What are Irrational Numbers?
Irrational numbers belong to the realm of real numbers, but they possess a unique characteristic: they cannot be neatly expressed as simple fractions. In contrast to rational numbers, which can be written as p/q (where both p and q are integers and q≠0), irrational numbers defy such ratio representation.
To further grasp the concept, think of irrational numbers as a contradiction to rationality. They resist being confined within the bounds of simple fractions.
A common way to express irrational numbers is through the notation R\Q, where the backward slash (\) symbol signifies ‘set minus.’ In simpler terms, it denotes the set of real numbers excluding the set of rational numbers. Another representation is R – Q, emphasizing the distinction between the set of all real numbers and the set of rational numbers.
The calculations based on these numbers are a bit complicated. For example, √5, √11, √21, etc., are irrational.
Irrational Number Definition
An irrational number is a type of real number that cannot be expressed as a simple fraction (ratio) of two integers. In other words, it’s a number that cannot be written in the form a/b, where “a” and “b” are integers and “b” is not equal to zero.
Irrational numbers have non-repeating, non-terminating decimal expansions. Examples of irrational numbers include the square root of non-perfect squares (like √2, √3), pi (π), and the mathematical constant “e”.
Examples of Irrational Numbers
Square Root of 2 (√2)
- The square root of 2 is irrational. Its decimal representation goes on forever without repeating, and it cannot be expressed as a fraction.
- Approximate decimal representation: 1.4142…
Pi (π)
- Pi is the ratio of the circumference of a circle to its diameter. It is an irrational number.
- Approximate decimal representation: 3.141…
Euler’s Number (e)
- Euler’s number is an irrational constant that is the base of natural logarithms.
- Approximate decimal representation: 2.718281828459045…
Rational and Irrational Numbers
Rational numbers and Irrational numbers are real numbers with unlike characteristics. Rational numbers are the numbers which can be represented in the A/B form where B ≠ 0. Irrational numbers are the numbers that cannot be represented in A / B form. In this article, we’ll learn the concepts of rational numbers and irrational numbers and explore the difference between them.
Table of Content
- What is Rational number?
- How to identify rational numbers?
- What are Irrational Numbers?
- How to Identify Irrational Numbers?
- How to Classify Rational and Irrational Numbers?
- Difference Between Rational and Irrational Numbers