How to Convert Decimal to Binary (With Easy Steps & Examples)
The concept of binary number system plays a key role in mathematics and is widely used in computer science, digital electronics, and various real-life applications. A clear understanding of the binary number system helps students solve exam questions quickly and builds a foundation for advanced topics like logic gates and programming.
What Is Binary Number System?
A binary number system is a way of representing numbers using only two digits: 0 and 1. It is called a base-2 system. You’ll find this concept applied in areas such as computer science, digital electronics, and Boolean algebra. In the binary number system, each digit is called a bit. For example, 1011 is a four-bit binary number.
Key Formula for Binary Number System
Here’s the standard formula to convert a binary number to decimal:
\( N = (b_n \times 2^n) + (b_{n-1} \times 2^{n-1}) + ... + (b_1 \times 2^1) + (b_0 \times 2^0) \)
where \( b \) is each binary digit (0 or 1).
Cross-Disciplinary Usage
The binary number system is not only useful in Maths but also plays an important role in Physics, Computer Science, and daily logical reasoning. Students preparing for JEE or NEET will see its relevance in topics like Boolean logic, digital circuits, and data representation.
Step-by-Step Illustration
- Convert binary 10112 to decimal.
Step 1: Write the place values (from right) as \(2^0, 2^1, 2^2, 2^3\).
Step 2: Multiply each bit by its place value:
- \(1 \times 2^3 = 8\)
- \(0 \times 2^2 = 0\)
- \(1 \times 2^1 = 2\)
- \(1 \times 2^0 = 1\)
Step 3: Add all results: 8 + 0 + 2 + 1 = 1110 - Convert decimal 13 to binary.
Step 1: Divide 13 by 2: quotient 6, remainder 1.
Step 2: Divide 6 by 2: quotient 3, remainder 0.
Step 3: Divide 3 by 2: quotient 1, remainder 1.
Step 4: Divide 1 by 2: quotient 0, remainder 1.
Step 5: Write remainders in reverse: 11012
Speed Trick or Vedic Shortcut
Here’s a quick shortcut to convert decimal numbers to binary quickly. This helps students save time during exams.
Example Trick: To convert 25 to binary:
- Keep dividing by 2, writing the remainder each time:
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
Arrange remainders bottom to top: 110012
Shortcuts like these are discussed in Vedantu’s live sessions to boost speed and build confidence for board and Olympiad exams.
Try These Yourself
- Write the binary form of 7, 15, and 20.
- What is the decimal value of 110102?
- Convert 23 from decimal to binary.
- Fill up the binary number table for numbers 1 to 10.
Binary Numbers Table
| Number | Binary | Number | Binary |
|---|---|---|---|
| 1 | 1 | 6 | 110 |
| 2 | 10 | 7 | 111 |
| 3 | 11 | 8 | 1000 |
| 4 | 100 | 9 | 1001 |
| 5 | 101 | 10 | 1010 |
Binary Arithmetic Operations
Adding, subtracting, multiplying, and dividing binary numbers follow simple rules:
- Addition: 0+0=0, 0+1=1, 1+0=1, 1+1=0 (carry 1)
- Subtraction: 0–0=0, 1–0=1, 1–1=0, 0–1=1 (borrow 1)
For example, add 1011 and 1101:
1. Align numbers:
1011
+1101
2. Add from right to left:
- 1+1=0 (carry 1)
- 1+0+1(carry)=0 (carry 1)
- 0+1+1(carry)=0 (carry 1)
- 1+1(carry)=1
So, 1011+1101 = 110002
Frequent Errors and Misunderstandings
- Mixing up place values (e.g., reading binary left-to-right instead of right-to-left for decimal conversion).
- Adding binary with incorrect carries or subtracting without borrowing.
- Thinking binary numbers only matter for computers, not realizing exam questions often test these conversions.
Relation to Other Concepts
The idea of binary number system connects closely with topics such as the decimal number system, octal number system, and concepts of number system conversion. Mastering binary helps in converting between number systems and understanding digital logic in future chapters.
Classroom Tip
A quick way to remember the binary number system is to think of place values as powers of 2 instead of 10. Each jump to the left doubles the value (1, 2, 4, 8, etc.). Vedantu’s teachers use visual diagrams and tables for clear understanding during live and recorded classes.
We explored the binary number system—from the definition, formulas, stepwise examples, and common mistakes. Continue practicing with Vedantu to become confident in binary conversions and ace your number system exams!
For more on related topics, see: Overview of Number System, Decimal to Binary Conversion Explained, Binary Addition Methods, and How to Convert Number Systems.
FAQs on Binary Number System Explained – Learn How to Convert, Calculate & Use
1. What is the binary number system?
The binary number system is a base-2 numeral system, using only two digits, 0 and 1, to represent all numbers. Unlike the decimal system (base-10), it's fundamental to computer science and digital electronics because it directly reflects the on/off states of electronic circuits.
2. Why do computers use the binary number system?
Computers use the binary system because it's easily implemented using electronic components. The binary digits (bits), 0 and 1, represent the two states of a transistor (on/off), making it efficient for storing and processing data. This simple system is the foundation of all digital technologies.
3. How do I convert a decimal number to its binary equivalent?
To convert a decimal number to binary, repeatedly divide the decimal number by 2 until the quotient becomes 0. Record the remainders at each step. The binary equivalent is the sequence of remainders read in reverse order (from bottom to top). For example, converting 13 to binary:
- 13 ÷ 2 = 6 remainder 1
- 6 ÷ 2 = 3 remainder 0
- 3 ÷ 2 = 1 remainder 1
- 1 ÷ 2 = 0 remainder 1
Therefore, 13 in decimal is 1101 in binary.
4. How do I convert a binary number to its decimal equivalent?
To convert a binary number to decimal, multiply each digit by the corresponding power of 2 (starting from the rightmost digit with 20) and sum the results. For example, converting 1101 to decimal:
(1 × 23) + (1 × 22) + (0 × 21) + (1 × 20) = 8 + 4 + 0 + 1 = 13
5. How does binary addition work?
Binary addition follows similar rules to decimal addition, but with only two digits.
- 0 + 0 = 0
- 0 + 1 = 1
- 1 + 0 = 1
- 1 + 1 = 0 (with a carry of 1)
The carry is added to the next column to the left, just like in decimal addition.
6. How does binary subtraction work?
Binary subtraction is similar to decimal subtraction.
- 0 - 0 = 0
- 1 - 0 = 1
- 1 - 1 = 0
- 0 - 1 = 1 (with a borrow of 1 from the next column to the left)
Borrowing from the next column means subtracting 1 from that column and adding 2 to the current column.
7. What are some real-world applications of the binary number system?
The binary number system is crucial in various applications, including:
- Computer hardware and software: Representing data, instructions, and memory addresses.
- Digital electronics: Designing logic circuits and digital devices.
- Telecommunications: Encoding and decoding digital signals.
- Data storage: Representing information on hard drives, SSDs, and other storage media.
8. What is the difference between binary, decimal, and hexadecimal number systems?
These are all positional number systems, but differ in their base (radix):
- Binary (base-2) uses only 0 and 1.
- Decimal (base-10) uses digits 0-9.
- Hexadecimal (base-16) uses digits 0-9 and letters A-F (representing 10-15).
Hexadecimal is often used as a shorthand for representing long binary numbers.
9. What are some examples of binary numbers?
Here are a few examples of binary numbers and their decimal equivalents:
- 1011₂ = 11₁₀
- 1000₂ = 8₁₀
- 1111₂ = 15₁₀
- 100000₂ = 32₁₀
10. How is binary used in Boolean logic?
Boolean logic uses binary values (TRUE/FALSE or 1/0) to represent logical operations such as AND, OR, and NOT. These operations are fundamental to the design of digital circuits and computer programming. Each binary input results in a binary output reflecting the logical operation.
11. What is a bit and a byte?
A bit (binary digit) is the smallest unit of data in a computer, representing either 0 or 1. A byte is a group of 8 bits, which is a more convenient unit for representing larger amounts of data. For example, one character in most computer systems is represented by one byte.
12. What is the significance of the most significant bit (MSB) and least significant bit (LSB)?
In a binary number, the Most Significant Bit (MSB) is the leftmost bit, holding the highest positional value. Conversely, the Least Significant Bit (LSB) is the rightmost bit, representing the smallest positional value. The MSB and LSB are important for understanding the magnitude and order of bits in a binary number. They are useful when carrying out binary arithmetic and are commonly used in computer architectures.
