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By Shailendra Singh
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Updated on 11 Nov 2025, 18:06 IST
Real numbers are all the numbers that can be found on the number line. This includes both rational and irrational numbers, positive and negative numbers, zero, integers, fractions, and decimals. Real numbers represent measurable quantities in the real world and form the foundation of most mathematical calculations.
A real number is any number that can express a measurable quantity along a continuous line. Real numbers can be rational (expressible as a fraction p/q where q ≠ 0) or irrational (non-terminating, non-repeating decimals).
The set of real numbers is denoted by ℝ or R.
Don't Skip: Real Numbers Important Questions
Understanding the hierarchy of real numbers helps in grasping their properties and relationships:
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Breakdown of Number Types
Real numbers follow several fundamental properties that make mathematical operations consistent and predictable.
Addition: If a, b ∈ R, then a + b ∈ R
Multiplication: If a, b ∈ R, then a × b ∈ R
Addition: a + b = b + a
Multiplication: a × b = b × a
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Addition: (a + b) + c = a + (b + c)
Multiplication: (a × b) × c = a × (b × c)
a × (b + c) = (a × b) + (a × c)
Additive Identity: a + 0 = a
Multiplicative Identity: a × 1 = a
Additive Inverse: a + (-a) = 0
Multiplicative Inverse: a × (1/a) = 1 (where a ≠ 0)
Between any two rational numbers, there exist infinitely many rational numbers.
| Aspect | Real Numbers | Complex Numbers |
| Definition | Numbers on the number line | Numbers of the form a + bi where i = √(-1) |
| Components | Single component | Two components (real and imaginary) |
| Examples | 5, -3.7, √2, π | 3 + 4i, -2 + 5i, 7i |
| Number Line | Can be plotted on a single line | Require a 2D plane (complex plane) |
| Square Roots | Negative numbers have no real square roots | All numbers have complex square roots |
| Symbol | ℝ | ℂ |
| Subset | Real numbers ⊂ Complex numbers | Complex numbers include all real numbers when b = 0 |
Real numbers represent all quantities that can be measured on a single continuous line, while complex numbers extend this concept to two dimensions, allowing solutions to equations like x² = -1, which have no real solutions.
This is one of the most famous proofs in mathematics, using the method of contradiction.
Theorem: √2 is an irrational number.
Proof:
Step 1: Assume, for contradiction, that √2 is rational.
Then √2 can be expressed as p/q where p and q are co-prime integers (no common factor other than 1), and q ≠ 0.
Step 2: If √2 = p/q, then squaring both sides:
Step 3: This means p² is even (since it equals 2q²).
Step 4: Substituting p = 2m into 2q² = p²:
Step 5: This means q² is even, so q must also be even.
Step 6: If both p and q are even, they have a common factor of 2.
Final Thought: Our assumption was wrong. Therefore, √2 cannot be expressed as p/q, which means √2 is irrational.
Using the same method of contradiction, we can prove:
Every real number corresponds to a unique point on the number line, and every point on the number line represents a unique real number.
Integers and Fractions:
Example: To represent 7/4:
Method to represent √2:
Method to represent √3:
This process can be continued to represent √4, √5, √6, etc.
Real numbers can be added, subtracted, multiplied, and divided (except division by zero).
| Operation | Closure Property | Example |
| Addition | ✓ Closed | 3.5 + 2.7 = 6.2 ∈ R |
| Subtraction | ✓ Closed | 8 - 3.2 = 4.8 ∈ R |
| Multiplication | ✓ Closed | 2.5 × 4 = 10 ∈ R |
| Division | ✓ Closed (except ÷ 0) | 10 ÷ 2.5 = 4 ∈ R |
What does "Closed" mean? A set is closed under an operation if performing that operation on any two elements of the set always produces another element of the set.
Rational + Rational = Rational
Irrational + Irrational = Can be Rational or Irrational
Rational + Irrational = Irrational
Rational × Irrational = Irrational (when rational ≠ 0)
Irrational × Irrational = Can be Rational or Irrational
Statement: Given any two positive integers a and b (a > b), there exist unique integers q and r such that:
a = bq + r, where 0 ≤ r < b
Where:
Algorithm Steps:
Example: Find HCF of 867 and 255
867 = 255 × 3 + 102 255 = 102 × 2 + 51 102 = 51 × 2 + 0
Since remainder = 0, HCF(867, 255) = 51
For any two positive integers a and b:
HCF(a, b) × LCM(a, b) = a × b
This formula is extremely useful when you know either HCF or LCM and need to find the other.
Every composite number can be expressed (factorized) as a product of prime numbers in a unique way, except for the order in which the prime factors occur.
1. Finding HCF: HCF = Product of smallest powers of common prime factors
Example: HCF(12, 15, 21)
2. Finding LCM: LCM = Product of greatest powers of all prime factors
Example: LCM(12, 15, 21)
Let x = p/q be a rational number such that the prime factorization of q is of the form 2^m × 5^n (where m, n are non-negative integers).
Then x has a decimal expansion which terminates.
Examples:
Let x = p/q be a rational number such that the prime factorization of q is not of the form 2^m × 5^n.
Then x has a decimal expansion which is non-terminating repeating.
Examples:
To check if p/q has a terminating decimal:
| Formula/Property | Mathematical Representation | Explanation |
| Euclid's Division Lemma | a = bq + r (0 ≤ r < b) | For any two positive integers a, b |
| HCF-LCM Relationship | HCF(a,b) × LCM(a,b) = a × b | Product relationship for two numbers |
| Square of any positive integer | n² = 3m or 3m+1 | Never of form 3m+2 |
| Rational numbers between a and b | (a+b)/2, (2a+b)/3, etc. | Infinitely many |
| Density property | Between any two rationals, infinite rationals exist | Fundamental property |
| Irrational + Irrational | Can be rational or irrational | √2 + √2 = 2√2; √2 + (-√2) = 0 |
| Rational + Irrational | Always irrational | 3 + √2 is irrational |
| Rational × Irrational | Irrational (if rational ≠ 0) | 2√3 is irrational |
| Terminating decimal condition | q = 2^m × 5^n | p/q terminates if denominator has only 2, 5 |
For positive integers a, b with a > b, there exist unique integers q, r such that a = bq + r where 0 ≤ r < b.
Every composite number can be uniquely expressed as a product of prime numbers (except for order).
Numbers like √2, √3, √5, π, e cannot be expressed as p/q and are irrational.
Real numbers form the backbone of mathematics and have vast applications in science, engineering, economics, and daily life. Understanding the classification, properties, and operations of real numbers is crucial for building a strong foundation in mathematics.
Important Notes:
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Real numbers are a combination of rational and irrational numbers. They include all numbers that can be plotted on a number line, such as natural numbers (1, 2, 3...), whole numbers (0, 1, 2, 3...), integers (...-3, -2, -1, 0, 1, 2, 3...), fractions, terminating decimals, non-terminating repeating decimals, and non-terminating non-repeating decimals. Real numbers encompass virtually every number used in everyday mathematics.
Rational numbers can be expressed as p/q where p and q are integers and q ≠ 0. Examples include 3/5, -17/19, and 7.123. They have either terminating decimal expansions (like 0.5) or non-terminating repeating decimal expansions (like 0.333...).
Irrational numbers cannot be written as simple fractions. They have non-terminating non-repeating decimal expansions. Common examples include √2, √3, √5, π, and e. These numbers continue infinitely without forming any repeating pattern.
A decimal is rational if it either terminates (ends) or has a repeating pattern. For example, 0.75 terminates, while 0.333... repeats. A decimal is irrational if it continues infinitely without any repeating pattern, such as 3.141592... (π) or 1.414213... (√2).
Euclid's Division Lemma states that for any two positive integers a and b (where a > b), there exist unique integers q and r such that a = bq + r, where 0 ≤ r < b. Here, 'a' is the dividend, 'b' is the divisor, 'q' is the quotient, and 'r' is the remainder.
To find the HCF (Highest Common Factor) of two numbers using Euclid's algorithm:
For example, to find HCF of 135 and 225: 225 = 135 × 1 + 90, then 135 = 90 × 1 + 45, then 90 = 45 × 2 + 0. Therefore, HCF is 45.
For any two numbers a and b: HCF(a,b) × LCM(a,b) = a × b. This relationship allows you to find one value if you know the other three. For instance, if you know two numbers and their HCF, you can calculate their LCM using this formula.
The Fundamental Theorem of Arithmetic states that every composite number can be expressed as a product of prime numbers in a unique way, except for the order in which the prime factors occur. For example, 90 = 2 × 3 × 3 × 5 = 2 × 3² × 5. This factorization is unique regardless of how you arrange the factors.
To find prime factorization:
Example: 1296 = 2 × 2 × 2 × 2 × 3 × 3 × 3 × 3 = 2⁴ × 3⁴
Using prime factorization:
For example, for 12 (2² × 3) and 18 (2 × 3²):
A rational number p/q (in lowest terms) has a terminating decimal expansion if and only if the prime factorization of q is of the form 2ⁿ × 5ᵐ, where n and m are non-negative integers. For example, 7/8 terminates because 8 = 2³, and 13/250 terminates because 250 = 2 × 5³.
A rational number p/q (in simplest form) has a non-terminating repeating decimal expansion if the prime factorization of q is NOT of the form 2ⁿ × 5ᵐ. For instance, 1/3 = 0.333... (repeating) because 3 is not expressible as 2ⁿ × 5ᵐ.
Yes, repeating decimals can always be expressed as fractions. For example, to convert 0.363636... to a fraction, let x = 0.363636..., then 100x = 36.363636.... Subtracting the first equation from the second: 99x = 36, so x = 36/99 = 4/11.
The proof uses contradiction:
If 3 + 2√5 were rational, then 2√5 would equal (rational number) - 3, which would also be rational. This would make √5 rational (by dividing by 2). However, √5 is irrational, creating a contradiction. Therefore, 3 + 2√5 must be irrational.
The number of trailing zeroes in n! depends on how many times 10 (2 × 5) divides the factorial. Since there are always more factors of 2 than 5, count the factors of 5. For example, 10! has 2 trailing zeroes because 10! contains two factors of 5 (from 5 and 10).
The units digit follows a pattern based on the base number raised to different powers. For numbers ending in specific digits, the pattern repeats every 4 powers. For instance, numbers ending in 3 cycle through 3, 9, 7, 1. To find the units digit of 3⁴⁷, divide 47 by 4 (remainder 3), so the units digit follows the 3rd position in the cycle: 7.
This is found by calculating the HCF of the differences between the numbers. For example, if you need the largest number dividing 398, 436, and 542 leaving remainders 7, 11, and 15 respectively, find HCF of (398-7), (436-11), and (542-15), which equals HCF(391, 425, 527).
No. Real numbers include both rational numbers (like 3/4, 0.5, 2) and irrational numbers (like √2, π, e).
Yes, irrational numbers can be geometrically constructed and represented on the number line using compass and ruler.
Recurring decimals repeat a pattern and are rational (e.g., 0.333...). Non-recurring decimals don't repeat and are irrational (e.g., 0.10110111011110...).
No, √4 = 2, which is a rational number.
Find HCF of first two numbers, then find HCF of this result with the third number.
