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Home » Algebra » Rational and Irrational Numbers

Rational and Irrational Numbers

In the article Classification of Numbers we have already defined Rational Numbers and Irrational Numbers. We also touched upon a few fundamental properties of Rational and Irrational numbers. In this article we shall extend our discussion of the same and explain in detail some more properties of Rational and Irrational Numbers.

Rational Numbers

In general the set of rational numbers is denoted as \mathbb{Q} .

\therefore \mathbb{Q} = \dfrac{a}{b} ; a,b \in Z and b \neq 0

  • The condition b \neq 0 is a necessary condition for \frac{a}{b} to be rational number, as division by zero is not defined. \frac{5}{11}, -\frac{4}{9}, \frac{3}{19} , etc. are rational numbers.
  • Just as, corresponding to any integer n , there is it’s negative integer -n ; similarly corresponding to every rational number \frac{a}{b} there is it’s negative rational number -\frac{a}{b} .
  • Two rational numbers \frac{a}{b} and \frac{c}{d} are equal if and only if ad=bc i.e., \frac{a}{b} = \frac{c}{d} or a \times d = b \times c . Also, \frac{a}{b} > \frac{c}{d} \Rightarrow a \times d > b \times c and \frac{a}{b} < \frac{c}{d} \Rightarrow a \times d < b \times c
  • In between any two rational numbers a and b , there exists another rational number \frac{a+b}{2} . If a>b then a > \frac{a+b}{2} >b and if a<b then a < \frac{a+b}{2} < b

Example: Find three rational numbers between 3 and 5.

Solution:

\because 3 < 5 \Rightarrow 3< \dfrac{3+5}{2} <5 (inserting one rational number between 3 and 5)

or, 3 < 4 < 5

or, 3 < \dfrac{3+4}{2} < 4 < \dfrac{4+5}{2} < 5

or, 3 < \dfrac{7}{2} < 4 < \dfrac{9}{2} < 5

\therefore \dfrac{7}{2}, 4, \dfrac{9}{2} are three rational numbers between 3 and 5.

Properties of Rational Numbers

  1. The sum of two or more rational numbers is always a rational number. For example: i) If a and b are any two rational numbers then a+b is also a rational number. ii) \because \frac{3}{5} \in \mathbb{Q} and \frac{4}{7} \in \mathbb{Q} then \frac{3}{5} + \frac{4}{7} = \frac{21+20}{35} = \frac{41}{45} , which is also a rational number.
  2. The difference of two rational numbers is always a rational number. For example: i) If a and b are any two rational numbers then each of a-b and b-a is also a rational number ii) \because \frac{3}{5} \in \mathbb{Q} and \frac{4}{7} \in \mathbb{Q} then \frac{3}{5} - \frac{4}{7} = \frac{21-20}{35} = \frac{1}{35} , which is also a rational number. Also, \frac{4}{7} - \frac{3}{5} = \frac{20-21}{35} = -\frac{1}{35} , which is also a rational number.
  3. The product of two or more rational numbers is always a rational number. For example: If a and b are two rational numbers, then ab \in \mathbb{Q}
  4. The division of a rational number by a non-zero rational number is always a rational number. For example: If a and b are two rational numbers and b \neq 0 , then \frac{a}{b} is always a rational number

Note: Since the sum of two rational numbers is always a rational number; we say the set of rational numbers is closed for addition.

In the same way the set of rational numbers is closed for:

  • Subtraction
  • Multiplication
  • Division (if divisor not equal to zero)

Decimal representation of Rational Numbers

Every rational number can be expressed either as a terminating decimal or as a non-terminating decimal.

Examine the following rational numbers:

\dfrac{1}{8} = 0.125

\dfrac{1}{25} = 0.04

In each example given above the division is exact. The quotients of such divisions are called terminating decimals.

Now examine the following rational numbers:

\dfrac{3}{7} = 0.42857

\dfrac{18}{23} = 0.7826086

In each example above the division never ends, no matter how long it continues. The quotients of such divisions are called non-terminating decimals.

Now examine the following divisions:

\dfrac{4}{9} = 0.444444444

\dfrac{11}{30} = 0.3666666666

These non terminating decimals in which a digit or a set of digits repeats continually, is called a recurring or a periodic or a circulating decimal. The repeating digit or the set of repeating digits is called the period of the recurring decimal.

Note: If the denominator of a rational number can be expressed as the power either of 2 or of 5 or of 2 and 5 both, the rational number is convertible into a terminating decimal. Otherwise, the rational number is convertible to a recurring decimal.

Irrational Numbers

Any real number that cannot be expressed as a ratio of integers, i.e., any real number that cannot be expressed as simple fraction is called an irrational number.

The square roots, cube roots, etc of natural numbers are irrational numbers, if their exact values cannot be obtained. \sqrt{2} is irrational since exact value of it cannot be obtained.

A non- terminating and non-recurring decimal is an irrational number.For example, 0.424344445

The number \pi = 3.1415926535 is also an irrational number.

Example: Identify the number as rational or irrational \sqrt{12}

Solution:

\sqrt{12} = \sqrt{2 \times 2 \times 3}= 2 \sqrt{3} , which is the product of a rational number 2 and an irrational  number \sqrt{3}

Example: Find two irrational numbers between 2 and 3.

Solution:

If a and b are two positive numbers such that ab is not a perfect square then :

i ) A rational number between a and b = \dfrac{a+b}{2}

ii) An irrational number between a and b = \sqrt{ab}

\because 2 and 3 are rational numbers and 2 \times 3 = 6 is not a perfect square

\therefore one irrational number between 2 and 3 = \sqrt{2 \times 3} = \sqrt{6}

An irrational number between 2 and \sqrt{6}= \sqrt{2 \times \sqrt{6}}

\therefore required rational and irrational numbers are:

\sqrt{6}

\sqrt{2 \times \sqrt{6}}

Example: Insert a rational and an irrational  number between 3 and 4.

Solution: Since, 3 and 4 are positive rational numbers and 3 \times 4 = 12 is not a perfect square, therefore:

i) A rational number between 3 and 4 = \dfrac{3+4}{2} = 3.5

ii) An irrational number between 3 and 4 = \sqrt{3 \times 4}= 2\sqrt{3}

More about irrational numbers

  1. The sum of two irrational  numbers may or may not be irrational.
  2. The difference of two irrational numbers may or may not be irrational.
  3. The product of two irrational numbers may or may not be irrational.
  4. The negative of an irrational number is always irrational.
  5. The sum of a rational and an irrational number is always irrational.
  6. The product of a non-zero rational number and an irrational number is always irrational.

Exercise

  1. Insert three rational numbers between:
    1. 4 and 4.5
    2. 5 and -2
  2. Represent as a decimal number:
    1. \dfrac{8}{13}
    2. \dfrac{4}{9}
  3. Find which of the following rational numbers have terminating decimal representation:
    1. \dfrac{32}{45}
    2. \dfrac{17}{40}
  4. Find the decimal representation of \dfrac{1}{7} and \dfrac{2}{7} . Deduce from the decimal representation of \dfrac{1}{7} , without actual calculation, the decimal representation of \dfrac{3}{7}, \dfrac{4}{7}, \dfrac{6}{7} .
  5. Prove that  $ \sqrt{m} $ is not a rational number, if m is not a perfect square.
  6. Compare : \sqrt{4} and \sqrt{3}
  7. Write in ascending order:
    1. 5\sqrt{3}, 4\sqrt{5}, 3\sqrt{7}
    2. 3\sqrt{2}, 2\sqrt{3}
  8. Write a pair of irrational numbers whose difference is irrational.
  9. Write a pair of irrational numbers whose product is rational.
  10. Insert two rational numbers and two irrational numbers between \sqrt{3} and \sqrt{7}
« Simple Formulae and their Application
Problems Leading to Simple Equations »


Filed Under: Algebra Tagged With: Irrational number, Rational Numbers

Comments

  1. mayur says

    November 11, 2014 at 6:53 am

    Very nice tricks…

    Reply
  2. Sam says

    January 16, 2018 at 5:52 pm

    Thanx guys yu helped me alot

    Reply

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Table of Content

  • Complex Numbers
  • Quadratic Equations
  • Logarithm
  • Permutation
  • Combination
  • More on Complex Numbers
  • Classification of Numbers
  • Positive and Negative Quantities
  • Understanding Simple Algebraic Formulas With Examples
  • Integers
  • Linear Inequalities
  • An Introduction to Fundamental Algebra
  • Basic Number Properties – Commutative, Associative and Distributive
  • Algebraic Multiplication and Division
  • Simple Equations in One Variable
  • Simple Formulae and their Application
  • Rational and Irrational Numbers
  • Problems Leading to Simple Equations
  • Simultaneous Equations
  • Mathematical Induction
  • Different Type of Sets
  • Indices
  • Framing Formulas
  • Sequences
  • Introduction to Matrices
  • Addition Of Matrices
  • Subtraction Of Matrix
  • Multiplication of Matrices
  • Determinant of Matrices
  • Co-factor of Matrices
  • Minor of Matrices
  • Transpose and Adjoint of Matrices
  • Inverse of a Matrix
  • System of Linear Equations in Matrices
  • Introduction to Polynomials
  • Classification of Polynomials
  • Addition and Subtraction of Polynomials
  • Multiplication of Polynomials
  • Factoring Polynomials
  • Zeroes of Polynomial
  • Remainder Theorem of Polynomials
  • Factor Theorem of Polynomial
  • Simplifying Polynomial Fractions
  • Roots of a Polynomial
  • Addition of Polynomial Fractions
  • Subtraction of Polynomial Fractions
  • Multiplying polynomial fractions
  • Division of Polynomial Fractions

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