How to Convert Between Binary, Decimal, Octal, and Hexadecimal Numbers?
The concept of number system conversion plays a key role in mathematics and is widely applicable to both real-life situations and exam scenarios. It helps students change numbers between different bases such as decimal, binary, octal, and hexadecimal. This is especially useful in digital electronics, computer science, and exams like JEE or competitive Olympiads. On Vedantu, you can easily master number system conversion using simple methods, tables, and calculator tools for accuracy and speed.
What Is Number System Conversion?
A number system conversion is defined as the process of changing a number from one base to another. For example: converting a number from decimal (base 10) to binary (base 2), octal (base 8), or hexadecimal (base 16). You’ll find this concept applied in areas such as digital electronics, computer programming, and mathematics problem solving.
Types of Number Systems
Let’s quickly look at the main types of number systems you’ll work with in base conversions:
| Number System | Base | Digits Used | Example |
|---|---|---|---|
| Decimal | 10 | 0-9 | 245(10) |
| Binary | 2 | 0, 1 | 1101(2) |
| Octal | 8 | 0-7 | 175(8) |
| Hexadecimal | 16 | 0-9, A-F | 2F(16) |
Key Formula for Number System Conversion
Here’s the standard formula to convert any number to decimal:
If \( (d_nd_{n-1}...d_1d_0)_b \) is a number in base “b”, then:
\( Number_{decimal} = d_n \times b^n + d_{n-1} \times b^{n-1} + ... + d_1 \times b^1 + d_0 \times b^0 \)
Number System Conversion Chart
Here’s a quick chart to help you remember the equivalent values in different number systems.
| Decimal | Binary | Octal | Hexadecimal |
|---|---|---|---|
| 0 | 0000 | 0 | 0 |
| 1 | 0001 | 1 | 1 |
| 2 | 0010 | 2 | 2 |
| 3 | 0011 | 3 | 3 |
| 4 | 0100 | 4 | 4 |
| 5 | 0101 | 5 | 5 |
| 6 | 0110 | 6 | 6 |
| 7 | 0111 | 7 | 7 |
| 8 | 1000 | 10 | 8 |
| 9 | 1001 | 11 | 9 |
| 10 | 1010 | 12 | A |
| 11 | 1011 | 13 | B |
| 12 | 1100 | 14 | C |
| 13 | 1101 | 15 | D |
| 14 | 1110 | 16 | E |
| 15 | 1111 | 17 | F |
Step-by-Step Illustration
Example 1: Convert (25)10 to Binary
1. Divide 25 by 22. 25 ÷ 2 = 12, remainder = 1
3. 12 ÷ 2 = 6, remainder = 0
4. 6 ÷ 2 = 3, remainder = 0
5. 3 ÷ 2 = 1, remainder = 1
6. 1 ÷ 2 = 0, remainder = 1
7. Write the remainders from last to first: 11001
So, (25)10 = (11001)2
Example 2: Convert (128)10 to Hexadecimal
1. Divide 128 by 162. 128 ÷ 16 = 8, remainder = 0
3. 8 ÷ 16 = 0, remainder = 8
4. Write the remainders from last to first: 80
So, (128)10 = (80)16
Speed Trick or Vedic Shortcut
Here’s a fast trick: To convert binary to octal or hexadecimal, group binary digits in sets of 3 (for octal) or 4 (for hex) from right and match each group to its equivalent. This saves time in exams, especially with big numbers.
Example Trick: Convert (1101011)2 to octal.
- Group into threes from right: 1 101 011.
- Left pad with zeros if needed: 001 101 011
- Convert each group: 001=1, 101=5, 011=3
- So, (1101011)2 = (153)8
Shortcuts like these are taught in Vedantu’s live sessions for both board and competitive exam prep.
Try These Yourself
- Convert (45)10 to binary and octal.
- Write (111101)2 in decimal.
- Change (7B)16 to decimal.
- Find the octal value for (101110)2.
Frequent Errors and Misunderstandings
- Forgetting to write remainders in reverse order while converting decimal to binary/octal/hex.
- Mixing up digits for octal (0-7) and hexadecimal (A-F after 9).
- Grouping wrong number of binary digits for octal/hex conversion.
- Arithmetic calculation slips during stepwise multiplication.
Relation to Other Concepts
The idea of number system conversion connects closely with number systems, decimal numbers, and digital logic. Mastering these conversions helps you work confidently with binary, octal, and hexadecimal topics, and understand their real-world applications in computing and electronics.
Classroom Tip
A quick mnemonic for hexadecimal: After 9, remember A=10, B=11, C=12, D=13, E=14, F=15. Also, always double-check your remainders and write groups clearly to avoid silly mistakes.
We explored number system conversion — from definition, stepwise examples, shortcut tricks, and common mistakes, to its connection with digital and computer systems. Keep practicing with Vedantu’s tools and live classes to master fast and error-free base conversions!
FAQs on Number System Conversion – Steps, Tricks & Calculator
1. What is number system conversion in Maths?
Number system conversion is the process of changing a number from one
2. How do you convert binary numbers to decimal?
To convert a binary number to decimal, multiply each binary digit by 2 raised to the power of its position (starting from the rightmost digit with position 0). Then, sum the results. For example: (1011)2 = (1 × 23) + (0 × 22) + (1 × 21) + (1 × 20) = 8 + 0 + 2 + 1 = (11)10
3. What are the four main types of number systems?
The four main types of number systems are:
Decimal (Base 10): Uses digits 0-9.Binary (Base 2): Uses digits 0 and 1.Octal (Base 8): Uses digits 0-7.Hexadecimal (Base 16): Uses digits 0-9 and letters A-F (A=10, B=11, C=12, D=13, E=14, F=15).
4. How to convert decimal to binary?
Repeatedly divide the decimal number by 2, recording the remainders. The remainders, read in reverse order, form the binary equivalent. For example, converting 1310 to binary: 13 ÷ 2 = 6 R 1; 6 ÷ 2 = 3 R 0; 3 ÷ 2 = 1 R 1; 1 ÷ 2 = 0 R 1. Therefore, 1310 = 11012
5. How to convert decimal to hexadecimal?
Repeatedly divide the decimal number by 16, noting the remainders. Convert remainders greater than 9 to their hexadecimal equivalents (A-F). The remainders, read in reverse order, form the hexadecimal equivalent. For example: 2610 = 1A16 (26 ÷ 16 = 1 R 10, which is A in hexadecimal).
6. How to convert binary to hexadecimal?
Group the binary digits into sets of four, starting from the right. Convert each group of four binary digits into its hexadecimal equivalent. For example, 110110112 is grouped as 1101 1011. This converts to D (1101) and B (1011) making the hexadecimal representation DB16.
7. Are number system conversions important for board or competitive exams?
Yes, number system conversions are essential for many board and competitive exams, particularly in subjects like computer science, mathematics, and digital electronics. Understanding these conversions is key to solving many problems related to binary arithmetic and computer architecture.
8. Is there a tool or calculator for number system conversion?
Yes, many online calculators and software tools are available for performing number system conversions. These tools can quickly and accurately convert between various number systems, helping you verify your manual calculations and save time during exams or problem-solving.
9. What are common mistakes to avoid in number system conversions?
Common mistakes include incorrect positional values, errors in arithmetic operations during conversion (especially when dealing with hexadecimal), and forgetting to reverse the order of remainders when converting from decimal to other bases. Double-check your work and use a calculator to verify your answers.
10. What are the applications of number system conversions in computer science?
Number system conversions are fundamental in computer science. They are used in representing data within computer systems (binary for machine level operations), in network communication (various protocols use different bases), and are key to understanding data structures and algorithms.
11. How can I quickly convert large numbers between bases?
For large numbers, using a calculator or programming code is recommended for accuracy and speed. Manual methods can become prone to error for large numbers. Consider using specialized software or algorithms designed for efficient base conversion.
