Class 10 Mathematics Β· Chapter 1 NotesReal Numbers
Study Class 10 Mathematics Real Numbers with clear notes on Euclid's division algorithm, the Fundamental Theorem of Arithmetic, HCF and LCM, irrational numbers andβ¦
Real Numbers is the first chapter of Class 10 Mathematics and it builds directly on the number work you did in earlier classes. Here you study two big ideas about positive integers: Euclid's division algorithm, which helps you find the HCF of two numbers, and the Fundamental Theorem of Arithmetic, which says every composite number can be written as a product of primes in only one way. Using these ideas, you will prove that numbers such as β2, β3 and β5 are irrational, and you will learn how the prime factorisation of the denominator of a fraction decides whether its decimal expansion terminates or repeats. These results connect divisibility, prime factorisation, HCF and LCM, irrational numbers and decimal expansions into one clear picture of the real number system.
What you'll learn
1Apply Euclid's division algorithm to find the HCF of two positive integers.
2State and use the Fundamental Theorem of Arithmetic to factorise composite numbers.
3Find the HCF and LCM of two or more numbers using the prime factorisation method.
4Verify the relation HCF(a, b) Γ LCM(a, b) = a Γ b for two positive integers.
5Prove that numbers such as β2, β3 and β5 are irrational using proof by contradiction.
6Show that expressions like 5 β β3 and 3β2 are irrational.
7Decide whether the decimal expansion of a rational number is terminating or non-terminating repeating by examining its denominator.
Chapter at a glance
01Euclid's Division Lemma and Algorithm
02The Fundamental Theorem of Arithmetic
03The Fundamental Theorem of Arithmetic
04Rational and Irrational Numbers
05Decimal Expansions of Real Numbers
06Operations and Properties of Real Numbers
Detailed chapter notes
01
Euclid's Division Lemma and Algorithm
Euclid's division lemma says that for any two positive integers a and b, there exist unique whole numbers q and r such that a = bq + r, where 0 β€ r < b. Here a is the dividend, b is the divisor, q is the quotient and r is the remainder. This is simply the rule behind the long division you already know: the remainder is always smaller than the divisor. Euclid's division algorithm uses this lemma again and again. To find the HCF of two positive integers, divide the larger by the smaller, then divide the previous divisor by the remainder, and keep repeating until the remainder becomes zero. The last non-zero remainder is the HCF of the two numbers. This method works without factorising the numbers, so it is very useful for large numbers.
Euclid's division lemmaa = bq + r, where 0 β€ r < b
Step 1divide the larger number by the smaller number.
Step 2divide the divisor by the remainder and repeat until r = 0.
The last non-zero remainder is the HCF.
02
The Fundamental Theorem of Arithmetic
The Fundamental Theorem of Arithmetic states that every composite number can be expressed as a product of primes, and this factorisation is unique except for the order in which the prime factors are written. For example, 32760 = 2Β³ Γ 3Β² Γ 5 Γ 7 Γ 13, and no other product of primes gives 32760. To factorise a number, you can use a factor tree and keep splitting the number into factors until only primes remain. Writing the primes in ascending order and combining equal primes as powers gives the standard prime factorisation. This theorem is the reason prime factorisation is reliable: it guarantees that a number has only one prime factorisation, so any method you use will give the same result.
Every composite number = product of primes, unique apart from order.
Example32760 = 2Β³ Γ 3Β² Γ 5 Γ 7 Γ 13
Use a factor tree to split a number into prime factors.
Combine repeated primes using powers.
03
HCF and LCM by Prime Factorisation
The Fundamental Theorem of Arithmetic gives a systematic way to find the HCF and LCM of numbers. Write each number as a product of powers of primes. The HCF is the product of the smallest power of each prime that is common to all the numbers. The LCM is the product of the greatest power of each prime that appears in any of the numbers. For example, for 6 = 2 Γ 3 and 20 = 2Β² Γ 5, the HCF is 2 and the LCM is 2Β² Γ 3 Γ 5 = 60. For any two positive integers a and b, HCF(a, b) Γ LCM(a, b) = a Γ b. This relation lets you find the LCM when the HCF is known, or the HCF when the LCM is known. Note that this product relation does not hold for three numbers.
HCF = product of the smallest powers of common prime factors.
LCM = product of the greatest powers of all prime factors involved.
For two positive integersHCF(a, b) Γ LCM(a, b) = a Γ b
For three numbers, HCF Γ LCM is not equal to the product of the numbers.
04
Revisiting Irrational Numbers
A number is irrational if it cannot be written in the form p/q, where p and q are integers and q β 0. Numbers such as β2, β3, β5 and Ο are irrational. To prove that β2 is irrational, we use proof by contradiction. We assume that β2 is rational, so β2 = a/b where a and b are coprime. Squaring gives 2bΒ² = aΒ², so 2 divides aΒ². A key theorem says that if a prime p divides aΒ², then p divides a. So 2 divides a, and we can write a = 2c. Substituting gives bΒ² = 2cΒ², so 2 divides b as well. This means a and b have 2 as a common factor, which contradicts the assumption that they are coprime. Therefore β2 is irrational. The same method proves that β3, β5 and, in general, βp are irrational for any prime p.
Irrational numbercannot be written as p/q with q β 0.
Theoremif a prime p divides aΒ², then p divides a.
Proof by contradictionassume the number is rational, then reach a contradiction.
β2, β3, β5 and βp (p prime) are irrational.
05
Irrationality of Sums, Differences, Products and Quotients
You can also prove that expressions built from rational and irrational numbers are irrational. The sum or difference of a rational number and an irrational number is irrational. The product or quotient of a non-zero rational number and an irrational number is irrational. For example, to show that 5 β β3 is irrational, assume it is rational and equal to a/b. Rearranging gives β3 = 5 β a/b, which is rational because a and b are integers. But β3 is irrational, so we have a contradiction. Similarly, to show that 3β2 is irrational, assume 3β2 = a/b. Then β2 = a/(3b), which is rational, contradicting the fact that β2 is irrational. These proofs all use the same contradiction technique.
rational + irrational = irrational
rational β irrational = irrational
non-zero rational Γ irrational = irrational
non-zero rational Γ· irrational = irrational
06
Decimal Expansions of Real Numbers
Every rational number p/q (q β 0) has either a terminating decimal expansion or a non-terminating repeating decimal expansion. The type depends only on the prime factorisation of the denominator q, after writing the fraction in lowest terms. If the prime factorisation of q is of the form 2βΏ Γ 5α΅, where n and m are non-negative integers, then the decimal expansion terminates. If q has any prime factor other than 2 or 5, the decimal expansion is non-terminating repeating. For example, 3/8 = 0.375 terminates because 8 = 2Β³, while 1/3 = 0.333... repeats because 3 is a prime factor other than 2 or 5. This result connects the Fundamental Theorem of Arithmetic with the decimal form of rational numbers.
Terminating decimaldenominator q = 2βΏ Γ 5α΅ after simplifying.
Non-terminating repeating decimalq has a prime factor other than 2 or 5.
Real numbers include both rational and irrational numbers. The rational numbers are those that can be written as p/q, where p and q are integers and q β 0. Irrational numbers cannot be written in this form. When you perform operations on real numbers, the nature of the result follows clear rules. The sum, difference, product or quotient of two rational numbers is rational, provided the divisor is not zero. The sum or difference of a rational and an irrational number is irrational. The product or quotient of a non-zero rational and an irrational number is irrational. However, operations between two irrational numbers can give a rational result; for example, β2 Γ β2 = 2, which is rational.
Rational numbersp/q form, q β 0.
Irrational numberscannot be written in p/q form.
Operations between rational and irrational numbers often give irrational results.
Two irrational numbers can combine to give a rational number.
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Euclid's division lemma: a = bq + r, where 0 β€ r < b.
Euclid's division algorithm finds HCF by repeated division until the remainder is zero.
Fundamental Theorem of Arithmetic: every composite number has a unique prime factorisation, apart from order.
HCF = product of smallest powers of common prime factors; LCM = product of greatest powers of all prime factors.
For two positive integers a and b, HCF(a, b) Γ LCM(a, b) = a Γ b.
If a prime p divides aΒ², then p divides a.
β2, β3, β5 and βp (p prime) are irrational; proofs use contradiction.
A rational number p/q has a terminating decimal only if q = 2βΏ Γ 5α΅ after simplifying.
If q has a prime factor other than 2 or 5, the decimal expansion is non-terminating repeating.
The sum or difference of a rational and an irrational number is irrational; the product or quotient of a non-zero rational and an irrational number is irrational.
According to Euclid's division lemma, for any positive integers a and b, there exist unique integers q and r such that a = bq + r. What is the condition on r?
A0 β€ r < b
B0 < r β€ b
C0 β€ r β€ b
D0 < r < b
Show answer
Answer: (A) 0 β€ r < b
Euclid's division lemma states that for positive integers a and b, there exist unique integers q and r satisfying a = bq + r, where 0 β€ r < b.
Question 02
According to the Fundamental Theorem of Arithmetic, every composite number can be expressed as:
Aa product of primes, and this factorisation is unique except for the order of factors
Ba product of any two natural numbers
Ca sum of primes in a unique way
Da product of primes, but the factorisation may not be unique
Show answer
Answer: (A) a product of primes, and this factorisation is unique except for the order of factors
The Fundamental Theorem of Arithmetic states that every composite number can be factorised as a product of primes, and this factorisation is unique apart from the order in which the prime factors occur.
Question 03
Which of the following is an irrational number?
Aβ2
B1/2
C0.75
D3
Show answer
Answer: (A) β2
β2 cannot be written in the form p/q where p and q are integers and q β 0, so it is irrational. The others are rational.
Question 04
What is the decimal expansion of the fraction 1/2?
A0.5
B0.2
C0.25
D0.05
Show answer
Answer: (A) 0.5
1/2 = 0.5. This is a terminating decimal because the denominator has only factors of 2 and 5.
Question 05
Which of the following is a real number?
Aβ4
Bβ(-4)
Cβ0/0
DNone of these
Show answer
Answer: (A) β4
β4 = 2, which is a real number. β(-4) is not real (imaginary), and β0/0 is undefined.
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Q1. State Euclid's division lemma and use it to find the HCF of 4052 and 12576.
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Model answer
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 is less than or equal to r and r is less than b. Applying the algorithm: 12576 = 4052 x 3 + 420; 4052 = 420 x 9 + 272; 420 = 272 x 1 + 148; 272 = 148 x 1 + 124; 148 = 124 x 1 + 24; 124 = 24 x 5 + 4; 24 = 4 x 6 + 0. Since the remainder is 0, HCF(4052, 12576) = 4.
Sample question3 marks
Q2. State the Fundamental Theorem of Arithmetic.
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Model answer
The Fundamental Theorem of Arithmetic states that 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.
Sample question3 marks
Q3. Prove that sqrt(2) is irrational.
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Model answer
Assume sqrt(2) is rational. Then sqrt(2) = a/b, where a and b are coprime integers with b β 0. Squaring gives 2b^2 = a^2, so 2 divides a^2. By Theorem 1.2, 2 divides a, so a = 2c. Substituting gives 2b^2 = 4c^2, so b^2 = 2c^2, hence 2 divides b. Thus a and b have a common factor 2, contradicting that they are coprime. Therefore, sqrt(2) is irrational.
Sample question3 marks
Q4. State the condition for the decimal expansion of a rational number p/q (q β 0) to be terminating. Give an example.
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Model answer
The decimal expansion of p/q (in lowest terms) is terminating if the prime factorisation of q is of the form 2^m * 5^n, where m and n are non-negative integers. Example: 3/8 has denominator 8 = 2^3, so its decimal expansion is terminating (0.375).
Sample question3 marks
Q5. State the Fundamental Theorem of Arithmetic and use it to explain why 7 Γ 11 Γ 13 + 13 is a composite number.
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Model answer
The Fundamental Theorem of Arithmetic states that every composite number can be expressed as a product of primes, and this factorisation is unique except for the order of factors. Now, 7 Γ 11 Γ 13 + 13 = 13(7 Γ 11 + 1) = 13(77 + 1) = 13 Γ 78 = 13 Γ 2 Γ 3 Γ 13 = 2 Γ 3 Γ 13^2. Since it has prime factors other than 1 and itself, it is composite.
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Euclid's division lemma states that for any two positive integers a and b, there exist unique whole numbers q and r such that a = bq + r, where 0 β€ r < b. It is the basis of Euclid's division algorithm, which is used to find the HCF of two numbers by repeated division.
What is the Fundamental Theorem of Arithmetic?
The Fundamental Theorem of Arithmetic says that every composite number can be expressed as a product of primes, and this factorisation is unique except for the order of the prime factors. For example, 32760 = 2Β³ Γ 3Β² Γ 5 Γ 7 Γ 13, and no other prime factorisation gives 32760.
How do you find the HCF and LCM using prime factorisation?
Write each number as a product of powers of primes. The HCF is the product of the smallest power of each common prime factor. The LCM is the product of the greatest power of each prime factor involved. For two numbers a and b, HCF(a, b) Γ LCM(a, b) = a Γ b.
Why is β2 irrational?
β2 is irrational because assuming it is rational leads to a contradiction. If β2 = a/b with a and b coprime, then 2bΒ² = aΒ², so 2 divides aΒ² and hence 2 divides a. Writing a = 2c gives bΒ² = 2cΒ², so 2 divides b too. This contradicts that a and b are coprime.
How do you know if a decimal expansion is terminating or non-terminating repeating?
Write the rational number in lowest terms. If the denominator q has only 2 and 5 as prime factors, that is q = 2βΏ Γ 5α΅, the decimal expansion terminates. If q has any other prime factor, the decimal expansion is non-terminating repeating.
What is the difference between rational and irrational numbers?
A rational number can be written in the form p/q, where p and q are integers and q β 0. An irrational number cannot be written in this form. Examples of rational numbers are 3/4 and 0.25; examples of irrational numbers are β2, β3 and Ο.