# Examples of Irrational numbers

It is called **irrational numbers** to elements of the real line which are usually expressed by the ratio of two integers. These numbers cannot be written as a fraction .

The **irrational numbers** within **mathematics** are those that can not be represented by **fractional numbers** ; In other words, they are those that have a **decimal expression, infinite NOT periodic** . Also, they can be defined as any **real number** that **is not rational** .

The first examples of **irrational numbers** can be seen here:

The fact that they cannot be expressed in the form of fractions, is because the decimal places continue indefinitely **without repeating themselves** .

Considering the set of **Irrational numbers** , it would be included in the **real numbers,** which are equivalent to all **numbers with decimal** , periodic and non-periodic **expression** .

Let’s build an example of any **irrational number** like this:

**2.90991999299993999994999999599999996 ………… ..**

You can see that it is built by making the list of figures in the decimal tail **infinite non-periodic** .

Let us remember that the **rational numbers** are also part of the **real numbers; **but **rational and irrational** have no elements in common.

This is: **I** are included in **R** ; **Q** is included in **R** and; **IUQ = ****R** with **I ∩Q = Ǿ (it** is empty)

**More on Irrational Numbers: Properties**

The origin of **irrational numbers** took place in the **Pythagorean School** where, according to what is said, a student of Pythagoras named Hipaso discovered **irrational numbers** while working on a way to write the square root of 2 as a fraction and when he failed to do so, he put to this number the title of **irrational** .

In this school it was also called **immeasurable numbers** . Irrational numbers have certain **properties** that start from the aforementioned, from the fact that they are **infinite non-periodic decimal numbers** .

Thus we find that there are properties: **commutative, associative** , opposite element and; in the case of **multiplications,** these are **distributive** in relation to the operations addition and subtraction. Unlike the rational numbers, these irrational numbers do not have **a neutral element** and **are NOT closed** , let’s see examples:

**-√2 + (-****√2) = 0****and “0” is not irrational;****√2 x (****√2) = 2 and “2” is not irrational;**

Also, the sum of a **rational plus an irrational** is an **irrational** and; the multiplication of **an irrational by a rational** is also **irrational. Examples:**

**√3 + (****2) = 2 +****√3**an irrational**2x (****√5) = 2****√5**is irrational

**10 Examples of Irrational Numbers**

**π****(Pi)**(3.141592653589…), one of the best known and most used. It has application to compare the**diameter of a circle (D)**with the length of the**circle (L)**. This is:**L =****π x D.**- Euler number or e number (2.718281828459…)
- Golden Number (1.618033988749…).
- √2;
- √7 (2.6457513110645905905016157536393…)
- √99 (9.9498743710661995473447982100121 …);
- √122 (11.045361017187260774210913843344 …);
- √999 (31.606961258558216545204213985699…)
**E:**it is the Euler number and it is the curve that is observed in electronic tissues.**Golden:**this type of number is known as the golden ratio and is used to express in irrational numbers the proportion between the two parts of a line.**2:**41421356328.**201:**177**19:**3588989.