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…).
- √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.