Irrational Number: A Special Type of Numbers
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.
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
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 $.
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. $e=2.7182818284 \ldots$
One common example of an irrational number is $\sqrt{2}=1.41421356237309540488\ldots $
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
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.
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.
Any irrational number multiplied by any non-zero rational number yields an irrational number.
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?
$0.14$
$0.14 \overline{16}$
$0 .\overline{1416}$
$0.4014001400014 \ldots$
Ans: (D)
Q 2. Choose the correct answer:
$\sqrt{10} \times \sqrt{15}$ is equal to
$6 \sqrt{5}$
$5 \sqrt{6}$
$3 \sqrt{5}$
$10 \sqrt{5}$
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.
FAQs on Irrational Numbers
1. How do rational and irrational numbers differ from each other?
Any whole number, fraction, or decimal that ends or repeats is rational. Any number that cannot be divided into a fraction and thus does not meet the definition of a rational number is considered to be irrational. Irrational numbers are those that neither repeat or terminate after a predetermined number of decimal places, whereas rational numbers are those that are terminating or non-terminating repeating numbers. Examples of irrational numbers include $\sqrt{5}$, $\sqrt{10}$, $\pi$, etc. Examples of rational numbers include $\dfrac{3}{2}=1.5, 3.76, 56, 0.66666$, and 3.25.
2. Why are irrational numbers referred to as Surds?
An expression that uses the square root, cube root, or another root symbol is referred to as a surd. The Latin word “surdus”, which means mute or deaf, is where the word surd originates. In the early days of Mathematics, irrational numbers were regarded as mute by the Arabs. This implies that these irrational numbers were worthless. All surds are irrational, even though not all irrational numbers are surds. Surds exclude irrational numbers that are not the roots of algebraic expressions and e.
3. Why is $\dfrac{22}{7}$ a rational number but Pi ($\pi$) is not?
$\pi$ is an irrational number, which means it cannot be expressed as a fraction of $\dfrac{a}{b}$, where a and b are integers. Pi's value is constant at 3.14. Pi ($\pi$) is a non-terminating, non-repeating decimal number that roughly equals 3.14159265359. $\pi$ is, therefore, an irrational number, and $\dfrac{22}{7}$ is equal to 3.14285714286; when these two numbers are subtracted, the result is 0.00126448927979323846... This indicates that the result is not zero. In fact, the value does not equal $\dfrac{22}{7}$. The value is roughly equivalent to the result of $\dfrac{22}{7}$. And since the actual value is neither terminating nor recurring, it is an irrational number. However, $\dfrac{22}{7}$ is a rational number.