Definition of Irrational Numbers

Different types of numbers in mathematics constitute number system. Some of them are whole numbers, real numbers, rational number, irrational numbers, integers, etc. In this topic, we will get to know about irrational numbers.

Irrational numbers: Irrational numbers are those which can’t be expressed in fractional form, i.e., in $\frac{p}{q}$ form. They neither terminate nor do they repeat. They are also known as non- terminating non-repeating numbers.

A number $\sqrt{x}$ (square root of x) where x is positive and x is not a perfect square of a rational number, is not a rational number. As such $\sqrt{x}$ cannot be put in the form $\frac{a}{b}$ where a ∈ Z, b ∈ Z, and b ≠ 0. Such numbers are called irrational numbers. 

Thus the numbers, derived form rational numbers, that cannot be put in the form $\frac{a}{b}$ where a ∈ Z, b ∈ Z, and b ≠ 0 are called irrational numbers.

For example:

Irrational numbers include ‘π’ which starts with 3.1415926535… and is never ending number, square roots of 2,3,7,11, etc. are all irrational numbers.

$\sqrt{2}$, $\sqrt{7}$, $\sqrt{13}$, $\sqrt{\frac{7}{3}}$, $\frac{\sqrt{7}}{5}$, 5 + $\sqrt{7}$ are all positive irrational numbers.

Similarly, - $\sqrt{3}$,  -$\sqrt{\frac{5}{2}}$, - $\frac{\sqrt{11}}{19}$, 1 - $\sqrt{7}$ are also irrational numbers which are negative irrational numbers.

But numbers such as $\sqrt{9}$, $\sqrt{81}$, $\sqrt{\frac{25}{49}}$ are not irrational because 9, 81 and $\frac{25}{49}$ are square root of 3, 9 and $\frac{5}{7}$ respectively.

The solution of x$^{2}$ = d are also irrational numbers if d is not a perfect square.

Euler’s number ‘e’ is also an irrational number whose value is 2.71828 (approx.) and is the limit of $(1 + \frac{1}{n})^{n}$. it can also be calculated as sum of infinite series.

Applications of irrational numbers:

1. In compound interest: Let us have a look at the following example to understand how irrational number helps us in case of calculating compound interest:

An amount of Rs. 2,00,000 is given to Animesh by his friend for a tenure of 2 years at interest of 2% per annum compounded annually. Calculate the amount which Animesh needs to return his friend after 2 years.

Solution: 

Principal = Rs 2,00,000

Time = 2 years

Interest rate (r) = 2% p.a.

Amount = p$(1 + \frac{r}{100})^{t}$

So, amount = 2,00,000$(1 + \frac{2}{100})^{2}$

                  = 2,00,000$(\frac{102}{100})^{2}$

                  = 2,00,000 × $\frac{10,404}{10,000}$

                  = 2,08,080

Hence, the amount that Animesh needs to return to his friend is Rs. 2,08,080.

So, compound interest is one of the applications of irrational numbers where we use sum of infinite series.

Another example where we use irrational numbers are:

(i) Finding area or perimeter (circumference) of any circular part: We know that area and circumference of a circular part is given by πr$^{2}$ and 2πr respectively, where ‘r’ is the radius of the circle and ‘pi’ is the irrational we use in finding area and circumference of the circle whose value is 3.14 (approx.).

(ii) Use of cube root: Cube roots are basically used in finding area and perimeter of three dimensional structures such as cubes and cuboids.

(iii) Used to find gravity equation: Equation for acceleration of gravity is given by:

g = $\frac{Gm}{r^{2}}$

where g = acceleration due to gravity

         m = mass of the object

          r = radius of earth

          G = gravitational constant

Here ‘G’ is the irrational number whose value is 6.67 x 10$^{-11}$.

Similarly, there are many such examples where we use irrational numbers.

In earlier days when people found difficulty in finding out the square and cube roots of numbers whose square and cube roots were not whole numbers, they developed concept of irrational numbers. They called this number as non- terminating non-repeating numbers.

Irrational Numbers

Definition of Irrational Numbers

Representation of Irrational Numbers on The Number Line

Comparison between Two Irrational Numbers

Comparison between Rational and Irrational Numbers

Rationalization

Problems on Irrational Numbers

Problems on Rationalizing the Denominator

Worksheet on Irrational Numbers

9th Grade Math

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