Unlocking the Mystery of Irrational Numbers: A Guide for Beginners

An irrational number is a number that cannot be expressed as a ratio of two integers. They are infinite and non-repeating decimals that do not terminate, such as pi and the square root of 2. Learn more about the fascinating world of irrational numbers and their properties in this comprehensive guide.

Irrational numbers have always been a mystery in the world of mathematics, and they have fascinated mathematicians for centuries. The previous articles briefly discussed rational numbers. In this article, we will briefly discuss what an irrational number is, its characteristics, properties, and methods to identify an irrational number. 

By the end, we will share some mathematical problems and the role of irrational numbers in their solution.

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Must Check: Difference Between Rational and Irrational Number

So, without further delay, let’s explore the article.

Table of Content

• What is an Irrational Number?
• Properties of Irrational Number
• Famous Mathematical Problems and the role of Irrational Numbers in their solutions
• Key Differences Between Rational and Irrational Number

What is an Irrational Number?

Any real number that cannot be expressed as p/q (where q is not equal to zero) and has a decimal representation which is non-terminating and non-repeating is called an Irrational Number. 

The irrational number was first discovered by a Greek Mathematician, Hippasus of Metapontum while finding the length of the hypotenuse of the isosceles right angle triangle.

When he uses the Pythagoras theorem over a triangle of base 1 unit and height 1 unit, the hypotenuse value will be sqrt(2), which can’t be expressed as p/q and can’t be represented in repeated or terminated decimal form.

This special type of number is referred to as an irrational number. Later, it is found that there exists an uncountable number of irrational numbers.

Examples of Irrational Number

Pi, sqrt(2), sqrt(3) + sqrt(5), e (euler number), Golden Ratio (1.618033988……), 3.14151926535…….

Properties of an Irrational Number

1. Irrational numbers can not be expressed as p/q (where p and q are integers and q is not equal to zero).
2. The decimal expression of irrational numbers is non-terminating and non-repeating.
3. Sum and Difference of Two Irrational Numbers may or may not be an Irrational Number.

4. Multiplication and division of two irrational numbers may or may not be irrational.

5. Any non-zero rational number multiplied by an irrational number always results in an irrational number.

6. The least Common Multiple (LCM) of any two irrational number may or may not exist.
7. Irrational numbers are dense in Real Numbers, i.e., there must be an irrational number between any two rational numbers.

Famous Mathematical Problems and the role of Irrational Numbers in their solutions

• Hilbert’s 7th Problem: At the beginning of the 20th century, mathematician David Hilbert proposes 23 mathematics questions. The 7th problem was related to the existence of an algorithm method to determine whether the given number is rational or irrational.
• Cantor’s Diagonal Argument: In the 19th century, mathematician Georg Cantor proved that there exist more irrational numbers than rational numbers. His proof is known as a diagonal argument.
• Transcendental Numbers: The numbers that can’t be the root of any algebraic equations with real coefficients. These transcendental numbers are frequently used in differential equations, cryptography, and number theory.
• Quadratic Equation: To find the roots of any quadratic equation, we use a quadratic formula that includes the square root of discriminants, which can be irrational.
• Geometry: One of the most used irrational numbers is pi, which is used to find the area and the volumes of geometries such as circles, spheres, cones, cylinders, etc.

Key Differences Between Rational and Irrational Number

• Dissimilar to rational numbers, irrational numbers can’t be expressed as p/q (where p and q are integers and q does not equal 0).
• There exists an uncountable number of irrational numbers, while rational numbers are countable.
• Arithmetic operations on irrational numbers may or may not be irrational, whereas arithmetic operations on rational numbers always result in a rational number.

Conclusion

Irrational numbers are the most amazing number that is subset of real numbers. In this article, we have briefly discussed what rational number is, its properties, and later in the article we have discussed some popular mathematical problem and the role of irrational numbers.

Hope you will like the article.

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