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I was asked to find the truth value of the statement:

$$ \pi + e \; \text{ is rational or } \pi - e\; \text{ is rational } $$

I am only allowed to use the fact that $\pi, e $ are irrational numbers and cannot use the theory of transcendental numbers.

Cannot proceed. any help would be appreciated.

SBF
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Ishfaaq
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    If you succeed, you'll be famous. Related http://math.stackexchange.com/questions/1013533/is-it-known-if-pi-e-is-transcendental-over-the-rational-numbers – Simon S Dec 04 '14 at 14:18
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    @SimonS: Guess I have a beef to pick with my teacher. – Ishfaaq Dec 04 '14 at 14:24
  • If you only know that two numbers $a$ and $b$ are irrational, and you are asked whether at least one of $a-b$ and $a+b$ is rational, the answer may be true or false depending on $a$ and $b$. Did you mean that you are actually allowed to use definitions of $\pi$ and $\mathrm e$, or only the fact that they are some irrational numbers? – SBF Dec 04 '14 at 14:34
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    I seems to me that the OR should be AND in the OP and title. The truth value is False as noted by Martin Brandenburg in his answer. – Fred Daniel Kline Dec 04 '14 at 14:39
  • @FredKline: Perhaps the exercise was given as stated and the author of that exercise confused "or" with "and"? – Martin Brandenburg Dec 04 '14 at 14:45
  • @MartinBrandenburg: It's from a past paper of an Intro to Analysis course. The exercise is as is. But I'll verify if there was a mistake in a day's time. Thanks for the answer. – Ishfaaq Dec 04 '14 at 14:52

2 Answers2

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This seems to be an open problem. It is a conjecture that the statement is false, i.e. that $\pi + e$ and $\pi - e$ are irrational. According to Wikipedia this remains unproven. (Just imagine the impact of the discovery of an equation such as $\pi=e+\frac{4233108252.........3123782}{31238295213.......0591231}$ ... unbelievable!)

Remark that at least one of those numbers is irrational, even transcendental (but this doesn't prove that both are irrational!). For if both would be algebraic, then their sum would be algebraic, which is $2 \pi$, a contradiction. Notice that this argument is not constructive at all, and again that it does not decide if "$\pi+e$ is rational or $\pi-e$ is rational" is false or not, it only proves that the stronger statement "$\pi+e$ is rational and $\pi-e$ is rational" is false.

Martin Brandenburg
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$2\sqrt2$ and $\sqrt2$ are two distinct irrational numbers s.t. the statement: '$2\sqrt2+\sqrt2$ is rational or $2\sqrt2-\sqrt2$ is rational' is not true.

$\sqrt2$ and $2-\sqrt2$ are two distinct irrational numbers s.t. the statement: '$\sqrt2+(2-\sqrt2)$ is rational or $\sqrt2-(2-\sqrt2)$ is rational' is true.

This illustrates that the statement cannot be proved purely based on the fact that $\pi$ and $e$ are irrational.

drhab
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