Are transcendental numbers a closed system?

Question

I need to know if the transcendental numbers closed under the operations of addition, subtraction, multiplication, and division. I also need to know why or why not. I also need to know if the imaginary numbers are closed under those operations. I also need to know where I can find the information on the internet.

Thank you.

Answer 1

Let's take each in turn.
Transcendental numbers: These are not closed
under any of the 4 operations.
Addition: Let n be any integer.
Then π + n - π = n.
Why is n - π transcendental?
If it were algebraic, n -(n - π), being the difference
of 2 algebraic numbers would be algebraic. So π
would also be algebraic, contradiction.
Subtraction: e-e = 0.
Multiplication: e*1/e = 1.
Division: π/π = 1.

Imaginary numbers are not closed under any
of the 4 operations either.
Addition: a+bi + a -bi = 2a is real.
Subtraction: bi-bi = 0 which is real
Multiplication: (a+bi)(a-bi) = a²+b² which is real.
Division: i/i = 1 which is real.
Look at Wikipedia for lots of fascinating information
about these kinds of numbers. There are still
many open questions about them.
For instance, we know that e^π is transcendental,
but we know nothing about π^e.

Answer 2

You lost me on the second word.`