Continuum

Real numbers


In my local chinese restaurant I found a book in calculus*. It was quite extensive, having 999 pages of text. I was just able to start reading it during my lunch. The first issue was defining the real numbers. This affected my digestion negatively. The definition was incomplete, and there is no need to define the concept. Thousands of years of mathematics have yielded lots of achievements without that definition. Calculus does not aquire a definition eighter. Numbers are basic pieces of arithmetics. They can be manipulated according to the rules. I think, not even Newton know the definition of real numbers. Checking some encyclopediae on the web I found incorrect or incomplete definitions of the subject. What metrology concerns: we never use real numbers. We use rational numbers, that are easy to define. If one measures the circumference of a circle, one will never get "pi" as a result. It is only in pure mathematics this mysterious constant exists. Written as a decimal fraction, no integer is a function of the preceeding ones. That kind of numbers are called "transcendental". They are many, but very hard to find.

• The natural numbers are 1, and its followers ad infinitem
• The rational numbers are x; a*x=b
• The algebraic numbers are x; P(x)=0, where P(x) is a polynom
• The non-algebraic number are the rest. Pi is one of them

Let's calculate!

x^3=-1 **

One root is real and two are "imaginary". Imaginary numbers have no order. You can't tell which is bigger, i or -i. Real numbers are ordered. Given two different numbers, one must be the least. As you can see, real numbers is not the fundament of arithmetics. They are just a part. The fundament is natural numbers. They are used by your computer and all other instruments. "1" can be interpreted as one gallon or one apple or one nanogramme and so forth. That is all that is needed for practical reckoning.

The continuum is "all numbers". We can not decide how big it is. This was proven in 1964. The proof was just a complement to Gödel's theorem, and nothing unexpected. Math is an open theory, that may let certain questions stay unanswered.

___________________

* Adams Robert A. Calculus A complete course, Toronto 2003 , Pearson education, ISBN 0-201-79131-5 "a real number is a number that can be represented by a decimal fraction" That's bullshit rather than a definition. I will demonstrate in a further blog, that definition of real numbers is not necessary for accomplishing calculus.

** It is possible your computer fails to do this operation. This is due to a "bug" in the system. You can certainly work it out in your brain. (r3/6; +-0,5) are what my head gave me as imaginary roots. I can have miscalculated. If so, please tell me! Yes, I was wrong.