What are Real Numbers in Maths?
The concept of real numbers plays a key role in mathematics and is widely applicable to both real-life situations and exam scenarios. Whether you're measuring, calculating, or solving equations, understanding real numbers is essential for Class 9, Class 10, and competitive exams.
What Is Real Numbers?
A real number is defined as any value on the number line, including all rational and irrational numbers. This means real numbers include natural numbers, whole numbers, integers, fractions, decimals, terminating and non-terminating decimals (repeating or not), and roots of positive numbers. They do not include imaginary numbers like √-1 or 2 + 3i. The set of real numbers is denoted by ℝ (R). You’ll find this concept applied in arithmetic, algebra, and geometry.
- Examples of real numbers: 3, 0, -7, 2/5, 1.75, π, √2
- Non-examples: √-4, 3 + 2i, 1/0
Categories of Real Numbers
All numbers except complex numbers are real numbers. These are divided into several key categories as shown in the table below.
| Number Set | Description | Examples |
|---|---|---|
| Natural Numbers (ℕ) | Counting numbers from 1 onwards | 1, 2, 3, ... |
| Whole Numbers (𝒶) | Natural numbers plus zero | 0, 1, 2, 3, ... |
| Integers (ℤ) | Positive, negative numbers & zero | ..., -3, 0, 2, ... |
| Rational Numbers (ℚ) | Can be written as p/q, q ≠ 0 | 1/2, -5, 3.75, 0.3̅ |
| Irrational Numbers (ℝ\ℚ) | Non-repeating, non-terminating decimals | π, √2, 0.1010010001... |
Symbol and Representation
The symbol for real numbers is ℝ (R). Every real number can be shown as a point on the number line—a visual way to see how negatives, zero, fractions, and irrationals all fit together.
Properties of Real Numbers
| Property | Example |
|---|---|
| Closure | 2 + 3 = 5 (real), 4 × π = 4π (real) |
| Commutative | a + b = b + a; 2 + 5 = 5 + 2 |
| Associative | (a + b) + c = a + (b + c); (1 + 2) + 3 = 1 + (2 + 3) |
| Identity | a + 0 = a, a × 1 = a |
| Distributive | a × (b + c) = a × b + a × c |
Examples and Non-Examples of Real Numbers
| Number | Type | Real/Not Real |
|---|---|---|
| 7 | Integer | Real |
| -4.5 | Rational | Real |
| 0 | Whole number | Real |
| √2 | Irrational | Real |
| π (pi) | Irrational | Real |
| √-3 | Imaginary | NOT Real |
| 1/0 | Undefined | NOT Real |
Step-by-Step Illustration: Classifying a Number
1. Check if 3/7 is a real number.2. 3/7 is a fraction where denominator ≠ 0, so it's a rational number.
3. All rational numbers are real numbers.
4. Final Answer: 3/7 is a real number.
Real Numbers in Daily Life
Real numbers show up everywhere: measuring a room in meters (decimal numbers), a bank balance (can include negative or fractional values), the value of π in engineering, and roots in science formulas. Almost every calculation in school or in jobs uses real numbers.
Try These Yourself
- List five irrational real numbers.
- Is 0.25 a real number?
- Identify which numbers below are not real: 2, -1, √5, √-7, 3 + i.
- Arrange these in increasing order: -2, 0, π, 1.6, √3.
Frequent Errors and Misunderstandings
- Confusing real numbers with only whole/integer values.
- Thinking π or √2 are not real because they aren’t fractions.
- Forgetting that negative numbers and zero are real.
- Believing imaginary numbers or undefined values (like 1/0) are real.
Relation to Other Concepts
Real numbers connect closely with rational numbers, irrational numbers, and the overall number system. Mastering this helps in topics like algebra, decimal expansion, the Fundamental Theorem of Arithmetic, and even coordinate geometry.
Classroom Tip
A good way to remember real numbers is to picture the number line—if you can point to a value on it, it’s a real number! Vedantu’s teachers use simple visual aids like Venn diagrams and color-coded number lines to help students recall the subdivisions quickly.
We explored real numbers—from basic definition, properties, types, and common mistakes, to how they connect to board exam topics. Continue practicing with Vedantu to strengthen your maths skills and feel confident in questions about real numbers for any test or real-world task!
Explore more: Rational Numbers | Irrational Numbers | Number Line | Types of Numbers
FAQs on Real Numbers
1. What are real numbers in mathematics?
In mathematics, real numbers include all rational and irrational numbers. They make up the complete set of numbers found on the number line, such as whole numbers, fractions, decimals, and numbers like $\pi$ and $\sqrt{2}$ that cannot be written as simple fractions.
2. How do we classify different types of real numbers?
We classify real numbers into the following categories:
- Natural numbers
- Whole numbers
- Integers
- Rational numbers
- Irrational numbers
3. What is the difference between rational and irrational numbers?
The key difference is that rational numbers can be written as a fraction (like $\frac{2}{3}$), while irrational numbers cannot be expressed as a simple fraction. Examples include $0.75$ (rational) and $\sqrt{5}$ (irrational).
4. Are all integers considered real numbers?
Yes, all integers are real numbers because they are included on the number line. Integers include both positive and negative whole numbers, as well as zero, and they belong to the real number system.
5. Can real numbers be negative or zero?
Real numbers can be positive, negative, or zero. The real number system covers all possible values on the number line, so numbers like $-3$, $0$, and $4.5$ are all examples of real numbers.
6. What is the symbol used to represent real numbers?
The standard symbol for the set of real numbers is $\mathbb{R}$. So, when you see $\mathbb{R}$ in math, it refers to all real numbers, including rational and irrational values found on the number line.
7. How would you identify if a given number is real or not?
A number is a real number if you can locate it on the number line. For example, $2$, $-4.8$, and $\pi$ are real, but $\sqrt{-9}$ is not, as it represents an imaginary number and cannot be plotted on the real line.
8. Do real numbers include imaginary numbers?
No, real numbers do not include imaginary numbers. Imaginary numbers, like $i = \sqrt{-1}$, do not exist on the real number line and form a different set called complex or imaginary numbers.
9. What is the closure property of real numbers?
The closure property states that when you add, subtract, multiply, or (sometimes) divide two real numbers, the result is also a real number. For example, $2 + \sqrt{3}$ is real. But division by zero is not defined in real numbers.
10. Where do we use real numbers in daily life?
We use real numbers in daily life for activities such as measuring length, tracking temperature, shopping (prices and discounts), and calculations involving distance and time. Any quantity that can be put on a scale or measured uses real numbers.
11. Can we graph real numbers on a number line?
Yes, you can graph every real number on a number line. Each real number corresponds to a unique point on the line, whether the number is positive, negative, zero, a fraction, or an irrational value like $\pi$.
