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Really nice answer to the question "in layman's terms, what exactly does it mean to be transcendental? How would a transcendental number be different from an ordinary number, say 5": https://math.stackexchange.com/a/1686299/18947 pic.twitter.com/IbOdO1Yklj
A 2nd order interesting thing is that the comments & other answers illustrate what a hard time people have coming up with explanations in layman's terms even when explicitly asked to come up with an explanation in layman's terms.pic.twitter.com/sViwL1eZph
oh, that is why exp^(i*pi) is so special ? 2 transcendental numbers that becomes algebraic thanks to a simple algebraic operation with an algebraic number ?
That's part of it. e^{I*pi} + 1 = 0 contains one addition, one multiplication and one exponential. It uses the unit of addition, the unit of multiplication, pi the fundamental unit of geometry, e the "unit" of differentiation and i for the fundamental theorem of arithmetic.
The 2nd & 3rd examples are not correct under the definition of the game: "...you can add, substract, multiply, or divide by any integer, except zero..." The word integer is not satisfied when you multiply by x and x is not an integer.
“You can also multiply by x“
So it's a value that can't be perfectly represented in the decimal system?
Not at all. The argument quoted has no dependence on the number system you use. Integers are integers in any number system, 0 is always 0 and x is always x. So, the argument applies in any number base.
I've also seen game semantics be used as a tool to explain (ε, δ)-definitions as a sort of game you play against a Limit Demon. https://math.mit.edu/~apm/ch03.pdf
If you don't know root 2 is the side (length) of a square whose field value is 2 you don't know difference between arithmetic & geometry.
@skdh @IndianInterest
i think there are little more than dozen popular transcendental numbers, eg, e, ln(2), hilberth, eulers g, etc
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