I have been studying the concept of transcedental numbers. Till now, I had taken it for granted that all important numbers like pi and e were transcedental. I have no reason for assuming this or for clustering them together.
It's just my intuition had placed numbers like pi, e and the golden ratio together and for some reason assumed they are all transcedental. This was before I was aware of a rigorous definition of a transcedental number.
I just remembered that the golden ratio is one less than it's square. So, it does satisfy an algebraic equation. Does this mean that the golden ratio is not a transcedental number?
Best Answer
You may find the wikipedia article "Algebraic Number" for useful and hopefully approachable understanding on how to classify (real) numbers into various classes. Here is a brief summary of the class definitions.
A natural number is any positive number that does not have a fractional part.
An integer is any negative or positive number, or zero that does not have a fractional part.
A rational number is any number that can be expressed as $a/b$ where $a,b$ are integers.
An irrational number is any number that is not rational.
An algebraic number is any number that can be expressed as the root of any proper single variable polynomial whose coefficients are all rational numbers.
Thus, the golden ratio $\phi$, is an algebraic number, as it the root of the equation $x^2-x-1=0$.
Although it is not obvious, all values of trigonometric functions whose angle is a rational multiple of $180^{\circ}$ (that is, $\pi$ radians) are algebraic numbers.
A transcendental number is a number that is not algebraic.
It is often extremely complex to determine if a number is algebraic or transcendental, and so there are many important numbers where it is currently not known if they are algebraic or transcendental. Often these numbers look seemingly very simple. For example, it is not known if Apery's constant is algebraic or not.
$$ \zeta(3) = \sum_{i=0}^{\infty} \frac{1}{i^3} $$