Irrational Numbers – Definition & Properties

Irrational Numbers

Any real number that you can’t show as the quotient of two integers is known as an irrational number. For example, there is no integer or fraction value that equals the square root of 2. A comparable measurement problem would be estimating the length of a square’s diagonal with a one-unit-long side; there is no subdivision of the unit length that splits evenly into the diagonal length.

Thus, early in the history of mathematics, it became necessary to broaden the definition of numbers to include irrational numbers. Every irrational integer has an infinite decimal expansion with no repeating digit or group of digits. When they are coupled with rational numbers, they generate real numbers.

What are Irrational Numbers & Who Invented Irrational Numbers?

Irrational numbers are real numbers that cannot be represented by a ratio or irrational numbers are real numbers that are not rational numbers. In the fifth century BC, Hippasus, a Pythagorean philosopher, found irrational numbers. Unfortunately, his hypothesis was mocked, and he was cast into the water. 

But irrational numbers do exist, visit the Cuemath website to acquire a better understanding of the notion, and don’t worry, you won’t be tossed into the sea. 

Let’s discuss the properties as well as the distinction between irrational and rational numbers.

Properties of Irrational Numbers

Irrational numbers are subsets of real numbers; irrational numbers have all of the attributes of the real number system. The following are the qualities of irrational numbers

  1. Adding an irrational number to a rational number yields an irrational number. For example, if x is an irrational number and y is a rational number, and the sum of both numbers, x+y, yields an irrational number.
  2. The product of two irrational numbers does not have to be an irrational number.
  3. When we multiply any irrational number by any non-zero rational number, we get an irrational number as the result.
  4. The product of two irrational numbers might be either a rational or an irrational number.

Difference Between Rational & Irrational Numbers

A rational number is one that can be represented as the ratio of two integers, with the denominator not equal to zero, whereas an irrational number cannot be expressed in fractions. Rational numbers have ending decimals, but irrational numbers do not. A rational number is 10/2, whereas an irrational number is the well-known mathematical value Pi(), which is equal to 3.141592653589……

What are Math Worksheets?

Math will be a part of everyone’s life from preschool through the last day of high school — and most likely well beyond that. As a result, there are numerous math worksheets accessible online that cover a wide range of arithmetic subjects for students of different ages and grade levels.

Math worksheets are essential for pupils in school and at home. Math worksheets are documents that may be accessed online or offline and contain a set of practice questions on a certain topic. They are motivated to augment a child’s school learning and assist him to develop his arithmetic abilities. The questions are given in a systematic manner to assist students in developing a crystal clear understanding.

The following are some of the advantages of using math worksheets:

  • It promotes active learning in the classroom and gives additional practice.
  • It will keep you entertained and learning for hours.
  • Engage pupils to encourage them.
  • Assists children in better preparing for school examinations.
  • It aids in the improvement of speed and accuracy.
  • It incorporates a step-by-step learning mechanism that assists students in approaching a problem strategically, recognizing their errors, and developing mathematical abilities.
  • Students have time flexibility.


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