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Mathematics is all about the magic of numbers. We already know that there are different types of numbers. Natural numbers, whole numbers, complex numbers, real numbers, and integers are all examples of different types of numbers that we are familiar with. Real numbers can also be divided into rational and irrational categories.
The irrational numbers cannot be expressed as the ratio of two integers. They can be expressed on the number line and are a subset of real numbers. Besides this, an irrational number’s decimal expansion is neither terminating nor repeating.
Contents
What is an Irrational Number?
In Mathematics, the real numbers that cannot be expressed as $\dfrac{p}{q}$, where p and q are integers and q is not equal to zero, are referred to as irrational numbers.
These numbers are represented using the symbol “Q”.
How to Identify Irrational Numbers?
We are aware that there are irrational numbers that cannot be stated using the formula $\dfrac{p}{q}$, where p and q are integers, and $q \neq 0$. Irrational numbers include those like $\sqrt{5}$, $\sqrt{3}$, etc. In contrast, the numbers that can be expressed as $\dfrac{p}{q}$, where p and q are integers and $q \neq 0$, are rational numbers. For example: $\dfrac{3}{10}$, $\dfrac{1}{4}$, etc.
Some Known Examples of Irrational Numbers
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There is a fact that Pi is an irrational number since it has been calculated to more than a quadrillion decimal places with no pattern ever forming. $\pi=3.1415926535\ldots $.
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Euler’s number(e) cannot be written as a fraction and does not terminate or repeat when written as a decimal, making it an irrational number. $e=2.7182818284 \ldots$
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One common example of an irrational number is $\sqrt{2}=1.41421356237309540488\ldots $
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In many disciplines, including computer science, design, art, and architecture, the golden ratio—an irrational number—is used. The first number in the Golden Ratio, represented by the symbol $\Phi=1.61803398874989484820 \ldots$
Properties of Irrational Numbers
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Decimals with non-terminating and non-repeating patterns make up irrational numbers. For example, $\pi=3.141592653…5\ldots $ has a no-repeating and non-terminating pattern.
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Two irrational numbers added together, subtracted from, multiplied, and divided may or may not result in a rational number. Let’s suppose that if ab=c is true, then $a = \dfrac{x}{b}$ is true, contradicting the assumption that x is irrational. The a*b product must be irrational.
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Any irrational number multiplied by any non-zero rational number yields an irrational number.
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It is not necessary for the result of multiplying two irrational numbers to always be an irrational number. Like, $\sqrt{2}+\sqrt{2}=2$ which is rational.
Interesting Facts
Hippasus, a former student of Pythagoras, reportedly discovered irrational numbers while attempting to express the square root of 2 as a fraction (using geometry). Sadly, his theory was mocked and cast into the water.
Solved Problems on Irrational Numbers
Example 1: Determine which of the following numbers are rational and irrational.
$\sqrt{12}, \sqrt{16}, \sqrt{5}, 1.23123123412…$.
Solution:
$\sqrt{12}=3.4641016151…$ non-terminating and non-recurring so it is an Irrational number.
$\sqrt{16}=\pm 4$ Rational number.
$\sqrt{5}=2.2360679995…$ non-terminating and non-recurring so it is an Irrational number.
1.23123123412… = non-terminating and non-recurring so it is an Irrational number.
Example 2. Simplify the expressions: $(\sqrt{5}+\sqrt{2})^2$.
Solution:
$(\sqrt{5}+\sqrt{2})^2 $
$\Rightarrow (\sqrt{5}+\sqrt{2})(\sqrt{5}+\sqrt{2})$
$\Rightarrow \sqrt{5}(\sqrt{5}+\sqrt{2})+\sqrt{2}(\sqrt{5}+\sqrt{2})$
$\Rightarrow \sqrt{5} \times \sqrt{5}+\sqrt{5} \times \sqrt{2}+\sqrt{2} \times \sqrt{5}+\sqrt{2} \times \sqrt{2}$
$\Rightarrow 5+\sqrt{10}+\sqrt{10}+2$
$\Rightarrow 7+2 \sqrt{10}$
Practice on Your Own
Q 1: Which of the following is irrational?
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$0.14$
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$0.14 \overline{16}$
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$0 .\overline{1416}$
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$0.4014001400014 \ldots$
Ans: (D)
Q 2. Choose the correct answer:
$\sqrt{10} \times \sqrt{15}$ is equal to
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$6 \sqrt{5}$
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$5 \sqrt{6}$
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$3 \sqrt{5}$
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$10 \sqrt{5}$
Ans: (B)
Conclusion
This article summarises that irrational numbers are defined as a kind of real numbers that cannot be stated as $\dfrac{a}{b}$, where a and b are integers and b is not equal to zero. They can’t be expressed because they are non-recurring and non-terminating decimals. Pi, $\sqrt{2}$, $\sqrt{5}$, the Golden Ratio, Euler’s number, and others are well-known examples of this type of number.
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