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Rational and irrational number - CSEET QT


What is a Rational Number?

A rational number, in Mathematics, can be defined as any number which can be represented in the form of p/q where q ≠ 0. Also, we can say that any fraction fits under the category of rational numbers, where the denominator and numerator are integers and the denominator is not equal to zero. When the rational number (i.e., fraction) is divided, the result will be in decimal form, which may be either terminating decimal or the repeating decimal.

How to identify rational numbers?

To identify if a number is rational or not, check the below conditions.

· It is represented in the form of p/q, where q≠0.

· The ratio p/q can be further simplified and represented in decimal form.

The set of rational numerals:

1. Include positive, negative numbers, and zero

2. Can be expressed as a fraction



Examples of Rational Numbers:

Types of Rational Numbers

A number is rational if we can write it as a fraction, where both denominator and numerator are integers and the denominator is a non-zero number.

Positive and Negative Rational Numbers

As we know that the rational number is in the form of p/q, where p and q are integers. Also, q should be a non-zero integer. The rational number can be either positive or negative. If the rational number is positive, both p and q are positive integers. If the rational number takes the form -(p/q), then either p or q takes the negative value. It means that

-(p/q) = (-p)/q = p/(-q).



Rational Numbers and Irrational Numbers

There is a difference between rational and Irrational Numbers. A fraction with non-zero denominators is called a rational number. The number ½ is a rational number because it is read as integer 1 divided by integer 2. All the numbers that are not rational are called irrational. Check the chart below, to differentiate between rational and irrational.

An irrational number cannot be written as a simple fraction but can be represented with a decimal. It has endless non-repeating digits after the decimal point. Some of the common irrational numbers are:

Pi (π) = 3.142857…

Euler’s Number (e) = 2.7182818284590452…….

√2 = 1.414213…

Solved Examples

Example 1:

Identify each of the following as irrational or rational: ¾ , 90/12007, 12 and √5.

Solution:

Since a rational number is the one that can be expressed as a ratio. This indicates that it can be expressed as a fraction wherein both denominator and numerator are whole numbers.

· ¾ is a rational number as it can be expressed as a fraction. 3/4 = 0.75

· Fraction 90/12007 is rational.

· 12, also be written as 12/1. Again a rational number.

· Value of √5 = 2.2360679775…….. It is a non-terminating value and hence cannot be written as a fraction. It is an irrational number.

Example 2:

Identify whether a mixed fraction, 11/2 is a rational number.

Solution:

The Simplest form of 11/2 is 3/2

Numerator = 3, which is an integer

Denominator = 2, is an integer and not equal to zero.

So, yes, 3/2 is a rational number.

Example 3:

Determine whether the given numbers are rational or irrational.

(a) 1.75 (b) 0.01 (c) 0.5 (d) 0.09 (d) √3

Solution:

The given numbers are in decimal format. To find whether the given number is decimal or not, we have to convert it into the fraction form (i.e., p/q)

If the denominator of the fraction is not equal to zero, then the number is rational, or else, it is irrational.


Frequently Asked Questions on Rational Numbers

Q1

What are rational numbers? Give Examples.

A rational number is a number that is in the form of p/q, where p and q are integers, and q is not equal to 0. Some of the examples of rational numbers include 1/3, 2/4, 1/5, 9/3, and so on.

Q2

What is the difference between rational and irrational numbers?

A rational number is a number that is expressed as the ratio of two integers, where the denominator should not be equal to zero, whereas an irrational number cannot be expressed in the form of fractions. Rational numbers are terminating decimals but irrational numbers are non-terminating and non-recurring. Example of the rational number is 10/2, and for an irrational number is a famous mathematical value Pi(π) which is equal to 3.141592653589…….

Q3

Is 0 a rational number?

Yes, 0 is a rational number because it is an integer that can be written in any form such as 0/1, 0/2, where b is a non-zero integer. It can be written in the form: p/q = 0/1. Hence, we conclude that 0 is a rational number.

Q4

Is 7 a rational number?

7 is a rational number because it can be written in the form of a ratio such as 7/1.

Q5

Is 3.14 a rational number?

Yes, 3.14 is a rational number because it is terminating. But is not a rational number because the exact value of is 3.141592653589793238…which is non terminating non recurring.

Q6

Find a rational number between 3 and 4.

Rational number between 3 and 4 = (3+4)/2 = 7/2

Q7

What is the denominator of the rational number?

The denominator of the rational number can be any real number except 0.

Q8

Is Pi(π) a rational number?

No, Pi (π) is not a rational number. It is an irrational number and its value equals 3.142857…


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