M1.1. Classification of Real Numbers

Classification of Real Numbers

Natural Numbers

Natural numbers are the numbers that are used for counting or enumerating items. Numbers like 1, 2, 3, 4, etc are examples of Natural Numbers.

A set of natural numbers is typically denoted by the symbol N and is written as

N = {1,2,3,4,5,6,7 …}

A natural number is an integer greater than 0. Natural numbers begin at 1 and increment to infinity: 1, 2, 3, 4, 5, etc. There are infinitely many natural numbers.

Representation of Natural Numbers on a Number Line :

Whole Numbers

The set of natural numbers that includes zero is known as the whole numbers. A set of whole numbers are typically denoted by W.

For example, the following is a set of whole numbers:

W = {0, 1, 2, 3, 4, 5, 6, 7…}

Representation of Whole Numbers on a Number Line :

Integers

The set of integers adds the opposites (negatives) of the natural numbers to the set of whole numbers: … but still no fractions allowed!

A set of integers are typically denoted by Z.

For example, the following is a set of integers :

Z = {…, −4, −3, −2, −1, 0, 1, 2, 3, 4, …}

Representation of Integers on a Number Line :

It is useful to note that the set of integers is made up of three distinct subsets : negative integers, zero, and positive integers (Natural Numbers).

Rational Numbers

The numbers of the form p/q, where p and q are integers and q ≠ 0 are known as rational numbers.

Every natural number, whole number, and integer is a rational number with a denominator of 1 Since q may be equal to 1. In other words, every integer is a rational number.

A set of rational numbers are typically denoted by Q.

$Thus,\; \frac{1}{3},\; \frac{5}{2},\; \frac{11}{78},\; -\frac{5}{7},\; 3,\; -5,\; 0\; etc., \; are\; all\; rational\; numbers.$

Equivalent Rational Numbers

Rational numbers do not have a unique representation in the p/q form, where p and q are integers and q≠0.

$Thus,\; \frac{1}{3}=\frac{2}{6}=\frac{3}{9}=\frac{-2}{-6}=\frac{9}{27}=......$

These all are equivalent rational numbers.

To find rational numbers equivalent to any given rational number, multiply both Numerator and Denominator of the given rational number by a non-zero integer. The resulting number will be a equivalent rational number.

$i.e.,\; \frac{a}{b}=\frac{a*m}{b*m},\; where\; m\in Z,\; a\neq 0,\;and\; m\neq 0.$

Simplest Form of a Rational Number

A rational number p/q is in its simplest form, if p and q are integers having no common factor other than 1 (that is, p and q are co-primes) and q≠0.

$\textsl{Thus, the simplest form of each of}\; \; \frac{6}{9},\frac{8}{12},\frac{10}{15},\; etc.,\; is\; \frac{2}{3}.$

Decimal Representation of Rational Numbers

Because rational numbers are fractions, these can also be expressed in decimal form. Any rational number can be represented as either :

(1) A Terminating Decimal : A decimal that ends after a finite number of digits. The remainder becomes zero and the division concludes after a finite number of steps. In this case, the decimal expansion obtained also terminates or ends.

$Examples :\; \frac{1}{4}=0.25,\; \frac{3}{8}=0.375,\; 1\tfrac{3}{5}=\frac{8}{5}=1.6$

Note : A rational number p/q (in simplest form) is expressible as a terminating decimal when prime factors of q are 2 and 5 only i.e., q can be expresses as 2^m5ⁿ where m and n are whole numbers.

(2) A Repeating (or Recurring) Decimal : A decimal in which a digit or a set of digits is repeated periodically, is called a repeating or a recurring decimal. The remainder never becomes zero and a repeating string of remainders is obtained. In this case, we get a digit or a block of digits repeating in the decimal expansion.

$Examples:\; \frac{3}{11}=0.2727.....=0.\overline{27},\; \frac{11}{6}=1.8333.....=1.8\overline{3}$

Irrational Numbers

Irrational numbers are the numbers which cannot be expressed in the form p/q, where p and q are integers and q ≠ 0.

Irrational numbers can neither be expressed as a terminating decimal nor as a repeating decimal.

$Examples \; \pi ,\; \sqrt{2},\; \sqrt{5},\; \sqrt[3]{7},\; 0.03003000300003.....\; etc.$

Note :

(1) If m is a positive integer which is not a perfect square, them √m is irrational.

(2) If m is a positive integer which is not a perfect cube, them ∛m is irrational.

Real Numbers

A number whose square is non-negative, is called a real number,

Given any number n, it can either rational or irrational. It cannot be both. The sets of rational and irrational numbers together make up the set of real numbers.

Any real number corresponds to a unique position on the number line.The converse is also true : Each location on the number line corresponds to exactly one real number. This is known as a one-to-one correspondence.