Real Numbers for Class 10 Mathematics
Real Numbers for Class 10 Mathematics is the chapter that quietly supports a lot of higher Mathematics. When you solve equations, draw graphs, compare measurements, use square roots, or work with decimals, you are using the Real Number system. So this chapter is not just a list of definitions. It teaches you how numbers are organised, how factors reveal hidden structure, and why some numbers can never be written as fractions. We will study Real Numbers in the same order a teacher would build it on the board: first the number family, then Euclid's division lemma, then HCF and LCM, then prime factorisation, then irrational numbers, and finally decimal expansions of rational numbers. By the end, you should be able to solve direct questions, prove standard irrationality results, and explain the reason behind every formula instead of memorising steps.
How This Lesson Helps You Learn
Understand
Definitions, formulas, proofs, explanations, and diagrams come first.
Clarify
Common confusion points are handled through direct rules, examples, and mistakes.
Practise
Worked examples, try-it-yourself tasks, quizzes, and questions help you test yourself.
1. Introduction
Real Numbers for Class 10 Mathematics is the chapter that quietly supports a lot of higher Mathematics. When you solve equations, draw graphs, compare measurements, use square roots, or work with decimals, you are using the Real Number system. So this chapter is not just a list of definitions. It teaches you how numbers are organised, how factors reveal hidden structure, and why some numbers can never be written as fractions.
We will study Real Numbers in the same order a teacher would build it on the board: first the number family, then Euclid's division lemma, then HCF and LCM, then prime factorisation, then irrational numbers, and finally decimal expansions of rational numbers. By the end, you should be able to solve direct questions, prove standard irrationality results, and explain the reason behind every formula instead of memorising steps.
2. Learning Objectives
- Understand natural numbers, whole numbers, integers, rational numbers, irrational numbers, and real numbers as connected sets.
- State and apply Euclid's division lemma: for positive integers a and b, a = bq + r, where 0 <= r < b.
- Find HCF using Euclid's division algorithm and explain why the last non-zero remainder is the HCF.
- Use the relationship HCF(a, b) x LCM(a, b) = a x b for two positive integers.
- Use the Fundamental Theorem of Arithmetic to write unique prime factorisations.
- Prove irrationality statements such as sqrt(2), sqrt(3), and sqrt(5) using contradiction.
- Decide whether the decimal expansion of p/q terminates by checking the prime factors of q in lowest form.
3. Prerequisites
- You should know the basic vocabulary used in Mathematics.
- You should be ready to read Real Numbers slowly and connect each idea to one example.
- You should keep a notebook for definitions, diagrams, solved examples, and mistakes.
4. What Is Real Numbers?
Real numbers are all numbers that can be represented on the number line. They include rational numbers and irrational numbers. A rational number can be written in the form p/q, where p and q are integers and q is not 0. Examples are 5, -3, 7/9, 0.25, and -11/4. An irrational number cannot be written in the form p/q. Its decimal expansion is non-terminating and non-recurring. Examples are sqrt(2), sqrt(3), sqrt(5), and pi.
The important idea is this: every point on the number line represents a real number, and every real number can be placed somewhere on the number line. Some points are easy to label, like 0, 1, and 1/2. Some points need construction or approximation, like sqrt(2). But they still belong to the same number line.
5. Key Definitions
- Real Numbers: the main chapter idea explained in Mathematics using meaning, reason, example, and practice.
- Every rational number can be written as p/q, where p and q are integers and q is not 0.
- An irrational number cannot be written as p/q and has a non-terminating, non-recurring decimal expansion.
- Euclid's division lemma is used to find the HCF of two positive integers.
- The Fundamental Theorem of Arithmetic says every composite number has a unique prime factorisation, apart from the order of factors.
- For a rational number p/q in lowest form, the decimal terminates only when q has no prime factors other than 2 and 5.
6. Concept Explanation
- Every rational number can be written as p/q, where p and q are integers and q is not 0.
- An irrational number cannot be written as p/q and has a non-terminating, non-recurring decimal expansion.
- Euclid's division lemma is used to find the HCF of two positive integers.
- The Fundamental Theorem of Arithmetic says every composite number has a unique prime factorisation, apart from the order of factors.
- For a rational number p/q in lowest form, the decimal terminates only when q has no prime factors other than 2 and 5.
- A non-terminating recurring decimal is rational; a non-terminating non-recurring decimal is irrational.
- Prime factorisation is the bridge between divisibility, HCF, LCM, and decimal expansion.
Topic-by-Topic Explanation
These explanations break the chapter into important topics, sub-topics, examples, and visual prompts so the lesson is easier to revise.
The Real Number Family
Start with natural numbers: 1, 2, 3, 4, and so on. Add 0 and you get whole numbers. Add negative numbers and you get integers. Now allow numbers of the form p/q, where p and q are integers and q is not 0, and you get rational numbers.
Irrational numbers are different. They cannot be written as p/q. Their decimal expansion goes on forever without repeating a fixed pattern. Examples include sqrt(2), sqrt(3), sqrt(5), and pi.
Real numbers are the union of rational and irrational numbers. So the number line is not only for integers or fractions. Every point on it represents a real number.
Draw a nested family: natural numbers inside whole numbers, whole numbers inside integers, integers inside rational numbers, and rational plus irrational together forming real numbers.
Euclid's Division Lemma and HCF
Euclid's division lemma says: for positive integers a and b, there exist unique whole numbers 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 HCF, divide the larger number by the smaller number. If the remainder is not 0, divide the previous divisor by the remainder. Keep repeating. The last non-zero remainder is the HCF.
Example: Find HCF of 867 and 255. 867 = 255 x 3 + 102. Then 255 = 102 x 2 + 51. Then 102 = 51 x 2 + 0. So HCF = 51.
Show 867 as three groups of 255 with 102 left over, then show 255 as two groups of 102 with 51 left over, then show 102 as two groups of 51.
Prime Factorisation, HCF, and LCM
The Fundamental Theorem of Arithmetic says every composite number can be expressed as a product of primes in a unique way, except for the order of factors. For example, 156 = 2 x 2 x 3 x 13 = 2^2 x 3 x 13.
For HCF, take the common prime factors with the smallest powers. For LCM, take all prime factors that appear, with the greatest powers.
For example, 12 = 2^2 x 3, 15 = 3 x 5, and 21 = 3 x 7. HCF = 3 because 3 is common to all three. LCM = 2^2 x 3 x 5 x 7 = 420.
Create two columns: HCF uses common factors with smaller powers; LCM uses all factors with larger powers.
Why Some Numbers Are Irrational
To prove sqrt(2) is irrational, assume sqrt(2) = p/q, where p and q are co-prime positive integers. Squaring both sides gives 2 = p^2/q^2, so p^2 = 2q^2. Therefore p^2 is even, which means p is even.
Let p = 2k. Then p^2 = 4k^2. Substitute into p^2 = 2q^2 to get 4k^2 = 2q^2, so q^2 = 2k^2. Therefore q^2 is even, which means q is even.
Now p and q are both even, so they have a common factor 2. But we started by saying p and q are co-prime. This contradiction proves sqrt(2) is irrational. The same structure helps prove sqrt(3), sqrt(5), and many similar results.
Draw a contradiction ladder: assume rational, square, prove p is even, prove q is even, contradiction, conclusion.
Decimal Expansions of Rational Numbers
Take a rational number p/q in lowest form. If q has only the prime factors 2 and 5, the decimal expansion terminates. This happens because q can be multiplied by enough 2s or 5s to become 10, 100, 1000, and so on.
For example, 13/3125 terminates because 3125 = 5^5. Multiply numerator and denominator by 2^5 to make the denominator 10^5. So the decimal must stop.
If q has any prime factor other than 2 or 5, the decimal expansion is non-terminating recurring. For example, 7/12 has denominator 12 = 2^2 x 3. The factor 3 prevents the decimal from terminating.
Show 10 = 2 x 5, then show denominators made only from 2s and 5s becoming 10, 100, or 1000.
Questions That Usually Appear in Tests
Pattern 1: HCF by Euclid's algorithm. These questions reward neat division steps. Pattern 2: LCM-HCF relation. These questions often give HCF and ask for LCM, or ask you to verify that HCF x LCM equals the product of two numbers.
Pattern 3: prime factorisation. You may be asked to express numbers like 140, 156, 3825, or 5005 as products of prime factors. Pattern 4: irrationality proofs. These require a clean contradiction argument, not a decimal approximation.
Pattern 5: decimal expansion. Always reduce p/q to lowest form first. Pattern 6: composite-number proofs. Factorise the expression to show it has factors other than 1 and itself.
Make a six-box practice map: HCF, LCM, prime factors, irrationality, decimals, composite numbers.
7. Rules / Properties
- Always start Real Numbers answers with the correct meaning before writing examples.
- Use the exact Mathematics term when the question asks for a definition, rule, law, property, process, or comparison.
- Write steps in order when the concept involves a method, proof, process, timeline, map, or grammar transformation.
- In HCF questions, show every division step clearly.
- In irrationality proofs, write "assume sqrt(2) is rational" and end with the contradiction.
- For decimal expansion questions, always mention prime factorisation of the denominator in lowest form.
- Use exact mathematical language: non-terminating recurring means rational; non-terminating non-recurring means irrational.
8. Formulae
Important Formulae and Statements
- Euclid division lemma: a = bq + r, 0 <= r < b
- HCF(a, b) x LCM(a, b) = a x b
- Terminating decimal condition: q = 2^m x 5^n after reducing p/q to lowest form
9. Visual Explanation
For Euclid's algorithm, draw the larger number as a long strip and cut it into equal smaller strips. The leftover part is the remainder. Now repeat the same cutting process with the previous smaller strip and the remainder. When there is no leftover part, the last strip size is the HCF.
For example, if you divide 225 by 135, the remainder is 90. Then divide 135 by 90, the remainder is 45. Then divide 90 by 45, the remainder is 0. That means 45 is the greatest length that can measure both 225 and 135 exactly.
For irrationality proofs, the visual idea is different. You begin by assuming the number is rational, so it can be written as p/q in lowest form. Then algebra shows p and q must both be divisible by the same number. That contradicts "lowest form", so the original assumption must be false.
Visual Explanation
Draw one number line. Mark integers such as -2, -1, 0, 1, and 2. Then mark rational numbers like 1/2 and 3/4 between integers. Finally show sqrt(2) between 1 and 2, because 1^2 = 1 and 2^2 = 4, so sqrt(2) must lie between them.
10. Worked Examples
Find the HCF of 135 and 225 using Euclid's division algorithm. Then find their LCM.
- 1Take the larger number first: 225. Divide it by 135.
- 2225 = 135 x 1 + 90
- 3Now divide the previous divisor 135 by the remainder 90.
- 4135 = 90 x 1 + 45
- 5Now divide 90 by 45.
- 690 = 45 x 2 + 0
- 7The last non-zero remainder is 45, so HCF = 45.
- 8For two positive integers, HCF x LCM = product of the numbers.
- 945 x LCM = 135 x 225 = 30375
- 10LCM = 30375 / 45 = 675
HCF = 45 and LCM = 675.
11. Applications
Try deciding whether 17/40 has a terminating decimal expansion before opening the answer.
- Check that 17/40 is in lowest form.
- Prime factorise the denominator: 40 = 2^3 x 5.
- Since the denominator has only 2 and 5 as prime factors, the decimal expansion terminates.
- In fact, 17/40 = 0.425.
12. Common Mistakes
- Writing Euclid's division lemma without the condition 0 <= r < b.
- Using HCF x LCM = a x b for more than two numbers without checking the condition.
- Forgetting to reduce p/q to lowest form before checking terminating decimals.
- Saying every non-terminating decimal is irrational; recurring decimals are rational.
13. Practice Questions
- Find the HCF of 867 and 255 using Euclid's division algorithm.
- Find the LCM of 96 and 404 if their HCF is 4.
- Show that sqrt(3) is irrational.
- Check whether 13/3125 has a terminating decimal expansion.
- Write the prime factorisation of 156 and use it to list all prime factors.
- Explain why 7 x 11 x 13 + 13 is a composite number.
- Find the least number that is divisible by 12, 15, and 21.
- Without division, decide whether 23/(2^3 x 5^2) has a terminating decimal expansion.
- Exam check: In HCF questions, show every division step clearly.
- Exam check: In irrationality proofs, write "assume sqrt(2) is rational" and end with the contradiction.
- Exam check: For decimal expansion questions, always mention prime factorisation of the denominator in lowest form.
- Exam check: Use exact mathematical language: non-terminating recurring means rational; non-terminating non-recurring means irrational.
- Exam check: When asked to prove a number is composite, factorise the expression instead of multiplying everything out.
14. Frequently Asked Questions
These questions help you revise Real Numbers in Mathematics with clear, test-ready understanding.
What is Euclid's division lemma?
For positive integers a and b, there exist unique whole numbers q and r such that a = bq + r, where 0 <= r < b.
How do I know whether a decimal expansion terminates?
Write the rational number p/q in lowest form. If q has only 2 and 5 as prime factors, the decimal terminates.
Why is sqrt(2) irrational?
Assuming sqrt(2) = p/q in lowest form leads to both p and q being even, which contradicts the assumption that they have no common factor.
15. Summary
- Real numbers include rational and irrational numbers.
- Euclid's division algorithm finds HCF through repeated division.
- For two positive integers, HCF x LCM = product of the numbers.
- Irrationality proofs often use contradiction and divisibility.
- A rational number p/q terminates only when q has prime factors 2 and/or 5 after simplification.
Quick Quiz
Try these before you move ahead. The goal is to catch small doubts early, while the explanation is still fresh.
Quick Self-Test
Revise in Two Minutes
What is the first thing to understand in Real Numbers?
The central relationship: what is involved, what changes, and why it matters in Mathematics.
What is a clear answer structure?
Idea, reason, example. Use it to make answers clear without sounding memorised.
How do I know I really understand Real Numbers?
You can explain it simply, draw a useful visual, solve a basic question, and spot one common mistake.
Before You Say Done
- I can explain Real Numbers in simple words.
- I can draw or describe the main visual model.
- I can solve one basic question step by step.
- I can identify one common mistake and avoid it.
- I can answer a why question, not only a what question.
Continue Learning
Use the visual prompt from this lesson first. Then move into practice and adaptive learning while the method is still fresh.