Contents

- Definition
- What Constitutes Irrational Numbers?
- What Sort of Numbers Fall Within this Category?
- How Can You Tell Whether a Number Is Irrational?
- Is the Number Pi an Irrational Value?
- Irrational Numbersâ€™ Inherent Characteristics
- What Applications Do Irrational Numbers Have In Everyday Life?
- Do Irrational Numbers Represent Real Numbers?
- Sum of an Irrational Number
- Product of an Irrational Number
- A Numerical Example of an Irrational Number

# Irrational Number|Definition & Meaning

## Definition

Any **integer** that cannot be **represented** as a fraction of any **integer** p and q is said to be irrational. The **decimal expansions** of irrational numbers are **neither** periodic nor do they come to an end. Every **number** that is transcendental is irrational.

**Irrational numbers** are those that cannot be expressed as fractions. An **irrational** number cannot be written as just a ratio, such as p/q, where p and q are positive **integers** and q is **greater** than zero.

It is inconsistent with **rational** numbers. **Irrational** numbers are typically written as R\Q, where the **backslash** signifies “set minus.” It is also **possible** to write the difference between the real **numbers** and the **rational** numbers as R – Q. The following figure shows the irrational number.

Figure 1 – Hypotenuse showing the irrational number

## What Constitutes Irrational Numbers?

**Irrational** numbers are real **numbers** that cannot be represented like a ratio of integers. For **example,** -2 is **indeed** an **irrational number** because it can’t be represented as a ratio of two integers. We cannot represent any irrational number as a ratio, such as p/q, **where** p and q are **integers** and q is less than zero.

## What Sort of Numbers Fall Within this Category?

Good **examples** of irrational numbers that are frequently used include pi **(=314159265…),** 2, 3, 5, Euler’s number (e = **2718281…),** 2.010010001…., etc.

## How Can You Tell Whether a Number Is Irrational?

**Irrational numbers** are real numbers that cannot be **stated** in the form of p/q, where **both** p and q are **integers** and q is less than zero. These real **numbers** are known as being irrational. **Irrational expressions** include things like 2 and 3, **amongst** others.

In **contrast,** the term **“rational number”** refers to any number that can be expressed as the **fraction** p/q, **provided** that **both** p and q are integers and that q is **not equal** to **zero.**

## Is the Number Pi an Irrational Value?

Pi is **considered** to be an **irrational number** because it does not **terminate** or repeat in **decimal** form. In addition, 22/7 is **indeed** a **rational** number, **whereas** pi is **indeed** an irrational number; hence, the two quantities cannot be **comparable.** The calculated value of pi is **3.141592653589…………..**

**An illustration** of pi, **which** is **irrational,** is **shown** below.

Figure 2 – Illustration of a pi

An illustration of Euler’s number e, which is irrational, is shown below.

Figure 3 – Irrational number

## Irrational Numbersâ€™ Inherent Characteristics

As a **result** of the fact that **irrational numbers** are subsets of real numbers, **irrational numbers** are subject to all of the **characteristics** that are **associated** with the real **number system. Irrational** numbers have the following characteristics as a result of their nature:

**Irrational numbers**can be created by adding them to rational ones. For**illustration,**suppose x is**irrational,**and y is rational, and that the sum of the two,**x + y**, is also irrational.- Any
**irrational number multiplied**by a**nonzero rational**number yields another**irrational**number. Let us**assume**that x is**irrational,**but if xy=z, then x = z/y is**rational.**As a result, we can conclude that xy is an**irrational**product. - There is no
**guarantee**that the**(LCM)**of any pair of**irrational**numbers even exists. - In
**contrast**to the set**containing rational**numbers, the set containing**irrational numbers**does not**become**closed when the process of**multiplication**is applied to it.

## What Applications Do Irrational Numbers Have In Everyday Life?

**Irrational numbers** are used in various **components** of life. Obviously, you may not be able to **visualize** it, but several **components** use **irrational** numbers.

**Trigonometric Ratios**need irrational numbers. The ratios are used in various height and distance**measurements**and also in several**calculations**in**physics.****Pi (Ï€)**– Circles make no sense**without****Ï€**. Trigonometric**Ratios**also use Ï€. Several Integrals use it, and it is also used in**Polar Coordinates****Eulerâ€™s Number e**–**Used extensively**in**Logarithms**and**Algebra.**

These **components** may not **directly** be used, but are used extensively in Engineering. **Engineering** revolves around **designing** things for real life, and several things like **Signal Processing,** Force **Calculations, speedometers,** etc., use irrational numbers. Calculus and other mathematical **domains** that use these irrational numbers are used a lot in real life. **Irrational Numbers** are used indirectly.

## Do Irrational Numbers Represent Real Numbers?

Most **irrational** numbers are **regarded** as real numbers in **mathematics,** which is not **rational.** This **implies** that **irrational numbers** can’t be stated as the ratio of two other **integers** in any way. Non-perfect square roots, for **example,** will **always** yield an **irrational** value.

## Sum of an Irrational Number

The **addition** of two **rationales** also **results** in a **rational** number. A rational number may be stated as a **fraction** having integer values in both the numerator and the denominator, according to the meaning of the term **(denominator** not zero).

Since integers are **generally closed** under addition and **multiplication,** adding two **rational numbers** is the same as adding two **fractions** of the same form, which will result in another **fraction** of the same form since integers have been closed under **addition.** The **addition** of two rational numbers results in the production of a third **rational** number.

Two **irrational integers** added together are still **irrational,** but only sometimes. When adding two **irrational integers** together, the result is **sometimes** irrational. The aggregate will be **rational, nevertheless,** if the irrational **components** cancel out one **another** to zero.

## Product of an Irrational Number

**Multiplying** two **rational** integers yields a **rational** result. A **rational** number can be written as a **fraction** with numerator and **denominator** values that are both integers **(denominator** not zero).

Since integers are generally closed under multiplication, **multiplying** two rationals is equivalent to **multiplying** two such **fractions,** leading to yet another fraction of this kind. As a result, a rational **number** may be obtained by **multiplying** two rational numbers.

It is not always the case, but **irrational** numbers **multiplying** together produce **irrational** results. An **irrational** result can occur by **multiplying** two irrational integers together. But it’s **feasible** that the product of two **irrational** integers might be **rational.**

## A Numerical Example of an Irrational Number

### Example

Find the **irrational numbers between** two numbers which are 5 and 6.

### Solution

We are **required** to find the 5 **irrational numbers** between the given two numbers, **which** are** 5 and 6**.

So we need to multiply the given number by 6.

** 5 = 5 Ã— (6/6) = 30/6**

**6=6 Ã— (6/6) = 36/6**

Thus, as a **result,** the **five rational numbers** that fall between 5 and 6 are as follows:

**31/6****32/6****33/6****34/6****35/6**

*All mathematical drawings and images were created with GeoGebra.*