives the decimal equivalent
Binary to Decimal Conversion 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 |
| 1100 | 12 |
| 1111 | 15 |
| 10000 | 16 |
Binary to Decimal Conversion Examples:
Example 1: Convert 1101₂ to Decimal
(1×23)+(1×22)+(0×21)+(1×20)=8+4+0+1=13(1×2^3) + (1×2^2) + (0×2^1) + (1×2^0) = 8 + 4 + 0 + 1 = 13(1×23)+(1×22)+(0×21)+(1×20)=8+4+0+1=13
Example 2: Convert 10110₂ to Decimal
(1×24)+(0×23)+(1×22)+(1×21)+(0×20)=16+0+4+2+0=22(1×2^4) + (0×2^3) + (1×2^2) + (1×2^1) + (0×2^0) = 16 + 0 + 4 + 2 + 0 = 22(1×24)+(0×23)+(1×22)+(1×21)+(0×20)=16+0+4+2+0=22
How to Convert Decimal to Binary: (H3)
When converting a decimal to binary, you repeatedly divide the decimal number by 2 and record the remainders. Reading the remainders in reverse gives you the binary number.
Decimal to Binary Conversion Formula: (H3)
There is no fixed equation, but the conversion follows this repeatable process:
Keep dividing the decimal number by 2 and collecting the remainders. The binary result is the remainders read from bottom to top.
Alternatively, you can use:
Binary=Remainders of (Decimal2)\text{Binary} = \text{Remainders of } \left( \frac{\text{Decimal}}{2} \right)Binary=Remainders of (2Decimal)
Decimal to Binary Conversion Table:
| Decimal | Binary |
| 0 | 0 |
| 1 | 1 |
| 2 | 10 |
| 3 | 11 |
| 4 | 100 |
| 5 | 101 |
| 6 | 110 |
| 7 | 111 |
| 8 | 1000 |
| 9 | 1001 |
| 10 | 1010 |
| 12 | 1100 |
| 15 | 1111 |
| 16 | 10000 |
Decimal to Binary Conversion Examples:
Example 1: Convert 13₁₀ to Binary
13 ÷ 2 = 6 remainder 1
6 ÷ 2 = 3 remainder 0
3 ÷ 2 = 1 remainder 1
1 ÷ 2 = 0 remainder 1
→ Binary = 1101
Example 2: Convert 25₁₀ to Binary
25 ÷ 2 = 12 remainder 1
12 ÷ 2 = 6 remainder 0
6 ÷ 2 = 3 remainder 0
3 ÷ 2 = 1 remainder 1
1 ÷ 2 = 0 remainder 1
→ Binary = 11001
Conclusion:
The ability to convert between binary and decimal systems is a fundamental skill in mathematics, programming, and digital electronics. Whether you are analyzing binary code, working with data storage, or learning how computers perform calculations, understanding these conversions builds a strong foundation. While the manual methods using powers of 2 (for binary to decimal) or division by 2 (for decimal to binary) help deepen your understanding, they can also become tedious, especially with large numbers.
That’s where a Binary to Decimal and Decimal to Binary Conversion Calculator becomes incredibly useful. It automates the process, saves time, and reduces errors, making it ideal for students, professionals, and developers alike. With tools, examples, and formulas at your fingertips, you can confidently tackle any number system problem and apply your knowledge in real-world tech scenarios.
Frequently Asked Questions (FAQs)
What is the easiest way to convert a decimal to binary?
The easiest way is to divide the decimal number by 2 repeatedly and write down the remainders. Read them in reverse to get the binary number. You can also use an online decimal to binary calculator for instant results.
How do I convert binary to decimal quickly?
Multiply each bit by 2 raised to the power of its position (starting from 0 on the right), then sum all the results. Or, use a binary to decimal converter tool to save time.
Why do computers use binary and not decimal?
Computers use binary because digital circuits have only two states: ON and OFF. Binary (0 and 1) maps directly to these two states, making it ideal for processing and storage.
What’s the difference between binary and decimal?
Decimal is base-10 and uses 10 digits (0–9), while binary is base-2 and uses only 0 and 1. Each system uses the powers of its base for place value.
Can I convert large numbers using a calculator?
Yes! A Binary to Decimal and Decimal to Binary Conversion Calculator can handle large numbers instantly and accurately, avoiding manual errors.