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?
Yes, the Golden ratio is an algebraic number (as is $\sqrt 2$), while $\pi$ and $e$ are transcendental. However, all of the numbers you've mentioned are irrational.