1. Real Numbers – The Number System
In our daily life, we use different types of numbers. Classifying them properly helps us understand their properties. Here are the main types of numbers you must know.
The counting numbers 1, 2, 3, 4, 5, 6, … are called Natural Numbers. They are used for counting objects. The set of Natural Numbers is denoted by \(\mathbb{N}\).
Natural Numbers together with 0 form the set of Whole Numbers: 0, 1, 2, 3, 4, …. The set is denoted by \(\mathbb{W}\).
Whole Numbers and their negatives form the set of Integers: …, −3, −2, −1, 0, 1, 2, 3, …. The set is denoted by \(\mathbb{Z}\).
A number that can be expressed in the form \(\dfrac{p}{q}\), where \(p\) and \(q\) are integers and \(q \neq 0\), is called a Rational Number.
Examples: \(\dfrac{2}{3},\; \dfrac{-5}{7},\; 0.5 = \dfrac{1}{2},\; 7 = \dfrac{7}{1}\)
Every integer is a rational number (take \(q = 1\)).
A number that cannot be expressed in the form \(\dfrac{p}{q}\) (where \(p, q\) are integers and \(q \neq 0\)) is called an Irrational Number.
Examples: \(\sqrt{2},\; \sqrt{3},\; \sqrt{5},\; \pi\)
All Rational Numbers and Irrational Numbers together form the set of Real Numbers, denoted by \(\mathbb{R}\). Every number on the number line is a Real Number.
Number System Hierarchy
The following diagram shows how the number systems are related:
- Real Numbers
- Rational Numbers
- Integers
- Whole Numbers
- Natural Numbers (1, 2, 3, 4, …)
- Whole Numbers
- Non-integer Rational Numbers (e.g. \(\frac{2}{3}, -\frac{5}{7}\))
- Integers
- Irrational Numbers (e.g. \(\sqrt{2}, \sqrt{3}, \pi\))
- Rational Numbers
Every Natural Number is a Whole Number. Every Whole Number is an Integer. Every Integer is a Rational Number. Every Rational Number is a Real Number. But the reverse is not always true.
2. Fundamental Theorem of Arithmetic
Fundamental Theorem of Arithmetic: Every composite number can be expressed (factorised) as a product of primes, and this factorisation is unique, apart from the order in which the prime factors occur.
Before we use this theorem, let us recall what prime and composite numbers are.
A number greater than 1 that has exactly two factors — 1 and itself — is called a Prime Number.
Examples: 2, 3, 5, 7, 11, 13, 17, 19, 23, …
A number greater than 1 that has more than two factors is called a Composite Number.
Examples: 4, 6, 8, 9, 10, 12, 14, 15, …
The number 1 is neither prime nor composite.
Prime Factorisation
Prime Factorisation means writing a composite number as a product of its prime factors. We find the prime factorisation by repeatedly dividing by the smallest prime that divides the number.
\(60 = 2 \times 2 \times 3 \times 5 = 2^2 \times 3 \times 5\)
\(84 = 2^2 \times 3 \times 7\)
\(72 = 2^3 \times 3^2\)
Prime factorisation is the foundation for finding HCF and LCM. The Fundamental Theorem of Arithmetic guarantees that every number has a unique prime factorisation, so the HCF and LCM we calculate are definite and reliable.
3. HCF Using Prime Factorisation
The Highest Common Factor (HCF) of two or more numbers is the largest number that divides all of them exactly.
HCF = Product of common prime factors with the smallest powers.
Method
- Find the prime factorisation of each number.
- Identify the prime factors that are common to all the numbers.
- For each common prime factor, take the smallest power appearing in the factorisations.
- Multiply these to get the HCF.
Prime factorisations:
\(12 = 2^2 \times 3\)
\(18 = 2 \times 3^2\)
Common prime factors: 2 and 3
Smallest power of 2: \(2^1\); Smallest power of 3: \(3^1\)
HCF = \(2 \times 3 = 6\)
Prime factorisations:
\(60 = 2^2 \times 3 \times 5\)
\(84 = 2^2 \times 3 \times 7\)
\(72 = 2^3 \times 3^2\)
Common prime factors: 2 and 3
Smallest power of 2: \(2^2\); Smallest power of 3: \(3^1\)
HCF = \(2^2 \times 3 = 4 \times 3 = 12\)
Prime factorisations:
\(90 = 2 \times 3^2 \times 5\)
\(126 = 2 \times 3^2 \times 7\)
Common prime factors: 2 and 3
Smallest power of 2: \(2^1\); Smallest power of 3: \(3^2\)
HCF = \(2 \times 9 = 18\)
Students often take the largest power instead of the smallest power for HCF. Remember: for HCF, always choose the smallest power of each common factor.
4. LCM Using Prime Factorisation
The Least Common Multiple (LCM) of two or more numbers is the smallest number that is exactly divisible by all of them.
LCM = Product of all prime factors with the greatest powers.
Method
- Find the prime factorisation of each number.
- List all prime factors that appear in any of the factorisations.
- For each prime factor, take the greatest power appearing in the factorisations.
- Multiply these to get the LCM.
\(12 = 2^2 \times 3\)
\(18 = 2 \times 3^2\)
All prime factors: 2 and 3
Greatest power of 2: \(2^2\); Greatest power of 3: \(3^2\)
LCM = \(2^2 \times 3^2 = 4 \times 9 = 36\)
\(60 = 2^2 \times 3 \times 5\)
\(84 = 2^2 \times 3 \times 7\)
\(72 = 2^3 \times 3^2\)
All prime factors: 2, 3, 5, 7
Greatest powers: \(2^3\), \(3^2\), \(5^1\), \(7^1\)
LCM = \(8 \times 9 \times 5 \times 7 = 2520\)
\(90 = 2 \times 3^2 \times 5\)
\(126 = 2 \times 3^2 \times 7\)
All prime factors: 2, 3, 5, 7
Greatest powers: \(2^1\), \(3^2\), \(5^1\), \(7^1\)
LCM = \(2 \times 9 \times 5 \times 7 = 630\)
For LCM, students sometimes omit prime factors that appear in only one of the numbers. Remember: for LCM, take all prime factors from all numbers.
5. HCF and LCM Relationship
For any two positive integers \(a\) and \(b\):
\[\text{HCF}(a, b) \times \text{LCM}(a, b) = a \times b\]
This is one of the most useful relationships in Real Numbers. If you know the HCF and can find one of the numbers, you can find the other.
HCF(12, 18) = 6, LCM(12, 18) = 36
HCF × LCM = \(6 \times 36 = 216\)
12 × 18 = 216
Both are equal. ✓
The HCF of two numbers is 12 and their LCM is 360. If one number is 72, find the other.
The other number is 60.
This formula works only for two positive integers. It does not extend directly to three or more numbers. For three numbers, you must find HCF and LCM using prime factorisation.
6. Applications of HCF and LCM
Many real-life problems can be solved by identifying whether we need HCF or LCM. Knowing which concept to apply is often the key step.
| Type of Problem | Concept |
|---|---|
| Largest possible equal groups | HCF |
| Greatest possible size that divides evenly | HCF |
| Smallest number divisible by given numbers | LCM |
| Events occurring together again | LCM |
| Finding the biggest piece / container / group | HCF |
| Finding when things line up / repeat / ring together | LCM |
There are 60 chocolates and 84 biscuits. They are to be packed in boxes such that each box has the same number of chocolates and the same number of biscuits. What is the greatest number of boxes possible?
Greatest number of boxes = 12 (5 chocolates and 7 biscuits per box).
Find the smallest number that is exactly divisible by 8, 12, and 15.
The smallest number = 120.
Three bells ring at intervals of 12 minutes, 18 minutes, and 24 minutes. If they ring together at 8:00 AM, when will they ring together again?
They will ring together again at 9:12 AM.
A rectangular garden is 96 m long and 72 m wide. It is to be divided into square plots of equal size. What is the greatest possible side length of each square plot?
Greatest possible side length = 24 m.
Two traffic lights change colours every 48 seconds and 72 seconds. If they change together now, after how many seconds will they change together again?
They will change together again after 144 seconds (2 minutes 24 seconds).
7. Irrational Numbers
A number is called an Irrational Number if it cannot be expressed in the form \(\dfrac{p}{q}\), where \(p\) and \(q\) are integers and \(q \neq 0\).
Irrational numbers cannot be written as fractions and have no exact fractional representation. Some important irrational numbers:
- \(\sqrt{2} \approx 1.41421356...\)
- \(\sqrt{3} \approx 1.73205080...\)
- \(\sqrt{5} \approx 2.23606797...\)
- \(\pi \approx 3.14159265...\)
Rational vs Irrational Numbers
| Feature | Rational Numbers | Irrational Numbers |
|---|---|---|
| Form | \(\dfrac{p}{q},\; q \neq 0\) | Cannot be written as \(\dfrac{p}{q}\) |
| Examples | \(\frac{1}{3},\; 0.5,\; -7\) | \(\sqrt{2},\; \sqrt{3},\; \pi\) |
The sum, difference, product, or quotient of a rational number and an irrational number is always irrational (provided \(q \neq 0\)). For example, \(2 + \sqrt{3}\) is irrational.
8. Proofs of Irrationality
Proving that a number is irrational is an important exam skill. The method used is called proof by contradiction.
- Assume the number is rational. So write it as \(\dfrac{p}{q}\) where \(p, q\) have no common factor (lowest terms) and \(q \neq 0\).
- Simplify to get an equation.
- Derive a contradiction — show that a prime divides both \(p\) and \(q\), contradicting the assumption that \(\dfrac{p}{q}\) is in lowest terms.
- Conclude: the assumption was wrong. Therefore the number is irrational.
Proof 1: \(\sqrt{2}\) is irrational
Proof 2: \(\sqrt{3}\) is irrational
Proof 3: \(\sqrt{5}\) is irrational
Always state the assumption clearly, show each step, and explicitly point out the contradiction. End with a concluding statement: "Therefore, our assumption is wrong, and the number is irrational."
9. Important Concepts to Remember
A rational number can be written as \(\dfrac{p}{q}\). An irrational number cannot. Together they form the Real Numbers.
A prime number has exactly two factors (1 and itself). A composite number has more than two factors. 1 is neither prime nor composite.
Writing a composite number as a product of prime numbers. By the Fundamental Theorem of Arithmetic, this factorisation is unique (apart from the order).
The product of common prime factors taken with the smallest powers. It gives the greatest number that divides all the given numbers.
The product of all prime factors taken with the greatest powers. It gives the smallest number that is divisible by all the given numbers.
For two positive integers \(a\) and \(b\): HCF(\(a, b\)) × LCM(\(a, b\)) = \(a \times b\). This does not work for three or more numbers.
Use proof by contradiction: assume the number is rational, express as \(\dfrac{p}{q}\) in lowest terms, derive that a prime divides both \(p\) and \(q\), and conclude the assumption is false.
10. Important Results
Every composite number can be expressed as a product of primes, and this factorisation is unique apart from the order of the prime factors.
Product of common prime factors using the smallest powers.
Product of all prime factors using the greatest powers.
For two positive integers \(a\) and \(b\):
\[\text{HCF}(a, b) \times \text{LCM}(a, b) = a \times b\]
A number that cannot be expressed as \(\dfrac{p}{q}\) (where \(p, q \in \mathbb{Z},\; q \neq 0\)) is irrational. Examples: \(\sqrt{2}, \sqrt{3}, \sqrt{5}\).
11. Solved Examples
Question: Which of the following are irrational? (a) \(\sqrt{9}\) (b) \(\sqrt{10}\) (c) \(\dfrac{22}{7}\) (d) \(1 + \sqrt{3}\)
Concept Used: Definition of irrational numbers.
Solution:
- (a) \(\sqrt{9} = 3\) — this is a rational number.
- (b) \(\sqrt{10}\) — 10 is not a perfect square, so \(\sqrt{10}\) is irrational.
- (c) \(\dfrac{22}{7}\) — this is in the form \(\dfrac{p}{q}\), so it is rational.
- (d) \(1 + \sqrt{3}\) — the sum of a rational number and an irrational number is irrational. So this is irrational.
Final Answer: \(\sqrt{10}\) and \(1 + \sqrt{3}\) are irrational.
Question: Find the prime factorisation of 240.
Concept Used: Prime factorisation.
Solution:
Final Answer: \(240 = 2^4 \times 3 \times 5\)
Question: Find the HCF of 180 and 252.
Concept Used: HCF using prime factorisation.
Solution:
\(180 = 2^2 \times 3^2 \times 5\)
\(252 = 2^2 \times 3^2 \times 7\)
Common factors with smallest powers: \(2^2 \times 3^2\)
Final Answer: HCF = \(4 \times 9 = 36\)
Question: Find the LCM of 24 and 36.
Concept Used: LCM using prime factorisation.
Solution:
\(24 = 2^3 \times 3\)
\(36 = 2^2 \times 3^2\)
All factors with greatest powers: \(2^3 \times 3^2\)
Final Answer: LCM = \(8 \times 9 = 72\)
Question: The HCF of two numbers is 14 and their LCM is 1512. If one number is 56, find the other.
Concept Used: HCF × LCM = product of the two numbers.
Solution:
Final Answer: The other number is 378.
Question: Three tankers contain 403 litres, 465 litres, and 496 litres of water respectively. These are to be filled into identical containers. What is the capacity of the largest container that can measure the water an exact number of times?
Concept Used: HCF gives the largest capacity that divides all quantities exactly.
Solution:
\(403 = 13 \times 31\)
\(465 = 3 \times 5 \times 31\)
\(496 = 2^4 \times 31\)
Common factor: 31
Final Answer: The largest container capacity is 31 litres.
Question: The length, breadth, and height of a room are 825 cm, 675 cm, and 450 cm respectively. Find the longest tape that can measure the dimensions of the room exactly.
Concept Used: HCF of three numbers.
Solution:
\(825 = 3 \times 5^2 \times 11\)
\(675 = 3^3 \times 5^2\)
\(450 = 2 \times 3^2 \times 5^2\)
HCF = \(3 \times 5^2 = 75\)
Final Answer: The longest tape = 75 cm.
Question: Prove that \(3 + 2\sqrt{5}\) is irrational.
Concept Used: Proof by contradiction + the fact that the sum of a rational and irrational number is irrational.
Solution:
Final Answer: \(3 + 2\sqrt{5}\) is irrational.
Question: Can two numbers have 18 as their HCF and 380 as their LCM? Give a reason.
Concept Used: The relationship HCF × LCM = product of two numbers.
Solution:
If two numbers have HCF 18 and LCM 380, then:
HCF × LCM = \(18 \times 380 = 6840\)
So the product of the two numbers must be 6840.
But the LCM must be a multiple of the HCF. Here, \(380 \div 18 = 21.11...\), which is not a whole number. So 18 does not divide 380.
Final Answer: No. The LCM must always be a multiple of the HCF. Since 18 does not divide 380, no such pair of numbers exists.
Question: Prove that \(\dfrac{1}{\sqrt{2}}\) is irrational.
Concept Used: Proof by contradiction.
Solution:
Final Answer: \(\dfrac{1}{\sqrt{2}}\) is irrational.
12. Common Mistakes to Avoid
HCF finds the largest number that divides all given numbers. LCM finds the smallest number divisible by all given numbers. Using the wrong one gives incorrect answers. Ask yourself: "Am I dividing or being divisible?"
For HCF, you must take the smallest power of each common factor, not the greatest. For example, HCF of \(2^3 \times 3^2\) and \(2^2 \times 3^3\) is \(2^2 \times 3^2\), not \(2^3 \times 3^3\).
For LCM, include all prime factors from all numbers. Students sometimes forget factors that appear in only one number.
Double-check your factorisation. A common error is stopping before reaching prime factors, e.g. writing \(60 = 4 \times 15\) instead of \(60 = 2^2 \times 3 \times 5\).
The relationship \(\text{HCF} \times \text{LCM} = a \times b\) works only for two numbers. It does not work for three or more numbers.
Always: (1) State the assumption clearly, (2) show the steps, (3) identify the contradiction explicitly, (4) conclude with a clear statement. Missing any step loses marks.
In irrationality proofs, you must state that \(p\) and \(q\) are coprime (have no common factor). This is what the contradiction refutes.
\(\dfrac{22}{7}\) is rational (it is in the form \(\dfrac{p}{q}\)), even though it is used as an approximation of \(\pi\). Don't assume a fraction-looking number involving 7 is irrational.
13. CBSE Exam Tips – Real Numbers
- Write prime factorisation clearly. Show the division steps; do not just write the answer.
- Show all important steps. CBSE awards marks for steps, not just the final answer.
- State whether you are using HCF or LCM. Write it explicitly: "We use HCF because…"
- In application problems, identify the concept first. Write one line explaining why you are finding HCF or LCM before starting calculations.
- For irrationality proofs: State the assumption, show the steps, point out the contradiction, and conclude clearly.
- Do not skip reasoning. Even if the answer is obvious to you, write the reasoning. CBSE marks depend on the steps shown.
- Check your final answer. Verify that HCF divides all given numbers. Verify that LCM is divisible by all given numbers. Verify that HCF × LCM = product (for two numbers).
- Use proper notation. Write \(2^2 \times 3 \times 5\), not \(2 \times 2 \times 3 \times 5\). Examiners appreciate concise mathematical notation.
14. Quick Revision Table
| Topic | Key Point |
|---|---|
| Real Numbers | Rational + Irrational numbers |
| Rational Number | Can be written as \(\dfrac{p}{q}\), \(q \neq 0\) |
| Irrational Number | Cannot be written as \(\dfrac{p}{q}\) |
| Prime Number | Exactly two positive factors (1 and itself) |
| Composite Number | More than two positive factors |
| Prime Factorisation | Product of prime numbers |
| HCF | Common prime factors with smallest powers |
| LCM | All prime factors with greatest powers |
| HCF × LCM | = Product of two positive integers |
| Proof of Irrationality | Assume rational → show contradiction |
15. Practice Your Understanding
Level 1 – Basic
Q1.
Classify the following as rational or irrational: (a) \(\sqrt{49}\) (b) \(\sqrt{50}\) (c) \(\dfrac{3}{8}\) (d) \(\sqrt{11}\)
Q2.
Find the prime factorisation of 210.
Q3.
State the Fundamental Theorem of Arithmetic in your own words.
Q4.
What is the HCF of 48 and 36?
Q5.
What is the LCM of 8 and 14?
Q6.
Is 1 a prime number? Is it a composite number? Give a reason.
Q7.
Write the number system hierarchy from Natural Numbers to Real Numbers.
Level 2 – Application
Q8.
The HCF of two numbers is 16 and their LCM is 1440. If one number is 192, find the other.
Q9.
Find the HCF and LCM of 84, 140, and 168 using prime factorisation.
Q10.
Two numbers are 24 and 36. Find their HCF × LCM and verify that it equals 24 × 36.
Q11.
A class has 72 students and another class has 108 students. The teacher wants to form groups of equal size from both classes, with no student left out. What is the greatest number of students that can be in each group?
Level 3 – Advanced
Q12.
Prove that \(\sqrt{7}\) is irrational.
Q13.
Prove that \(5 - 3\sqrt{2}\) is irrational.
Q14.
Prove that \(\dfrac{2}{\sqrt{3}}\) is irrational.
Competency-Based
Q15.
Three lights at a traffic intersection change at intervals of 30 seconds, 45 seconds, and 60 seconds. If they all change together now, after how many seconds will they all change together again?
Q16.
A rectangular floor is 540 cm long and 360 cm wide. It is to be covered with square tiles of equal size. What is the largest possible side length of each tile, and how many tiles will be needed?
Q17.
Can the HCF of two numbers be greater than their LCM? Justify your answer.
Q18.
A farmer has 120 apples and 180 oranges. He wants to pack them in boxes such that each box has only one type of fruit and all boxes in a set have the same number of fruits. What is the greatest number of fruits per box? How many boxes will he need in total?
16. Answers
Level 1
A1.
(a) Rational (\(\sqrt{49} = 7\)). (b) Irrational (50 is not a perfect square). (c) Rational. (d) Irrational.
A2.
\(210 = 2 \times 3 \times 5 \times 7\)
A3.
Every composite number can be written uniquely as a product of prime numbers (apart from order).
A4.
\(48 = 2^4 \times 3\), \(36 = 2^2 \times 3^2\). HCF = \(2^2 \times 3 = 12\).
A5.
\(8 = 2^3\), \(14 = 2 \times 7\). LCM = \(2^3 \times 7 = 56\).
A6.
1 is neither prime nor composite. It has exactly one factor (itself), so it does not satisfy the definition of prime (exactly two factors) or composite (more than two factors).
A7.
Natural Numbers ⊂ Whole Numbers ⊂ Integers ⊂ Rational Numbers ⊂ Real Numbers, with Irrational Numbers also being a subset of Real Numbers.
Level 2
A8.
HCF × LCM = 16 × 1440 = 23040. So \(192 \times x = 23040\), \(x = 120\). The other number is 120.
A9.
\(84 = 2^2 \times 3 \times 7\), \(140 = 2^2 \times 5 \times 7\), \(168 = 2^3 \times 3 \times 7\).
HCF = \(2^2 \times 7 = 28\).
LCM = \(2^3 \times 3 \times 5 \times 7 = 840\).
A10.
\(24 = 2^3 \times 3\), \(36 = 2^2 \times 3^2\). HCF = \(2^2 \times 3 = 12\). LCM = \(2^3 \times 3^2 = 72\).
HCF × LCM = 12 × 72 = 864. Product = 24 × 36 = 864. ✓
A11.
HCF of 72 and 108: \(72 = 2^3 \times 3^2\), \(108 = 2^2 \times 3^3\). HCF = \(2^2 \times 3^2 = 36\).
The greatest number of students per group is 36.
Level 3
A12.
Assume \(\sqrt{7} = \dfrac{p}{q}\) (coprime). Then \(p^2 = 7q^2\). So \(7 \mid p\). Let \(p = 7k\). Then \(49k^2 = 7q^2\), so \(7 \mid q\). Both divisible by 7 — contradiction. Therefore \(\sqrt{7}\) is irrational.
A13.
Assume \(5 - 3\sqrt{2} = \dfrac{p}{q}\) (coprime). Then \(3\sqrt{2} = 5 - \dfrac{p}{q} = \dfrac{5q - p}{q}\), so \(\sqrt{2} = \dfrac{5q - p}{3q}\) (rational). But \(\sqrt{2}\) is irrational — contradiction. Therefore \(5 - 3\sqrt{2}\) is irrational.
A14.
Assume \(\dfrac{2}{\sqrt{3}} = \dfrac{p}{q}\) (coprime). Then \(2q = p\sqrt{3}\), so \(\sqrt{3} = \dfrac{2q}{p}\) (rational). But \(\sqrt{3}\) is irrational — contradiction. Therefore \(\dfrac{2}{\sqrt{3}}\) is irrational.
Competency-Based
A15.
LCM of 30, 45, 60: \(30 = 2 \times 3 \times 5\), \(45 = 3^2 \times 5\), \(60 = 2^2 \times 3 \times 5\). LCM = \(2^2 \times 3^2 \times 5 = 180\) seconds = 3 minutes.
A16.
Largest tile = HCF of 540 and 360. \(540 = 2^2 \times 3^3 \times 5\), \(360 = 2^3 \times 3^2 \times 5\). HCF = \(2^2 \times 3^2 \times 5 = 180\) cm. Number of tiles = \(\dfrac{540}{180} \times \dfrac{360}{180} = 3 \times 2 = 6\) tiles.
A17.
No. Since HCF divides both numbers and LCM is a multiple of both, HCF ≤ each number ≤ LCM. So HCF ≤ LCM always. They are equal only when both numbers are the same.
A18.
Greatest number of fruits per box = HCF of 120 and 180 = 60. Apples: 120 ÷ 60 = 2 boxes. Oranges: 180 ÷ 60 = 3 boxes. Total = 5 boxes.