How To Convert Decimal To Binary With Step By Step Method and Solved Examples
The concept of decimal to binary conversion plays a key role in mathematics, computer science, and electronics. Learning to convert from decimal (base 10) to binary (base 2) helps you understand how numbers are represented inside computers and digital devices. This is an important topic for school exams, engineering, and coding interviews, as well as for daily logical reasoning.
What Is Decimal to Binary?
A decimal to binary conversion is the process of changing a number written with base 10 digits (0-9) into a number using only base 2 digits (0 and 1). You’ll find this concept applied in areas such as computing, digital logic circuits, and digital communication. Knowing decimal to binary helps you understand how data, images, and programs are stored and processed by computers.
Key Formula for Decimal to Binary
Here’s the standard formula:
Divide the decimal number by 2, keep note of the remainder, and continue dividing the quotient by 2 until you reach 0. The binary number is formed by reading the remainders backwards (from bottom to top).
For a decimal number \( N \):
Repeatedly do: \( N \div 2 \). The remainder at each step becomes the next binary digit (starting from the least significant bit).
Cross-Disciplinary Usage
Decimal to binary is not only useful in Maths but also plays an important role in Computer Science (for coding in Python, C++), Physics (for digital signals), and in electronics where circuits use binary logic. Students preparing for exams like JEE, NEET, Olympiads or boards encounter number system conversions frequently.
Step-by-Step Illustration
- Let’s convert decimal 29 to binary.
1. 29 ÷ 2 = 14 remainder 1
2. 14 ÷ 2 = 7 remainder 0
3. 7 ÷ 2 = 3 remainder 1
4. 3 ÷ 2 = 1 remainder 1
5. 1 ÷ 2 = 0 remainder 1
Now, write the remainders from last to first: 11101
29 in binary is 11101.
Speed Trick or Vedic Shortcut
Here’s a quick shortcut that helps solve problems faster when working with decimal to binary conversion. Many students use this trick during timed exams to save crucial seconds.
Example Trick: Instead of dividing each time, you can list powers of 2 (128, 64, 32, 16, 8, 4, 2, 1), subtract the largest possible from your number, and put '1' at those places.
- For 12: Largest power of 2 ≤ 12 is 8 (so, first bit is 1, 12-8=4).
- Next, 4 fits in 4 (second '1'), 4-4=0.
- So, bits for 8, 4, 2, 1: 1 1 0 0 (since 2 and 1 are not used, write 0).
- 12 in binary: 1100
Tricks like this help in quick quizzes and mental maths. Vedantu’s live classes share more such easy methods to build speed for exams.
Try These Yourself
- Convert 174 to binary.
- What is the binary for 100?
- Express 45 in binary using repeated division.
- If a number is 0 in decimal, what is it in binary?
- Find the binary for 256.
Frequent Errors and Misunderstandings
- Placing remainders in the wrong order (it should be from bottom to top).
- Stopping before the quotient reaches 0 (always continue till quotient is zero).
- Mixing up decimal to binary with binary to decimal conversions.
- Confusing the place values in binary (rightmost is 2⁰, next left is 2¹, and so on).
Decimal to Binary Table (1–20)
| Decimal | Binary |
|---|---|
| 0 | 0 |
| 1 | 1 |
| 2 | 10 |
| 3 | 11 |
| 4 | 100 |
| 5 | 101 |
| 6 | 110 |
| 7 | 111 |
| 8 | 1000 |
| 9 | 1001 |
| 10 | 1010 |
| 11 | 1011 |
| 12 | 1100 |
| 13 | 1101 |
| 14 | 1110 |
| 15 | 1111 |
| 16 | 10000 |
| 17 | 10001 |
| 18 | 10010 |
| 19 | 10011 |
| 20 | 10100 |
Relation to Other Concepts
The idea of decimal to binary connects with Number System basics, binary numbers, and is the foundation for learning hexadecimal and octal representations. It also helps in programming, algorithms, and solving base conversion problems in Olympiad/NTSE and school exams.
Classroom Tip
A quick way to remember decimal to binary is to always write your list of remainders down the side of your notebook and then read them from bottom to top at the end. Teachers at Vedantu often use chips or colored beads for hands-on practice and to make binary numbers more visual in live classes.
Decimal to Binary in Python and C++
If you want to use a program to convert decimal to binary, here is a simple code in Python:
print(bin(num)) # Output: 0b11101
In C++:
while(n > 0) {
cout << n % 2;
n = n / 2;
} // Remember to reverse the output to get the correct order!
Wrapping It All Up
We explored decimal to binary—from what it means, how to do the conversion with steps, some easy mental tricks, and how not to make mistakes. Keep practicing with sample questions and using quick reference tables. Visit Vedantu for live lessons and full practice sets on decimal to binary conversion and more.
Related Topics:
Number System |
Binary Number System |
Hexadecimal Number System |
Octal Number System
FAQs on Decimal To Binary Conversion Explained Clearly
1. What is decimal to binary conversion?
Decimal to binary conversion is the process of converting a number from the base-10 (decimal) system to the base-2 (binary) system. In the decimal system, digits range from 0 to 9, while in binary, only 0 and 1 are used. Binary numbers represent values using powers of 2 instead of powers of 10, making them essential in computer science and digital electronics.
2. How do you convert a decimal number to binary?
To convert a decimal number to binary, repeatedly divide the number by 2 and record the remainders. Follow these steps:
- Divide the decimal number by 2.
- Write down the remainder (0 or 1).
- Divide the quotient again by 2.
- Repeat until the quotient becomes 0.
- Read the remainders from bottom to top.
Example: Convert 13 to binary → Remainders: 1, 0, 1, 1 → Binary = 1101.
3. What is the binary equivalent of 10?
The binary equivalent of 10 is 1010. Converting 10 to binary:
- 10 ÷ 2 = 5 remainder 0
- 5 ÷ 2 = 2 remainder 1
- 2 ÷ 2 = 1 remainder 0
- 1 ÷ 2 = 0 remainder 1
Reading remainders from bottom to top gives 1010.
4. What is the formula for converting decimal to binary?
The formula for decimal to binary conversion is expressing the number as a sum of powers of 2. Any decimal number N can be written as:
N = a₀×2⁰ + a₁×2¹ + a₂×2² + ... + aₙ×2ⁿ, where each coefficient a is 0 or 1.
For example, 13 = 1×2³ + 1×2² + 0×2¹ + 1×2⁰, so binary form is 1101.
5. How do you convert decimal fractions to binary?
To convert a decimal fraction to binary, repeatedly multiply the fractional part by 2 and record the integer parts. Steps:
- Multiply the fraction by 2.
- Write down the integer part (0 or 1).
- Repeat with the new fractional part.
- Continue until the fraction becomes 0 or reaches desired precision.
Example: 0.625 → 0.625×2=1.25 (1), 0.25×2=0.5 (0), 0.5×2=1.0 (1) → Binary = 0.101.
6. Why do computers use binary instead of decimal?
Computers use binary because digital circuits have two stable states represented by 0 and 1. These states correspond to off/on electrical signals, making the binary number system simple, reliable, and efficient for electronic processing. Decimal numbers are converted to binary before being processed by computer hardware.
7. What is the binary equivalent of 25?
The binary equivalent of 25 is 11001. Conversion steps:
- 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
Reading from bottom to top gives 11001.
8. What is the difference between decimal and binary number systems?
The main difference is that the decimal system uses base 10 while the binary system uses base 2. Key differences:
- Decimal digits: 0–9
- Binary digits: 0 and 1
- Decimal place values: powers of 10
- Binary place values: powers of 2
Binary is mainly used in computing, while decimal is used in everyday arithmetic.
9. How do you check if a binary conversion is correct?
To check a binary conversion, convert the binary number back to decimal using powers of 2. Steps:
- Write each digit with its corresponding power of 2.
- Multiply each digit by its power of 2.
- Add all the results.
Example: 1101 = 1×2³ + 1×2² + 0×2¹ + 1×2⁰ = 8 + 4 + 0 + 1 = 13, so the conversion is correct.
10. What are common mistakes when converting decimal to binary?
Common mistakes in decimal to binary conversion include incorrect remainder order and calculation errors. Key points to remember:
- Always read remainders from bottom to top.
- Do not stop division before the quotient becomes 0.
- For fractions, multiply only the fractional part by 2.
- Double-check by converting the binary result back to decimal.
Careful step-by-step calculation ensures accurate binary conversion.
