# how is pi an irrational number when it can be expressed as 22/7.....

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sunnyraghvendraposted 7 years ago

how is pi an irrational number when it can be expressed as 22/7.....
22/7 is a rational number since it can be written in the form p/q.....
any thought processes are welcome......

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(Q)posted 7 years ago in reply to this

Pi is an irrational number simply because it does not have an exact decimal equivalent. It can only be derived to the infinite.

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seanorjohnposted 7 years ago in reply to this

22/7 is a rational number but pi is not because 22/7 is only a rough or approximate value of pi

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lovelylizzyposted 6 years ago in reply to this

i.d.k.,isn't pi 3.14?at least that's what my math teacher told me. some say pi has no end.serously.why i replyed to this?your question is complicated for the average 12 year old mind....

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RychardeManneposted 7 years ago

22/7 does not equal pi, it is merely an approximation. It dates back to Archimedes as the lower-bound value, the upper-bound being 223/71. Other popular ancient approx values of pi include square-root of 10 and 25/8. It is not clear how these two were derived.

Note that sqrt(10) is also irrational like pi, but pi is also transcendental, meaning that there is no polynomial equation with natural number coefficients of which pi is a solution.

Hope that helps.

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sneakorocksolidposted 7 years ago

A math question! Are you serious? Alright I'll need to take my socks off first.

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Obscurely Diverseposted 7 years ago in reply to this

Did you at least get through pre-algebra?

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zampanoposted 7 years ago

22/7 is not a number.
It's the fractional expression of a number.
And it does not express pi.
pi = Perimeter / Diameter for any circle

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Obscurely Diverseposted 7 years ago in reply to this

They are both infinite decimals, so in that sense, the poor guy has a point.
Divide 22 by 7 and tell me when it ends......

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Obscurely Diverseposted 7 years ago in reply to this

But either way, 22/7 is not pi...  This is so silly, ha-ha!

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Ron Montgomeryposted 7 years ago

mmmmmmmmm, pi

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Obscurely Diverseposted 7 years ago in reply to this

Hungry?

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Ron Montgomeryposted 7 years ago in reply to this

Yes, I'd like 22/7 of a pi.

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Obscurely Diverseposted 7 years ago in reply to this

I usually just eat half.  BUT, since you're so nice, bake us all up some damn pies and we'll split it like savage beasts...  Ha-ha!

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VillagePeopleposted 7 years ago

22/7 is a recurring decimal and therefore not equal to pi, which does not have any such recurring pattern.

If pi did recur in this way, then it would be rational as all recurring decimals can be easily shown to be rational.

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Non-offensiveUserposted 7 years ago

I think Beavis and Butthead hit this one on the head when they said,"I'm angry at numbers, let's go look at some boobs and butts..."

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thewinnerishereposted 6 years ago

I agree with RychardeManne. it is just an approximation.
it is circumference/diameter.
π is an irrational number, which means that it cannot be expressed as a simple fraction such as 1/5 or 3/4. However, you can get pretty close.

A fraction that is often used is 22/7. This is not all that good:

22/7     =    3.14285714...
π     =    3.14159265...

Although within 0.04% of the correct answer, 22/7 is only correct to 2 decimal places. We can do better than this.

355/113 is correct to 6 decimal places. It's within 0.000009% of π.

355/113     =    3.1415929203...
π     =    3.1415926535...

355/113 is such a good approximation to π, that there is not a more accurate fraction until 52163 / 16604, and that is only marginally closer to π, still only correct to 6 decimal places.

52163/16604     =     3.1415923874...
355/113     =    3.1415929203...
π     =    3.1415926535...

To be accurate to 7 decimal places we need to go as far as 86953 / 27678

86953/27678     =     3.1415926006...
52163/16604     =     3.1415923874...
355/113     =    3.1415929203...
π     =    3.1415926535...

The importance of 355/113 has been recognized, giving it the name Milü.

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kathryn1000posted 6 years ago in reply to this

I am glad to see you all taking such an interest in Pi.....send me a slice!
but those numbers are really interesting.

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jhyun95posted 6 years ago

If you really want to go into depth on how to express pi as a bunch of fractions, you should look into the power series for arctan (or other arc trig functions, but arctan seems the most convenient)

In deriving the power series. Note that,
d/dx(arctan(x)) = 1 / (1+x^2) = 1 / (1-(-x^2)), which is the sum of a power series with first term 1, multiplier -x^2. So,

d/dx(arctan(x)) = 1 - x^2 + x^4 - x^6 ...
∫(d/dx(arctan(x)))dx = ∫(1 - x^2 + x^4 - x^6 ...)dx
arctan(x) = x - x^3/3 + x^5/5 - x^7/7 ...
Since arctan(1) = pi/4, subbing that in gets
pi/4 = 1 - 1/3 + 1/5 - 1/7 ...
pi = 4(1 - 1/3 + 1/5 - 1/7 ... )

Sadly, even though this is "exact", it takes far too many terms to get anything close to pi (the last term added is about how "off" this series will be when not added infinitely). So you would need well over a hundred terms to even get 3.14...

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sukipanesarUKposted 6 years ago

Yes I agree with thewinnerishere a RychardeMannend 22/7 is just an appromination for pi. If you really want to prove that pi is irrational you can prove it by contradiction i.e. you assume it is rational and through a logical process you will arrive at a contradiction meaning that your initial assumption must be false. See book Theory of Numbers (Hardy and Wright) or Calculus (Spivak). Alternatively you can use knowledge from fields and Galois theory.

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brianzenposted 6 years ago

I suppose it is neither rational nor irrational, only we can call it such when it has presented it's self to us. (what were we discussing?)

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