Real Numbers — Study Notes

1. Introduction

Real Numbers is the opening chapter of Class 10 Mathematics and forms the foundation of the entire Number System branch of study. This chapter revisits the concepts of Euclid's division lemma and the Fundamental Theorem of Arithmetic, which the students first encountered in Class 9. The chapter deals with the properties of natural numbers, whole numbers, integers, rational numbers, and irrational numbers, and establishes the deep relationships among them.

The central theme of this chapter is to understand how every composite number can be expressed uniquely as a product of primes, and how this simple yet powerful idea helps us prove the irrationality of certain numbers like the square root of 2. Along the way, students also learn the decimal expansions of rational numbers and how they relate to the denominators of the fractions, together with concepts of HCF, LCM, and their interrelationship.

Real Numbers is not only crucial for board examinations but also for competitive examinations and higher studies. A clear command of divisibility rules, prime factorization, and the decimal expansion behaviour of numbers prepares the student to handle more advanced topics in algebra and number theory with confidence.

2. Euclid's Division Lemma

Statement

For any two positive integers a and b, there exist unique integers q and r satisfying: a = bq + r, where 0 <= r < b

Here, q is called the quotient and r is called the remainder. This lemma is a restatement of the long division method we have used since primary school, but it gives it a precise mathematical form.

Application: Euclid's Division Algorithm

This lemma is used to compute the HCF of two positive integers through the following systematic steps:

1. Apply the division lemma to the two integers a and b, assuming a > b, to get a = bq + r.
2. If r = 0, then b is the HCF.
3. If r is not zero, apply the lemma to b and r, that is, b = rq1 + r1.
4. Continue the process until the remainder becomes zero. The divisor at that stage is the HCF.

Example: HCF of 455 and 42. 455 = 42 x 10 + 35 42 = 35 x 1 + 7 35 = 7 x 5 + 0 Hence, HCF(455, 42) = 7.

This algorithm is extremely efficient and always terminates because the remainders keep decreasing and are non-negative, eventually reaching zero.

3. The Fundamental Theorem of Arithmetic

Statement

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.

Example: 32760 = 2^3 x 3^2 x 5 x 7 x 13.

This theorem is also called the Unique Prime Factorisation Theorem. Its usefulness is enormous because it guarantees that regardless of how we begin the factorisation process, we always end up with the same set of prime factors.

HCF and LCM using Prime Factorisation

If a and b are two positive integers, then: HCF(a, b) x LCM(a, b) = a x b

Example: For 12 and 18: 12 = 2^2 x 3 18 = 2 x 3^2 HCF = 2 x 3 = 6 LCM = 2^2 x 3^2 = 36 Check: HCF x LCM = 6 x 36 = 216 = 12 x 18. Hence verified.

The relationship HCF x LCM = a x b holds only for two numbers at a time. For three or more numbers, this simple product rule does not apply, and we must use prime factorisation directly.

4. Irrational Numbers

A number that cannot be written in the form p/q, where p and q are integers and q is not zero, is called an irrational number. Examples include sqrt(2), sqrt(3), sqrt(5), pi, and e.

Proving Irrationality

The standard method to prove that a number like sqrt(2) is irrational uses proof by contradiction:

1. Assume sqrt(2) is rational, so sqrt(2) = p/q in the lowest terms where p and q are coprime integers.
2. Squaring both sides: 2 = p^2/q^2, so p^2 = 2q^2, meaning p^2 is divisible by 2, so p is divisible by 2.
3. Let p = 2k. Then (2k)^2 = 2q^2 gives 4k^2 = 2q^2, so q^2 = 2k^2, meaning q is also divisible by 2.
4. This contradicts the assumption that p and q are coprime. Therefore sqrt(2) is irrational.

The same technique works for sqrt(3), sqrt(5), and in fact for sqrt of any prime number.

Important Result

If p is a prime and p divides a^2, then p divides a as well. This lemma is frequently used in irrationality proofs.

5. Rational Numbers and Their Decimal Expansions

Let x = p/q be a rational number whose decimal expansion terminates. Then the denominator q is of the form 2^n x 5^m, where n and m are non-negative integers.

Conversely, if the denominator of a rational number in its simplest form is of the form 2^n x 5^m, then the rational number has a terminating decimal expansion.

Examples

• 7/8 = 7/2^3: denominator is of the form 2^n, so it terminates. Indeed 7/8 = 0.875.
• 13/40 = 13/(2^3 x 5): terminates. 13/40 = 0.325.
• 3/14 = 3/(2 x 7): denominator contains 7, which is neither 2 nor 5, so the decimal expansion is non-terminating and repeating. 3/14 = 0.2142857142857...

Summary of Expansion Types

• Terminating: q = 2^n x 5^m (in lowest terms).
• Non-terminating repeating: q has a prime factor other than 2 and 5.
• A rational number can never have a non-terminating non-repeating expansion; such expansions belong only to irrational numbers.

6. Full Expansion Form of a Rational Number

Given any rational number with denominator of the form 2^n x 5^m, we can obtain its exact terminating decimal by multiplying numerator and denominator by suitable powers of 2 and 5 so that the denominator becomes a power of 10.

Example: 17/8 = 17/2^3 = (17 x 5^3)/(2^3 x 5^3) = (17 x 125)/1000 = 2125/1000 = 2.125.

Similarly, 1/20 = 1/(2^2 x 5) = 5/(2^2 x 5^2) = 5/100 = 0.05.

This technique shows that the number of digits in the decimal part equals the greater of n and m.

Quick Revision Tables

Table 1: Classification of Numbers

Table 2: Key Formulas and Theorems

Mind Map

Important Diagrams (SVG)

Diagram 1: Number System Classification

Diagram 2: Euclid's Division Algorithm for HCF

Common Mistakes

1. Writing the HCF as the last remainder instead of the last divisor. The HCF is the divisor at the stage where the remainder becomes zero, never the final zero remainder itself.
2. Forgetting the condition 0 <= r < b in Euclid's division lemma and writing a remainder larger than or equal to the divisor.
3. Assuming HCF x LCM = a x b holds for three numbers as well. This relation is valid only for two numbers.
4. Stating that a rational number like 3/14 has a terminating decimal expansion because it looks "clean", without checking that the denominator in its simplest form has prime factors only 2 and 5.
5. Concluding that sqrt(4) or sqrt(9) is irrational just because the number is written with a square root symbol. sqrt(4) = 2 is rational; only square roots of non-perfect squares are irrational.
6. Writing a terminating decimal as non-terminating, or failing to reduce the fraction to its simplest form before checking the denominator's prime factors.
7. Assuming every irrational number is of the form sqrt(n). Numbers like pi and 0.1010010001... are also irrational without being square roots.

Exam Tips

1. Always state Euclid's division lemma (a = bq + r, 0 <= r < b) before using it; examiners award marks for the correct statement.
2. For HCF and LCM problems, show the full factorisation in ascending order of primes, e.g., 72 = 2^3 x 3^2.
3. In irrationality proofs, begin with "Let us assume, to the contrary, that the number is rational" and clearly reach the contradiction.
4. Remember the shortcut: a rational number terminates if and only if its reduced denominator is of the form 2^n x 5^m.
5. Use the identity HCF x LCM = a x b to find LCM when HCF is given, saving time in one-mark questions.
6. Practice writing decimal expansions by multiplying numerator and denominator to get powers of 10, as this is a frequent two-mark question.
7. Revise the proof of sqrt(2) being irrational thoroughly, as it is the most commonly asked proof in board exams.

Conclusion

Real Numbers consolidates the arithmetic backbone of mathematics that students will use in almost every later chapter, from polynomials to statistics. The Fundamental Theorem of Arithmetic, Euclid's division lemma, and the classification of decimal expansions together give a complete picture of how numbers behave. Mastery of this chapter ensures a strong start to the board year, and the skills of proving irrationality and finding HCF by the division algorithm will recur throughout higher mathematics. A regular practice of solved examples and previous year questions is the most reliable path to full marks in this chapter.