Chapter 1

Introduction to Real Numbers

Real numbers: The real number system consists of whole numbers, fractions, and irrational numbers. These numbers may be positive, zero, or negative. Real numbers may be represented by infinite decimals.

Or, the real numbers may be thought of as points on an infinitely long number line.

Rational number: A rational number is a number which can be expressed as a quotient of two integers;

Or, a number is called rational if it can be written in the form, where p and q are integers and q ≠ 0.

Irrational number: A number is called irrational if it cannot be written in the form, where p and q are integers and q ≠ 0.

Algorithm: A series of well defined steps which gives a procedure for solving a type of problem is called an algorithm.

Lemma: A proven statement used for proving another statement is known as a lemma.

Euclid’s division algorithm: Any positive integer ‘a’ can be divided by another positive integer ‘b’ in such a way that it leaves a remainder ‘r’ that is smaller than ‘b’.

Euclid’s division lemma: Given positive integers a and b, there exist unique integers q and r satisfying a = bq + r, 0 ≤ r < b, where q means quotient and r means remainder.

Euclid’s division algorithm is a technique to compute the Highest Common Factor of any two given positive integers.

Highest common factor (HCF): The HCF of two positive integers a and b is the largest positive integer d that divides both a and b.

Finding HCF of two positive integers, say a and b, with a > b by Euclid’s division algorithm:

(1) Apply Euclid’s division lemma, to a and b. By this we find whole numbers, q and r such that a = bq + r, 0 ≤ r < b.

(2) If r = 0, b is the HCF of a and b.

And if r ≠ 0, again apply the division lemma to b and r

(3) Continue the process till the remainder becomes zero. The required HCF will be the divisor at this stage.

This algorithm works because HCF (a, b) = HCF (b, r)

Fundamental theorem of arithmetic: According to fundamental theorem of arithmetic every composite number can be written as the product of powers of primes.

Two main uses of fundamental theorem of arithmetic:

(i) It is useful to prove the irrationality of many of the numbers such as.

(ii) It is applied to explore when exactly the decimal expansion of a rational number, say (q≠ 0), is terminating and when it is non-terminating repeating.

Highest common factor (HCF): The HCF of two positive integers a and b is the largest positive integer d that divides both a and b.

Note:

(a) For any two positive integers a and b, HCF (a, b) × LCM (a, b) = a × b.

HCF (a, b) =

LCM (a, b) =

(b) HCF (a, b, c) × LCM (a, b, c) ≠ a × b × c, where a, b, c are positive integers

However, the following results hold good for three numbers a, b and c:

LCM (a, b, c) =

HCF (a, b, c) =

Theorem: Let ‘a’ be a prime number. If a divides x², then a divides x, where x is a positive integer

Decimal Expansion of Rational numbers:

Theorem: A real number is a rational number of the form, where the prime factorisation of q is of the form 2ⁿ5^m, and n, m are some non-negative integers. Then the rational number has a decimal expansion which terminates.

Theorem: A real number is a rational number of the form, where the prime factorisation of q is not of the form 2ⁿ5^m, and n, m are some non-negative integers. Then the rational number has a decimal expansion which is non-terminating repeating.

Some properties of rational numbers:

(1) The sum or difference of a rational and an irrational number is irrational.

(2) The product and quotient of a non-zero rational and irrational number is irrational.

(3) Rational numbers have either a terminating decimal expansion or a non-terminating repeating decimal expansion.