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What is the positive real number (say less than one) that is not a rational nor an irrational number?
I have encountered a mathematical problem that confused me about the definition of real numbers, so I thought there may be some other set of numbers that is not well defined !
I'll give you the standard, schematic answer, but be aware that this question opens a Pandora's box.
Irrational numbers are those that cannot be indicated by a fraction of two integers. Because it is a negative definition, it can be a bit misleading, and unhelpful. See, all numbers that are not rational have to be... irrational.
There is, however, an "outer circle", a set of numbers that aren't rational, but that can be positively defined as those that are the solution of some algebraic equation. These are called the algebraic numbers.
The next circle outside is the set of computable numbers. These are, roughly speaking, the numbers that can be approximated to an arbitrary precision by a finite calculation process.
Numbers that are not algebraic are called transcendental. Most of these numbers are also not computable. Another negative definition, as you can see. There are (and probably will always be) more things that we do not know about these numbers than things that we do know, so if you are perplexed by them, you are in good company.
If $x$ is positive rational number less than $\frac{1}{2}$, can the following logarithmic expression be equivalent to any real number , say $r$?
$$\frac{\log(1-x)}{\log x} = r$$, Where $r$ is positive real number less than one,
Assume first $r$ is a rational number say $n/m$, then you would get this equation, $x^n=(1-x)^m$, where solution for $x$ will be clearly algebraic - irrational number contradicting the assumption of rationality of $x$, hence $r$ cannot be rational number & must be irrational number, if so, you get this following equation, $x^r = 1-x$, the right hand side is a rational number, but the left hand side is transcendental number according to - Gel-schneider theorem, that contradicts again the assumption of irrationality of $r$, so what is that number suppose to be?
I guess someone will simply say it is a transcential number given by definition, if so assume $x$ an integer power of a rational number, say $x=a^p$, where $p$ is odd prime number, then by substitution you get this simple equation which is a reduced form of FLT for rational numbers , $a^p + (a^r)^p = 1$, but we already know that $a^r = b$, where $b$ is a rational number that doesn't exist from the proof of Andrew Wiles & Taylor to Fermat's Last Theorem, thus $log(b) / log(a)$ is not Transcendental number, it is a kind of numbers that have to be redefined again & precisely
