How to Convert Between Decimal, Binary, Octal, and Hexadecimal Numbers?
The concept of number system plays a key role in mathematics and is widely applicable to both real-life situations and exam scenarios.
What Is Number System?
A number system is a mathematical notation for expressing numbers using a specific set of digits or symbols, according to defined rules. You’ll find this concept applied in areas such as decimal numbers, binary code in computers, and measurement conversions.
Types of Number Systems
| Name | Digits Used | Base | Example | Where Used? |
|---|---|---|---|---|
| Decimal | 0–9 | 10 | 528 | Everyday maths |
| Binary | 0, 1 | 2 | 1011 | Computers |
| Octal | 0–7 | 8 | 157 | Programming |
| Hexadecimal | 0–9, A–F | 16 | 9AF | Digital systems |
Key Concepts in Number System
In maths, number systems help us represent and manipulate numbers conveniently. Key points include:
- The base (or radix) tells how many symbols are used. For example, decimal is base-10.
- Any number can be written as a sum using place value (like 327 = 3×100 + 2×10 + 7×1).
- We often convert numbers between different number systems for use in maths or computer science.
Number System Chart
| Decimal | Binary | Octal | Hexadecimal |
|---|---|---|---|
| 0 | 0 | 0 | 0 |
| 1 | 1 | 1 | 1 |
| 2 | 10 | 2 | 2 |
| 10 | 1010 | 12 | A |
Number System Conversion: Step-by-Step Illustration
- Convert decimal 25 to binary:
1. Divide 25 by 2 → Quotient = 12, Remainder = 1
2. Divide 12 by 2 → Quotient = 6, Remainder = 0
3. Divide 6 by 2 → Quotient = 3, Remainder = 0
4. Divide 3 by 2 → Quotient = 1, Remainder = 1
5. Divide 1 by 2 → Quotient = 0, Remainder = 1
Read the remainders from bottom to top: 11001
Classification of Numbers
| Type | Definition/Examples |
|---|---|
| Natural Numbers | Counting numbers: 1, 2, 3, ... |
| Whole Numbers | 0 and all natural numbers |
| Integers | ..., -3, -2, -1, 0, 1, 2, 3, ... |
| Rational Numbers | Fractions and integers: 2/3, 7, -5 |
| Irrational Numbers | π, √2 (cannot be written as fraction) |
| Real Numbers | Rational + Irrational numbers |
| Complex Numbers | a + bi (where i = √-1) |
Speed Trick or Vedic Shortcut
Here’s a quick shortcut for multiplying two 2-digit numbers near 100 using the base method, which is helpful for exams:
- Let’s multiply 97 × 98:
100 − 97 = 3, 100 − 98 = 2
97 − 2 = 95
3 × 2 = 6 (write as 06)
Answer = 9506
Tricks like this are explained step-by-step in Vedantu’s number system practice questions to help you with competitive exams.
Try These Yourself
- Write 57 in binary, octal, and hexadecimal.
- Check if 129 is an odd or even number.
- Convert 11011 (binary) into decimal.
- List all whole numbers between 10 and 20.
Frequent Errors and Misunderstandings
- Mixing up binary values with decimal values when converting numbers.
- Forgetting place value for bases other than 10.
- Misinterpreting the symbols used in hexadecimal (A–F = 10–15).
Relation to Other Concepts
The idea of number system connects closely with topics such as types of numbers and number system conversion. Mastering this helps with understanding arithmetic, algebra and even data handling in advanced maths chapters.
Classroom Tip
To easily remember number system types, use the base as a clue—binary (2), octal (8), decimal (10), and hexadecimal (16). Vedantu’s teachers often suggest drawing a quick chart or table as you solve problems for faster conversion.
We explored number system—from definition, types, conversion, examples, tricks, and connections to other maths concepts. Practice more using Vedantu’s worksheets for quick revision before exams. Keep practicing for confidence!
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FAQs on Number System in Maths: Definition, Types & Conversion
1. What is a number system in Maths?
A number system in mathematics is a method of representing numbers using digits or symbols. It defines the rules for representing, naming, and performing operations on numbers. Common number systems include the decimal (base 10), binary (base 2), octal (base 8), and hexadecimal (base 16) systems. Understanding number systems is crucial for various mathematical calculations and computer science applications.
2. What are the different types of number systems?
The most common types of number systems are:
- Decimal: Base 10, using digits 0-9.
- Binary: Base 2, using digits 0 and 1.
- Octal: Base 8, using digits 0-7.
- Hexadecimal: Base 16, using digits 0-9 and letters A-F (A=10, B=11, C=12, D=13, E=14, F=15).
3. How do I convert a decimal number to binary?
To convert a decimal number to binary, repeatedly divide the decimal number by 2, recording the remainders. The binary equivalent is the sequence of remainders read 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
4. How do I convert a binary number to decimal?
To convert a binary number to decimal, multiply each digit by the corresponding power of 2 (starting from the rightmost digit with 20, then 21, 22, and so on). Sum the results. For example, converting 11012 to decimal: (1 × 23) + (1 × 22) + (0 × 21) + (1 × 20) = 8 + 4 + 0 + 1 = 1310
5. What is the difference between rational and irrational numbers?
Rational numbers can be expressed as a fraction p/q where p and q are integers, and q is not zero. Irrational numbers cannot be expressed as such a fraction; their decimal representation is non-terminating and non-repeating (e.g., π and √2).
6. What are real numbers?
Real numbers encompass all rational and irrational numbers. They can be represented on a number line and include positive numbers, negative numbers, and zero.
7. What are complex numbers?
Complex numbers are numbers of the form a + bi, where 'a' and 'b' are real numbers, and 'i' is the imaginary unit (√-1). They extend the real number system to include numbers with an imaginary component.
8. What is the significance of the number system in computer science?
The binary number system (base 2) is fundamental in computer science because computers use electronic circuits that represent data as sequences of 0s and 1s (on/off states). Hexadecimal (base 16) is often used as a shorthand notation for binary data due to its compactness.
9. How are number systems used in daily life?
We use the decimal number system daily for counting, measuring, and performing calculations. Other number systems might be less visible but are present in various technologies (e.g., binary in computers and digital devices).
10. What is the base of a number system?
The base (or radix) of a number system is the number of unique digits it uses to represent numbers. For example, the decimal system has a base of 10, the binary system has a base of 2, and the hexadecimal system has a base of 16.
11. How do I convert between octal and hexadecimal numbers?
Conversion between octal and hexadecimal is best done via the decimal system as an intermediary. First, convert the octal or hexadecimal number to decimal, then convert the resulting decimal number to the target base (octal or hexadecimal).
