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I tried to find what makes a number $\pi$ special.

$22/7$, $355/113 \approx\pi$, which is an irrational number.

Why is this constant is used for defining any cyclic function? Why is it that this constant (that we can calculate for a life time) is calculated by mathematicians and computers to their limits?

What makes this value irrational and non repeatable decimal places?

As I consider the world is, in itself, is a cyclic process. Does this infinite value really have significance when calculating the REAL scenarios.

(Edited After Knowledge Upgrade)

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Just adding to the answers by others below: 22/7 is also, by definition, not irrational. –  Hery Feb 22 '13 at 13:30
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Pi = Greek letter used to denote the ratio between the circle's circumference and its diameter. Pie = delicious cake. $\pi e\approx 8.53$. –  Asaf Karagila Feb 22 '13 at 13:31
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@AsafKaragila Pie is not cake. Pie>cake :) –  David Mitra Feb 22 '13 at 13:32
    
@David: So $\mathbf{cake}<8.5$? –  Asaf Karagila Feb 22 '13 at 13:36
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$\mathbf{cake \approx 299 792 458\times 3.14 \times 2.71 \times a}$ –  Parth Kohli Feb 22 '13 at 14:05
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4 Answers

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When I was young, I was taught that $\pi$ was $22/7$ and that its decimal expansion didn't repeat (which is equivalent to "it is irrational.") Since we had been taught long division, I tried dividing $22$ by $7$ and it definitely repeated. It led to what I still consider my first "proof" - I was able to convince a friend that any long division of whole numbers would always repeat (or terminate, which I later realized was the same as repeating $000000\dots$.)

Why is $\pi$ irrational? There are plenty of proofs of the fact, but, given the nature of your question, I'm not sure you are looking for proofs. You seem to want a deeper explanation, and I can't give you one, but I can tell you the following:

Almost all real numbers are irrational.

There is a technical sense in which this is true, but the important thing is that there are way more irrational numbers. Staggeringly more. It is probably too soon for you to understand this technical definition. I can recommend a popular treatment of this idea in a book by George Gamow called "One, Two, Three, ..., Infinity." Not sure if there is anything more recent.

Because of this startling fact, mathematicians are not surprised when a numeric problem results in an irrational number. Instead, mathematicians tend to be surprised when an expression that is not clearly a rational number turns out to be rational. When some complicated expression ends up rational, that is when mathematicians tend to ask "Why?"

So the fact that $\pi$ is irrational does not make it special. $\sqrt{2}$ is irrational, as is $\sqrt{3}$, $\sqrt[3]{7}$ and any root that is not an integer.

So why is $\pi$ important? At heart, it is probably because, in some deep way, circles are important. So there are lots of problems that don't appear to be about circles, but when we look deeper, we find that we can get an answer in terms of $\pi$. In some fundamental way, $\pi$ crops up a lot. There are some other constants like $\pi$ that crop up a lot, like $e\approx 2.71828$. $e$ is also irrational (and there is a startling relationship between $e$ and $\pi$ that is one of those "deep" facts about circles.)

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$\frac{22}7$ is not $\pi$. It is just a good approximation by a rational number.

In fact it is the best rational approximation you can get of $\pi$ with denominator below $100$.

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Too bad you edited out the pie comment. I kind of liked that sarcasm. :) –  Thomas E. Feb 22 '13 at 13:32
    
Then why commonly it is used for calculation when we need an answer in solid numbers and not like 10Pi. –  Aura Feb 22 '13 at 13:34
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@Aura: Because if I tell you to measure $10\pi$ meters you will only be able to measure a rational approximation of it. Roughly $31$ meters, or rather $31.4$ meters. You will never measure exactly $10\pi$, so for practical purposes (e.g. engineering) approximations are sufficient. –  Asaf Karagila Feb 22 '13 at 13:39
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It is false that $\frac{22}{7}=\pi$. Instead, $\frac{22}{7}$ is a rational number that is commonly used as an approximation to $\pi$.

You may find these other math.SE questions helpful:

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Ok...I got it then what really is Pi. –  Aura Feb 22 '13 at 13:32
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Did you read the Wikipedia article on $\pi$? If not, then do so; if so, then please explain what it is (specifically) that you do not understand. –  Zev Chonoles Feb 22 '13 at 13:34
    
Why only this number results as an answer for a ratio of the distance around a circle to the circle's diameter. –  Aura Feb 22 '13 at 13:45
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$$\pi = \dfrac{\rm Circumference}{\rm Diameter} = 3.14159265\cdots \approx \dfrac{22}{7}$$Ideally, $\pi$ is irrational. $22/7$ is only a rational approximation which makes it convenient for us to work with. We can, for example, approximate the area of a circle with a radius $49$ to be $484$ square units. This gives us a good idea of the most.

The question why $\pi$ is irrational can be thought of by the following approach: Take a thread and make a circle with it. Cut out another thread which is equal to the radius of your circle. Now challenge yourself to divide the circular thread in $N$ parts and take $X$ out of those $N$ parts such that all the $X$ parts make up a thread congruent to the thread which represents the radius.

You will not be able to do that (why?)

There are many proofs that happens.

The reason why $\pi$ is called so amazing is that it pops up everywhere in mathematics.

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I'm not sure that the string argument really lets you "resolve" that $\pi$ is irrational. –  Thomas Andrews Feb 22 '13 at 14:15
    
@ThomasAndrews it is not a proof; it's just an intuition. –  Parth Kohli Feb 22 '13 at 14:15
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Yeah, was objecting to the word "resolve" in your original post, which you have edited out. It gave the impression that the threads somehow would give you an intuition for why $\pi$ was irrational, rather than giving an intuition for what it means for $\pi$ to be irrational. –  Thomas Andrews Feb 22 '13 at 14:23
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