To say that “something exists because otherwise it does not” is a statement of pure irrationality, or, my idea of existentialism.
To many students of mathematics, the statement “An irrational number is any number that is not rational; it cannot be represented as a quotient of two integers.” is a message of pure irrational logic meant to confuse the best of us into believing that mathematics and mathematical statements are best rendered to logicians and not independent thinkers.
I’m sorry but that I cannot and will not accept!
In all of my scholarly readings, I have yet to come across a definitive statement of exactly what is an irrational number. Only what it is not. In fact, what obtains is a lesson in contradiction…an approved method of proof.
Proof by Contradiction is a standard logical and mathematical argument. Assume something is false as a premise. The collection of logical statements that follow then leads to a natural fallacy or “contradiction”. Hence the original premise must be false. And, “voila!” you have then “proved” something to be true…”not false equals true”.
A classic example used in many text books is the proof that √2 is irrational. I’ve even seen a publication by Ed Sandifer (in his “How Euler did it” series) establishing Euler’s proof that the constant e is irrational by use of “continued fractions” (not being infinite).
Then again, I can’t expect anything less. After all it’s all about filling in the blanks.
The basis for any theory of numbers began with the need to count objects. Hence the counting numbers, aka integers starting from 1. Then came the need to depict fractional portions of something, but simple fractions of 1. Along came the representation of nothing, aka zero.
As to when came the need to represent the unknown? That’s the basis for finding X…aka the birth of algebra and solving equations. Linear then quadratic then cubic…it never ends. Following this same path…there must be room in the natural scheme of things to include solutions involving negative quantities such as x^2 + 1 = 0.
The birth of ‘imaginary’ (as opposed to ‘real’) numbers such as √-1…later represented by the letter “i”, and the theory of complex numbers came into being.
So now, in addition to ‘irrational’ numbers we now have ‘complex’ ones. And, to think that these ideas come into being not overnight but over centuries of mathematical thought. Because for anything to make sense, everything must ‘fit’ into a comprehensive and all-embracing ‘theory’. The integers are a subset of the rationals which are a subset of the reals which are a subset of complex numbers (since a complex number is defined to consist of both real and imaginary parts.
So you see. Things are put together to “fill a need”, aka to satisfy an equation, aka to explain a result.
Now, I’m not being facetious here but just exploring my own journey along the way to getting at the truth.
I recently followed a course at “complexityexplorer.org” called Dynamical Systems and Chaos. An excellent view of order and disorder existing together as one. This introductory course looked at two types of dynamical systems…iterated functions and differential equations; one dimensional Logistic Equation; the two dimensional Henon Map and the three dimensional Lorenz Equations were highlighted in the lecture series. The lecturer, David Feldman, was able to show, via the use of computer software using numerical algorithms , how these systems exhibit strange and unpredictable long-term behavior. Sensitive dependence on initial conditions, the “butterfly effect”, attracting cycles , pattern formations, with both periodic and aperiodic orbits. It is both interesting and amazing that to this day “Euler’s Method” is still being used as the algorithmic basis for finding those numerical solutions. Score yet another one for Leonhard Euler!
Irrational numbers exhibit strange behavior. Their decimal representation is “aperiodic”. Is there any connection with dynamical systems and chaos? I wonder!
In my simplistic and maybe naive view, I do not think it is appropriate to teach number theory with the real numbers as existing solely on a one dimensional “line”. This leads to a conceptual fallacy that narrows the mind into thinking only in one dimensional terms. If we extend the view to two dimensions we can then incorporate the “natural” fit in which complex numbers reside…the complex plane. And yet, dynamical systems in the “complexity” realm introduces “strange and aperiodic” behavior in the 3D space.
So do we then consider the possibility that all numbers should exist and be represented in 3-dimensional space?