Number Theory Level 4

\(\pi\) and \(e\) are both transcendental numbers. Transcendental numbers are numbers which can't be roots of a non-zero polynomial equation with rational coefficients. Keeping that in mind, read the following statements.

\([1]\). It is possible for \(\pi+e\) to be a rational number in base \(10\).

\([2]\). At least one of \(\pi+e\) and \(\pi \times e\) is irrational in base \(10\).

\([3]\). \(\pi\) is an irrational number in base \(10\).

Which of these statements are correct?

Note: This problem is a part of the set "I Don't Have a Good Name For This Yet". See the rest of the problems here. And when I say I don't have a good name for this yet, I mean it. If you like problems like these and have a cool name for this set, feel free to comment here.


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