How to Convert Between Binary Decimal Octal and Hexadecimal Systems
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 in Maths
1. What is number system conversion in mathematics?
Number system conversion is the process of changing a number from one base to another, such as from binary (base 2) to decimal (base 10). In mathematics, different number systems use different bases to represent numbers.
- Binary uses digits 0 and 1.
- Octal uses digits 0 to 7 (base 8).
- Decimal uses digits 0 to 9 (base 10).
- Hexadecimal uses digits 0–9 and A–F (base 16).
Number system conversion helps in computing, digital electronics, and understanding place value in different bases.
2. How do you convert a binary number to a decimal number?
To convert a binary number to decimal, multiply each digit by powers of 2 and add the results.
- Write the binary number.
- Assign powers of 2 from right to left starting at 2⁰.
- Multiply each digit by its corresponding power.
- Add all the products.
Example: Convert 1011₂ to decimal.
1×2³ + 0×2² + 1×2¹ + 1×2⁰ = 8 + 0 + 2 + 1 = 11.
So, 1011₂ = 11₁₀.
3. 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.
- 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.
- Write the remainders in reverse order.
Example: Convert 13₁₀ to binary.
13 ÷ 2 → remainder 1
6 ÷ 2 → remainder 0
3 ÷ 2 → remainder 1
1 ÷ 2 → remainder 1
So, 13₁₀ = 1101₂.
4. How do you convert decimal to octal?
To convert a decimal number to octal, repeatedly divide the number by 8 and write the remainders in reverse order.
- Divide the decimal number by 8.
- Note the remainder (0–7).
- Divide the quotient again by 8.
- Continue until the quotient is 0.
- Write the remainders from bottom to top.
Example: Convert 65₁₀ to octal.
65 ÷ 8 = 8 remainder 1
8 ÷ 8 = 1 remainder 0
1 ÷ 8 = 0 remainder 1
So, 65₁₀ = 101₈.
5. How do you convert binary to hexadecimal?
To convert binary to hexadecimal, group the binary digits into sets of 4 bits from right to left and convert each group to its hexadecimal equivalent.
- Group digits into 4-bit sections.
- Add leading zeros if needed.
- Convert each 4-bit group to decimal.
- Write the corresponding hexadecimal digit (0–9, A–F).
Example: 11010110₂ → 1101 0110
1101 = 13 (D), 0110 = 6
So, 11010110₂ = D6₁₆.
6. What is the difference between binary, octal, decimal, and hexadecimal number systems?
The main difference between number systems is their base and the digits they use.
- Binary (base 2): digits 0 and 1.
- Octal (base 8): digits 0–7.
- Decimal (base 10): digits 0–9.
- Hexadecimal (base 16): digits 0–9 and A–F.
Each system uses place values based on powers of its base, such as 2ⁿ in binary or 16ⁿ in hexadecimal.
7. What is the formula for converting any base to decimal?
The formula to convert a number from base b to decimal is ∑ (digit × bⁿ), where n starts from 0 at the rightmost digit.
- Write the number in expanded form.
- Multiply each digit by its base raised to the power of its position.
- Add all the terms.
Example: Convert 243₅ to decimal.
2×5² + 4×5¹ + 3×5⁰ = 50 + 20 + 3 = 73₁₀.
8. How do you convert hexadecimal to decimal?
To convert hexadecimal to decimal, multiply each digit by powers of 16 and add the results.
- Replace A–F with values 10–15.
- Assign powers of 16 from right to left.
- Multiply and add.
Example: Convert 2F₁₆ to decimal.
2×16¹ + 15×16⁰ = 32 + 15 = 47₁₀.
9. Can you give an example of converting decimal to hexadecimal?
To convert decimal to hexadecimal, repeatedly divide the number by 16 and write the remainders in reverse order.
- Divide the decimal number by 16.
- Note the remainder (0–15).
- Convert remainders 10–15 to A–F.
- Repeat until quotient becomes 0.
Example: Convert 254₁₀ to hexadecimal.
254 ÷ 16 = 15 remainder 14 (E)
15 ÷ 16 = 0 remainder 15 (F)
So, 254₁₀ = FE₁₆.
10. What are common mistakes in number system conversion?
Common mistakes in number system conversion include using the wrong base or incorrect place values.
- Forgetting to start powers from 0 on the right.
- Not reversing remainders in repeated division method.
- Confusing hexadecimal letters (A = 10, F = 15).
- Grouping binary digits incorrectly when converting to octal (3 bits) or hexadecimal (4 bits).
Always check the base and verify the final answer by reconverting if needed.
