Real Numbers | Class X Mathematics | Facts and MCQ Quiz

Introduction: Real Numbers (Class X)

Real Numbers is the first chapter of Class X Mathematics and forms the foundation for many important concepts used throughout the syllabus. This chapter helps students understand different types of numbers such as natural numbers, whole numbers, integers, rational numbers, and irrational numbers.

Two key ideas discussed in this chapter are Euclid’s Division Lemma and the Fundamental Theorem of Arithmetic. Euclid’s Division Lemma provides a systematic method to find the Highest Common Factor (HCF) of numbers, while the Fundamental Theorem of Arithmetic explains how every composite number can be expressed uniquely as a product of prime numbers.

A strong understanding of Real Numbers is essential for solving problems related to HCF, LCM, decimal expansions, and proving the irrationality of numbers. This chapter plays a crucial role in board examinations and competitive exams.

Real Numbers – 50 Important Facts (Class X)

• Real numbers include both rational and irrational numbers.
• Natural numbers, whole numbers, and integers are subsets of real numbers.
• Rational numbers can be expressed in the form p/q where q ≠ 0.
• Irrational numbers cannot be expressed as p/q.
• Examples of irrational numbers include √2, √3, and π.
• Euclid was a Greek mathematician known as the “Father of Geometry”.
• Euclid’s Division Lemma is used to find the HCF of two numbers.
• Euclid’s Division Lemma states: a = bq + r.
• Here, a is the dividend and b is the divisor.
• q is the quotient and r is the remainder.
• The remainder r always satisfies 0 ≤ r < b.
• Euclid’s Division Lemma applies only to positive integers.
• It is the basis of Euclid’s Division Algorithm.
• Euclid’s Division Algorithm is used to find HCF.
• HCF means Highest Common Factor.
• HCF is also known as GCD (Greatest Common Divisor).
• Euclid’s algorithm involves repeated division.
• The algorithm stops when the remainder becomes zero.
• The last non-zero remainder is the HCF.
• This method is faster than prime factorization for large numbers.
• The Fundamental Theorem of Arithmetic deals with prime factorization.
• It states that every composite number can be expressed as a product of primes.
• This factorization is unique, apart from the order of primes.
• Prime numbers have exactly two factors.
• Examples of prime numbers are 2, 3, 5, and 7.
• 1 is neither prime nor composite.
• Composite numbers have more than two factors.
• Prime factorization helps find HCF and LCM.
• LCM means Least Common Multiple.
• HCF and LCM are important for solving word problems.
• Using prime factorization, HCF is found by common prime factors.
• LCM is found by taking highest powers of all prime factors.
• The Fundamental Theorem of Arithmetic is used in proving irrationality.
• It helps prove √2 is irrational.
• This theorem is applicable only to natural numbers greater than 1.
• Real numbers are represented on the number line.
• Between any two real numbers, infinite real numbers exist.
• Real Numbers is the first chapter of Class X Mathematics.
• This chapter forms the base for many other chapters.

Class X Mathematics – Chapter 1: Real Numbers

1. Euclid’s division algorithm is used to find:

A) LCM
B) HCF
C) Prime numbers
D) Factors

Answer: B

2. According to Euclid’s division lemma:

A) a = bq
B) a = b + q
C) a = bq + r
D) a = b − r

Answer: C

3. In Euclid’s division lemma, r satisfies:

A) r ≥ b
B) r < 0
C) 0 ≤ r < b
D) r = b

Answer: C

4. HCF of two numbers is the:

A) Greatest number dividing both
B) Least number dividing both
C) Product of numbers
D) Sum of numbers

Answer: A

5. Which is always a positive integer?

A) LCM
B) HCF
C) Both A and B
D) None

Answer: C

6. HCF of two prime numbers is:

A) 0
B) 1
C) Smaller prime
D) Larger prime

Answer: B

7. Decimal expansion of a rational number is:

A) Always terminating
B) Always non-terminating
C) Terminating or recurring
D) Non-recurring

Answer: C

8. Terminating decimals occur when denominator has factors:

A) Only 2
B) Only 5
C) 2 and 5 only
D) Any prime

Answer: C

9. Which has a terminating decimal?

A) 1/3
B) 2/7
C) 5/8
D) 7/11

Answer: C

10. Decimal expansion of an irrational number is:

A) Terminating
B) Recurring
C) Non-terminating non-recurring
D) Whole number

Answer: C

11. Which is an irrational number?

A) √4
B) √9
C) √2
D) 5/2

Answer: C

12. Product of HCF and LCM equals:

A) Sum
B) Difference
C) Product of numbers
D) Square

Answer: C

13. If HCF = 12 and LCM = 180, product of numbers is:

A) 2160
B) 180
C) 12
D) 360

Answer: A

14. HCF of 306 and 657 is:

A) 9
B) 3
C) 6
D) 18

Answer: A

15. Which is a rational number?

A) √3
B) √5
C) √25
D) √7

Answer: C

16. LCM of two coprime numbers equals:

A) Sum
B) Difference
C) Product
D) HCF

Answer: C

17. Coprime numbers have HCF:

A) 0
B) 1
C) 2
D) Same number

Answer: B

18. Non-terminating recurring decimal:

A) 0.25
B) 0.125
C) 0.333…
D) 0.5

Answer: C

19. 0.375 is equal to:

A) 3/8
B) 5/8
C) 7/8
D) 1/3

Answer: B

20. In a = bq + r, q is called:

A) Dividend
B) Divisor
C) Quotient
D) Remainder

Answer: C

21. Smallest composite number is:

A) 1
B) 2
C) 3
D) 4

Answer: D

22. Prime factorisation of 60:

A) 2×2×3×5
B) 2×3×10
C) 4×15
D) 6×10

Answer: A

23. Non-terminating recurring decimal is:

A) 1/8
B) 3/5
C) 7/20
D) 2/3

Answer: D

24. HCF of 26 and 91:

A) 7
B) 13
C) 26
D) 91

Answer: B

25. Decimal expansion of 13/125:

A) 0.104
B) 0.108
C) 0.112
D) 0.125

Answer: A

26. Which is not a prime number?

A) 2
B) 3
C) 5
D) 9

Answer: D

27. HCF of two consecutive even numbers:

A) 1
B) 2
C) 4
D) 0

Answer: B

28. Number of prime factors of 72:

A) 2
B) 3
C) 4
D) 5

Answer: B

29. True statement:

A) Every irrational number is real
B) Every real number is irrational
C) Every integer is irrational
D) Every rational is irrational

Answer: A

30. Which is a real number?

A) √2
B) −5
C) 3/7
D) All of these

Answer: D