NCERT Solutions for Class 10 Maths Chapter 1- Real Numbers

The entire chapter deals with the concepts and theorems related to Real numbers, their types, and the operation of real numbers. Euclid's division algorithm and the Unique Prime Factorization Theorem, also known as the Fundamental Theorem of Arithmetic, are the important topics to be considered in this chapter. Euclid's algorithm given in Maths Chapter 1 real numbers deals with the divisibility of the integers, and it is used to compute the Highest Common Factor (HCF), whereas the Fundamental Theorem of Arithmetic is used to express the composite numbers as the multiple of prime numbers. They are considered for positive integers only with a condition of divisor not equal to zero.

Euclid's division theorem states that if "a = bq + r", where the range of r lies between 0 and b. When Euclid's division algorithm is applied on given two positive integers, if the remainder (r) is zero, then "b" is the HCF of "a". The Fundamental Theorem of Arithmetic states that every composite number can be written as the product of the powers of primes.

The NCERT Maths Chapter 1 real numbers further discusses the concept of rational and irrational numbers. A number is called irrational if it cannot be represented in the form of a/b where a and b both are integers, and b is not equal to zero.

In the case of rational numbers, these are the numbers that can be expressed in the form of a fraction, i.e. in the form of a/b, where both the numerator and denominator are integers, the denominator is not equal to zero. It can have either a terminating decimal expansion or a non-terminating repeating decimal expansion.

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Frequently Asked Question (FAQs)

Q. What is Euclid's Division Lemma?

A. Euclid's division theorem states that if "a = bq + r", where the range of r lies between 0 and b. When Euclid's division algorithm is applied on two positive integers, if the remainder (r) is zero, then "b" is the HCF of "a". The Fundamental Theorem of Arithmetic states that every composite number can be written as the product of the powers of primes.

Q. What is HCF? How to Find HFC?

A. The highest common factor (HCF) of two or more numbers & is the largest number that divides all of those numbers without leaving any remainder. Therefore, the Highest Common Factor (HCF) is the product of the lowest power of common prime factors. 

Q. What is LCM? How to find it?

A. The lowest common multiple (LCM) of two or more numbers is the smallest number that is a multiple of all numbers.

LCM is the product of the greatest power of all the prime factors found. 

Q. What are irrational Numbers?

A. A number is called irrational if it cannot be represented in the form of a/b where a and b both are integers, and b is not equal to zero.

Q. What are Rational Numbers?

A. A rational number is a type of real number, which is in the form of p/q where q is not equal to zero. 

Q. What number is formed with the addition of a rational and an irrational number?

A. When a rational number is added to an irrational number, the sum of the two forms an irrational number. For example, ½ is a rational number, while √2 is an irrational number. If both are added an irrational number is formed.