Approach to Irrational Numbers

Irrational numbers

Introduction

As in the case of rational numbers, this article begins with a visit to a dictionary. The word irrational in The American Heritage Dictionary gives (as adjective): 1.a. Not endowed with reason. b. Affected by loss of usual or normal mental clarity; incoherent, as from shock. c. Marked by a lack of accord with reason or sound judgement: an irrational dislike. ... 3.Mathematics. Of or relating to an irrational number.

There is also a definition for irrational number: Any real number that cannot be expressed as an integer or as a ratio between two integers.

Historically, the adjective irrational corresponds to the pythagorean reject to numbers different from integers or quotients (ratios) between them.

Geometric theorems  like the following could lead to think that relations between two lengths can always be expressed by means of a quotient between integers, This was a belief  for pythagoreans.

If D, E are mid points of sides of triangle ABC, then DE = (1/2)AB, or DE/AB = 1/2

If lines FL and HJ are medians for the triangle FGH, then HK = (2/3)HJ, or HK/HJ = 2/3.

However, a situation like the following contradicts such assumption: If polygon ABCD is a square, and BD is a diagonal then...

 

BD/AB = ?

The so called Pythagorean Theorem establish that  

Now, the question is if the square root of 2 is a rational number or not.

It will be proved, by contradiction, that the square root of 2 is not a rational number. 

Suppose that square root of 2 is rational. Then it can be expressed as the quotient of two integers , a, b. The fundamental theorem of arithmetic states that an integer can be expressed in terms of their prime factors, in a unique way. If the considered integer is prime, say p, it can be expressed as 1 * p.

In above equation the left side and the right side are expressions of the same number. In the left side there are a total of 2s +1 prime factors (since the prime factor 2 is included). In the right side there are a total of 2t prime factors. In this way it has obtained two different expressions for the same integer: one with an odd number of prime factors and the other with an even number of prime factors.

This contradiction with the fundamental theorem of arithmetic comes from the supposing about rationality of √2. The conclusion is that √2 is not a rational number. This result was catastrophic for the pythagorean brotherhood, but it opened a window to a new kind of numbers.

Above kind of proof of irrationality for square root of 2 can be used to demonstrate that if p is a prime number then square root of p is irrational..

Decimal representation of irrational numbers

Using an algorithm to obtain rational approximations to square root of two it can be obtained decimal expressions of this irrational number.

To obtain √2, an archaic method is

A calculator gives directly 1.41421356...

A computer program can do an impressive result like

1.41421356237309504880168872420969807856967187537694807317667973799073 ...

where it cannot ever found a repeating period. If it happened the number could be represented in quotient form a/b.

Some notable irrational numbers

As well as √2, there are other notable irrational numbers.

π = 3.141592653589793238462643383279502884197169399375... is associated with C/D, where C is the length of a circle and D is the length of its diameter.

e = 2.718281828459045235360287471352662497757247093700 ...is associated with the limit when n tends to infinity in

 For n = 1 the value is 2. For n = 2 the value is 9/4 = 2.25. For n = 3 the value is 64/ 27 =2.3703. Former values are rational, but as n ttakes greater values periodicity tends to disappear.

The so called golden ratio is an irrational number too: (1 + √5)/2 = 1.61803398...

Some operations with irrational and rational numbers

It is clear that addition, multiplication, subtraction and division  between two rational numbers give a rational number as a result. But, what happen when rational and irrational numbers are involved?

Let A be a rational number and B an irrational. If A + B = C then C has to be rational or irrational. If C is rational, then C - A = B and B would be rational which gives a contradiction. The conclusion is, C is irrational. Example: 4 + √3 is irrational.

What happen with A + B if both of them are irrationals? The result could be or not a rational. 

Example (1): A = 2/3 + √5, and B = 1/2 - √5. A + B = 7/6. 

Example (2): A = 1 + √3, and B = -1 -2√3, A + B = -√3 

In example (2) above it is accepted that 2√3 is irrational. This will be justified.

If M is rational and J is irrational then M * J  has to be irrational. If M * J = T and T were rational, then T/M = J would be rational, which is contradictory.

The product of two irrationals could be rational.

Example: (3 + √2) * (3 - √2) = 7

Construction of irrational numbers

Construction or design of irrational numbers represented in decimal non periodic form is possible if a rule is given that allows to know the next digit in any given place.

Example: In a decimal representation, after a given place there is: 2 and one 8, 2 and two 8s, 2 and 3 8s, ...:

3.428288288828888...

Clearly there are not period in that development and so, this is an irrational number located between 3.4 and 3.5 on the number line.

In the same way irrationals can be placed in points which does not correspond to any rational number. 

Those numbers are filling the points which are free in the rational number line, despite the fact that the set of rationals is dense everywhere.

Density of irrational numbers

The set of irrational numbers is dense in the same way as rationals. Between any two irrational numbers there is at least one irrational. For instance between two designed irrationals like

3.428288288828888... and 3,537377377737777,,,

1t can be found , for instance , 3.4320200200020000...

Real numbers

The theory on rational and irrational numbers constitutes a challenge for the intuition since the model of points on a line to represent them has advantages at the beginning but becomes far from intuition in grasping the concept of density.

The set of rational numbers together with the set of irrational numbers constitute the set of real numbers which fills completely the number line.