What Is Number System Definition Types Properties and Solved Examples
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 Mathematics Complete Guide
1. What is a number system in mathematics?
A number system is a structured way of representing and expressing numbers using digits or symbols according to fixed rules. In mathematics, number systems classify numbers into different types such as:
- Natural numbers (N): 1, 2, 3, ...
- Whole numbers (W): 0, 1, 2, 3, ...
- Integers (Z): ..., -2, -1, 0, 1, 2, ...
- Rational numbers (Q): Numbers in the form p/q
- Irrational numbers: Non-terminating, non-repeating decimals
- Real numbers (R): All rational and irrational numbers
2. What are the types of number systems?
The main types of number systems are natural, whole, integers, rational, irrational, and real numbers. They are classified as follows:
- Natural Numbers (N): Counting numbers starting from 1
- Whole Numbers (W): Natural numbers including 0
- Integers (Z): Positive and negative whole numbers including 0
- Rational Numbers (Q): Numbers expressed as p/q where q ≠ 0
- Irrational Numbers: Numbers that cannot be written as p/q (e.g., √2, π)
- Real Numbers (R): All rational and irrational numbers combined
3. What is the difference between rational and irrational numbers?
The key difference is that rational numbers can be written as a fraction p/q, while irrational numbers cannot be expressed as a simple fraction. Differences include:
- Rational numbers have terminating or repeating decimals (e.g., 1/2 = 0.5, 1/3 = 0.333...)
- Irrational numbers have non-terminating, non-repeating decimals (e.g., √2, π)
- Every integer is rational, but not every real number is rational
4. How do you convert a rational number into decimal form?
A rational number is converted into decimal form by dividing the numerator by the denominator. Steps:
- Write the number in p/q form
- Divide p by q using long division
- If division ends, it is terminating; if it repeats, it is recurring
5. What are real numbers in the number system?
The real numbers (R) include all rational and irrational numbers on the number line. This means real numbers consist of:
- Integers
- Fractions
- Terminating and repeating decimals
- Non-terminating, non-repeating decimals
6. What is the difference between natural numbers and whole numbers?
The main difference is that whole numbers include 0 while natural numbers start from 1. Specifically:
- Natural Numbers (N): 1, 2, 3, ...
- Whole Numbers (W): 0, 1, 2, 3, ...
7. How do you represent numbers on a number line?
Numbers are represented on a number line by marking points at equal intervals from zero. Steps:
- Draw a horizontal line and mark 0 at the center
- Mark positive numbers to the right of 0
- Mark negative numbers to the left of 0
- Fractions and decimals are placed between integers proportionally
8. What is a prime number in the number system?
A prime number is a natural number greater than 1 that has exactly two factors: 1 and itself. Examples include:
- 2 (smallest prime number)
- 3, 5, 7, 11, 13
9. Is zero a rational number?
Yes, 0 is a rational number because it can be written in the form p/q where q ≠ 0. For example:
- 0 = 0/1
- 0 = 0/5
10. What are the properties of real numbers?
The properties of real numbers include closure, commutative, associative, distributive, identity, and inverse properties. Key properties are:
- Closure: a + b and a × b are real numbers
- Commutative: a + b = b + a
- Associative: (a + b) + c = a + (b + c)
- Distributive: a(b + c) = ab + ac
- Identity: a + 0 = a, a × 1 = a
