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Can the sum of a finite series equal $\pi$? I'm assuming of course that no element in the series is some fraction of $\pi$. I'm wondering since all methods I've seen of calculating $\pi$ involve infinite series or infinte products or some limit as an index goes to infinity.

From Wikipedia I find that "π cannot be expressed using any finite combination of rational numbers and square roots or n-th roots". But what about a finite combination of other irrational numbers?

Jens
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    $\left(\pi - 1\right) + (1) = \pi$. – Christopher Carl Heckman May 07 '16 at 00:13
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    See my assumption in the question. – Jens May 07 '16 at 00:14
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    @Jens neither $\pi - 1$ not $1$ are fractions of $\pi$. – AJY May 07 '16 at 00:15
  • I see three elements, one of which is $\pi$. Anyway, the intent of the question is that no element contains $\pi$ in any way. – Jens May 07 '16 at 00:16
  • @MathematicsStudent1122 The irrationality of $\pi + e$ is an open problem: http://math.stackexchange.com/questions/159350/why-is-it-hard-to-prove-whether-pie-is-an-irrational-number – Travis Willse May 07 '16 at 00:18
  • @Travis Okay, thanks for pointing that out – MathematicsStudent1122 May 07 '16 at 00:19
  • I'm sure the answer to the question is yes, but it is difficult to find a good choice of two numbers to express this since most irrational numbers are difficult to express (for instance, the set of all irrationals is uncountable, but the algebraic numbers, including as many variations on roots as one could ever think of, is only countable). – Chill2Macht May 07 '16 at 00:42

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Taking into account what you've stated in the comments (you don't want to involve $\pi$ in any way) I would say that the answer is no, but it is no for non-mathematical reasons, and this question kind of leaves the realm of the mathematical and becomes one about our practices when it comes to expressing numbers. By asking for a number that "does not involve $\pi$ in its expression", I don't really think that is a meaningful property of a number. I think that is a matter of how the number would be presented in practice.

Let's say we have a finite set $P$ of real numbers, whose sum is $\pi$. Since the algebraic numbers are closed under addition, we can say that there is definitely a transcendental number in $P$, let's call it $p$. Without loss of generality, we can say that there is exactly one of these transcendental numbers as follows: if there are multiple instances of transcendental numbers, add them all together and consider that to be the one transcendental number of the set. We know that the result of adding all of the transcendental numbers in $P$ will also be transcendental, because if it were not, we would have a set of algebraic numbers whose sum is $\pi$, which is a contradiction.

Alright, so we have this transcendental number $p$. How would you express it, present it, or define it? Well, it seems only natural that anyone would do so as $\pi - a$, where $a$ is the sum of all algebraic numbers in $P$.

Do you see the issue with the question as posed? You're asking for a set of numbers that do a thing, with the caveat that none of the numbers are written in a certain way, but that certain way is really the only way we'd ever write these numbers. In other words, $\pi + 22.1$ is the best way to write $\pi + 22.1$, and asking for it to be written in a way that doesn't involve $\pi$ is kind of weird, and not really about the number itself.

Jonathan Hebert
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