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 and can you give some examples?
Real numbers are the set of numbers that includes all rational and irrational numbers. Essentially, any number that can be represented on a number line is a real number. The symbol for the set of real numbers is ℝ. Examples include:
- Integers: -5, 0, 10
- Fractions: 1/2, -7/3
- Decimals: 1.5, -0.875
- Irrational Numbers: π (pi), √2 (the square root of 2)
2. What is the main difference between real numbers and integers?
The main difference is that integers are a specific subset of real numbers. Integers are whole numbers, including positive numbers, negative numbers, and zero (e.g., -3, -2, -1, 0, 1, 2, 3). Real numbers, however, encompass integers and all the values in between them, such as fractions (1/2), decimals (0.75), and irrational numbers (π). Therefore, every integer is a real number, but not every real number is an integer.
3. What are the key properties that define real numbers?
Real numbers follow several fundamental properties for addition and multiplication that are crucial in algebra:
- Commutative Property: The order does not matter (a + b = b + a; a × b = b × a).
- Associative Property: The grouping does not matter (a + (b + c) = (a + b) + c).
- Distributive Property: Multiplication distributes over addition (a × (b + c) = a × b + a × c).
- Identity Property: There is an additive identity (0) and a multiplicative identity (1).
- Inverse Property: Every real number has an additive inverse (-a) and every non-zero real number has a multiplicative inverse (1/a).
4. How are real numbers represented on a number line?
Every real number corresponds to a unique point on the number line. The line is infinitely long in both directions with zero (0) at the center. Positive real numbers are located to the right of zero, and negative real numbers are to the left. This visual representation helps in comparing numbers and understanding their order and magnitude.
5. What kinds of numbers, like the square root of a negative number, are not real numbers?
Numbers that cannot be located on the number line are not real numbers. These are primarily imaginary numbers, which arise from taking the square root of a negative number (e.g., √-1, which is denoted as 'i'). Numbers that combine a real part and an imaginary part are called complex numbers (e.g., 5 + 2i) and are also not part of the real number set.
6. How does Euclid's Division Lemma, a concept for integers, relate to the broader set of real numbers in Class 10?
Euclid's Division Lemma states that for any two positive integers 'a' and 'b', there exist unique integers 'q' and 'r' such that a = bq + r, where 0 ≤ r < b. This lemma is a tool that applies specifically to the subset of integers within the real number system. In the Class 10 curriculum, its primary application is to find the Highest Common Factor (HCF) of two integers, which is a key concept within the chapter on Real Numbers. It does not apply to irrational or fractional real numbers.
7. Why are irrational numbers like π (pi) considered real numbers if their decimals go on forever without repeating?
The defining characteristic of a real number is that it must have a specific location on the number line. Although the decimal representation of an irrational number like π (approx. 3.14159...) is non-terminating and non-repeating, it corresponds to a precise, unique point on the number line. This existence on the number line is what qualifies it as a real number, distinguishing it from imaginary numbers which have no place on this line.
8. How can the decimal expansion of a number help determine if it is rational or irrational?
The decimal form of a number is a clear indicator of its type:
- Rational Numbers: Their decimal expansion either terminates (e.g., 0.5) or is non-terminating but repeating (e.g., 0.666...).
- Irrational Numbers: Their decimal expansion is both non-terminating and non-repeating (e.g., the value of √3 = 1.7320508...).
This distinction is a fundamental concept for classifying numbers within the real number system as per the CBSE syllabus.
9. Why is proving a number irrational (e.g., √2) a key topic in the Class 10 Real Numbers chapter?
Proving numbers like √2 are irrational is a crucial topic because it introduces students to the method of proof by contradiction. This is a fundamental technique in higher mathematics and helps develop logical reasoning skills. As per the NCERT syllabus for the 2025-26 session, this exercise solidifies a student's understanding of the fundamental difference between the properties of rational and irrational numbers.
10. Beyond basic measurements, what is a real-world example that illustrates the concept of an irrational number?
A great real-world example is in finance, specifically in calculating compound interest over certain periods, which can result in irrational growth factors. Another powerful example is the Golden Ratio (φ ≈ 1.61803...), an irrational number that appears frequently in nature, describing patterns in things like the spiral arrangements of sunflower seeds, the structure of seashells, and the branching of trees.
