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 are the 4 types of number system?
The four types of number systems commonly taught in mathematics are:
- Binary Number System: Uses only two digits, 0 and 1. Widely used in computers and digital systems.
- Octal Number System: Consists of eight digits, from 0 to 7. It is less common but sometimes used in computing.
- Decimal Number System: Uses ten digits, from 0 to 9, and is the standard system for everyday counting.
- Hexadecimal Number System: Has sixteen symbols (0-9 and A-F), often used in programming and digital electronics.
2. What is the definition of a number system?
A number system is a systematic way of expressing numbers using specific symbols and rules. It enables us to represent, read, write, and perform mathematical operations on numbers efficiently. The most common example is the decimal number system, which is based on ten digits. At Vedantu, students learn how various number systems are structured and how to convert numbers from one system to another.
3. Who invented 1, 2, 3, 4, 5, 6, 7, 8, 9?
The numerals 1, 2, 3, 4, 5, 6, 7, 8, 9 are known as Hindu-Arabic numerals. They were developed in ancient India, and later transmitted to the Islamic world, where mathematicians improved them further. Eventually, they spread to Europe and became the standard numerals used worldwide today. Vedantu’s expert educators explore this rich history, connecting the evolution of numerals with current mathematical learning.
4. What is the number system used today called?
The number system used today is called the decimal number system or the base-10 system. It employs ten digits, from 0 to 9, and is most widely used for mathematical computations, transactions, and everyday counting. Vedantu provides detailed lessons to help students master decimal system concepts and operations, essential for academic success.
5. What is the importance of number systems in mathematics?
Number systems are fundamental to all branches of mathematics. Their importance includes:
- Enabling representation and classification of numbers.
- Facilitating various arithmetic operations ($+, -, \times, \div$) and mathematical calculations.
- Supporting problem-solving in fields like computer science, physics, and engineering.
- Providing the foundation for understanding advanced mathematical concepts taught at Vedantu, such as algebra, programming, and data analysis.
6. How do you convert numbers between different number systems?
To convert numbers between different number systems, follow these steps:
- Identify the base (radix) of both the source and target number system.
- For converting from decimal to another system, use successive division by the target base and record remainders.
- For converting to decimal, multiply each digit by its base raised to the appropriate power and sum the results: $$N = \sum_{i=0}^{n-1} d_i \times b^i$$ where $d_i$ is the digit and $b$ is the base.
7. What is the difference between binary and decimal number systems?
Binary number system (base-2) uses only two digits: 0 and 1, primarily for digital and computer-based applications. In contrast, the decimal number system (base-10) uses ten digits: 0 to 9, and is used in everyday mathematics. Each binary position represents a power of 2 ($2^n$), while each decimal position represents a power of 10 ($10^n$). Vedantu’s digital math courses help students practice and differentiate between these two essential systems.
8. What is the role of place value in a number system?
Place value is a key concept in any positional number system. The value of a digit depends on its position within the number, with each position representing a specific power of the base. For example, in 345 (decimal system):
- 3 is in the hundreds place: $3 \times 10^2 = 300$
- 4 is in the tens place: $4 \times 10^1 = 40$
- 5 is in the ones place: $5 \times 10^0 = 5$
9. How are number systems applied in computer science?
In computer science, number systems such as binary, octal, and hexadecimal play an essential role. Computers utilize binary code for data processing, memory storage, and communications. Hexadecimal is used as a human-friendly representation of binary-coded values, especially in programming. Vedantu’s computer science modules introduce students to practical applications of these number systems through engaging projects and coding exercises.
10. Which number system concepts are most important for competitive exams?
For competitive exams like JEE, NEET, and Olympiads, key number system concepts include:
- Conversion between number systems
- Understanding binary, decimal, octal, and hexadecimal operations
- Application of place value and number representation
- Problem-solving with number bases and digit manipulation
