Irrational numbers

In the set of real numbers, there are rational and irrational numbers. The latter cannot be expressed as a fraction and can be of two types, algebraic or transcendental. The irrational numbers are those that can not be expressed in fractions that contain decimal indeterminate elements and are used in complex mathematical operations as algebraic equations and physical formulas.

What are irrational numbers?

Irrational numbers are part of the set of real numbers that is not rational, that is, it cannot be expressed as a fraction. This set of numbers is formed by all the decimal numbers whose decimal part has infinite digits . They are represented by the letter I or with the representation RQ (This is the subtraction of the real numbers minus the rational numbers). They can be algebraic or transcendent .

  • What are the irrational numbers
  • What are they for
  • features
  • History
  • How they are represented
  • Classification of irrational numbers
  • Operations
  • Properties
  • Examples of irrational numbers

What are the irrational numbers

The set of irrational numbers is made up of algebraic numbers and transcendent numbers .

What are they for

They are used to carry out operations in factual sciences such as physics , chemistry , mathematics, among others.


Of the most representative characteristics of irrational numbers we can cite the following:

  • They are part of the set of real numbers .
  • They can be algebraic or transcendent .
  • They cannot be expressed as a fraction.
  • They are represented by the letter I .
  • They have infinite decimal places .
  • It has commutative and associative properties .
  • They cannot be represented as a division of two whole numbers .


The mathematician who first identified this set of numbers is presumed to be a disciple of Pythagoras named Hipaso . This character trying to describe the root of the number 2 as a fraction showed that there are numbers that are not rational because they cannot be expressed with fractions.

It is said that this discovery was not well accepted by Pythagoras, who claimed that all numbers have perfect values. As the Pythagorean master could not disprove the discovery of Hypasus, they threw Hippasus over the side of a ship and he drowned.

There is also another version of the story of irrational numbers that comes from Ancient Greece. It states that in the practice of measuring lengths of a segment of a line that could only be fractioned, the Greeks identified numbers that could not be fractionated . This discovery is attributed to Pythagoras, when determining the existence of segments of the line that are incommensurable in relation to a segment that is taken as a unit in a measurement system.

How they are represented

They are represented by the uppercase letter I because the lowercase i represents imaginary numbers. They are also usually represented in the following way RQ (this means Real Numbers – Rational Numbers ). However, it is important to mention that there are irrational numbers that have their own symbols . This is the case of the number Pi or the golden number .

Classification of irrational numbers

Irrational numbers are classified into algebraic numbers and transcendental numbers.

The algebraic numbers are those from solving any algebraic equation and finite numbers of free or nested radicals. Example: the roots are not exact.

The transcendent numbers are those that come from the trigonometric, logarithmic and exponential transcendent functions . These numbers are not finite numbers of nested or free radicals. Example: the number Pi = 3.141592653589…; the golden number = 1.618033988749…; Euler’s number = 2.718281828459…


The operations of addition, subtraction, multiplication and division are not well defined because these when applied to irrational numbers do not tend to result in irrational numbers . Taking this into account, the following observations are important:

  • If a rational number is added with an irrational one , the result will always be irrational .
  • If a rational number (other than zero) is multiplied by an irrational number the product will be irrational .
  • It has commutative and associative properties .
  • The multiplication is distributive with respect to the operations of addition and subtraction.
  • It has its opposite or negative element that cancels it out.


Irrational numbers have the following properties:

  • Commutative: irrational numbers can be added or multiplied.
  • Associative: they can be grouped.
  • Closed: any irrational number added, subtracted, multiplied or divided will not always result in an irrational number. This is not true in the case of filing.

Examples of irrational numbers

As an example of irrational numbers we can mention the following:

Examples of algebraic irrational numbers

  • √7
  • 0.1961325454898161376813268743781937693498749 …

Example of transcendental irrational numbers

  • Pi = 3.141592653589…
  • Golden Number = 1.618033988749…
  • Euler number = 2.718281828459…

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