We know that pie is a irrational number but still we can write pie=22/7 ?

Question

if we are able to write pie=22/7,which is a rational number then how can we say that pie is irrational?

Answer 1

22/7 is approximately pi, but is not equal to pi

numerical calculations must use approximations of π. For many purposes, 3.14 or 22/7 is close enough, although engineers often use 3.1416 (5 significant figures) or 3.14159 (6 significant figures) for more accuracy. The approximations 22/7 and 355/113, with 3 and 7 significant figures respectively, are obtained from the simple continued fraction expansion of π. The approximation 355/113 (3.1415929…) is the best one that may be expressed with a three-digit numerator and denominator.

Answer 2

If you know some trigonometry, we can prove that pi is less than 22/7. Take a circle C of diameter 1. If P is a regular n-gon circumscribed about C, then each side of P has length tangent(pi/n). Therefore the perimeter of P is n*tan(pi/n). Since it is circumscribed about C, the perimeter of P is greater than the perimeter of C, which is pi. So pi is less than n*tan(pi/n). This is true for all integers n=3 or greater. If you put in n=91 you get that pi is less than 3.142841..., which is less than 22/7=3.142857...

In particular pi is not 22/7.

Edit: To volter. This is a good point, that I didn't prove why the circumscribed perimeter must be greater. It really gets down to how one defines the length of a curve; Archimedes addressed it in the following way. First we define the notion of a curve being "concave" in a certain direction, in that secant lines lie on the same side of it. Now suppose that we have two curves with common endpoints which are both concave in the same direction. Then both are on the same side of the straight line joining the points. Now a line is the shortest distance between two points, so intuitively the curve closer to the line must have shorter length than the other one. The closer one is "closer to being a line". In our example, we take as endpoints the n points where the polygon intersects the circle, and compare the arcs to the parts of the polygon. These are concave in the same direction and the polygon part is on the other side from the secant line.

An area argument as you suggest would work but it adds a level of complexity (what is area? why is the area of the circle pi r squared?) contrary to my taste for this problem.

Answer 3

It's pi anyway and we approximate it to 22/7 for simple work, most lower grade math problems are designed to cancel off perfectly if we approximate pi to 22/7, but this should be heavily frowned upon. Infact there are a bunch of fractional approximations closer to pi than 22/7 is. In advanced work we denote pi with just the symbol.
Pi is actually smaller than 22/7. Geometrical proofs which require both values to be evaluated are many, but the most elegant one's are based on calculus. In that a positive definite integral is considered that would contain an inverse trig function. Check the references for a link to a proof on wikipedia.

Hope this helps!

Answer 4

22/7 is an approximation of pi (and not a very good one), but is not equal to pi. Pi was, is, and always will be an irrational number, no matter how rational the approximation one uses for it.

Answer 5

pie is delicious, PI is irrational. 22/7 is what people thought is equalled a very long time ago, but it is still very close to it can be used

22/7 = 3.1428

pi = 3.1415...and a bunch more digits

But, I do beleive it should be a law that pie at all restaurants should cost $3.14.

Answer 6

22/7 is an approximation of the irrational number pi, in the same way that 3.14 is an approximation.

Answer 7

Because 22/7 is only a bad approximation.

Answer 8

22/7 is not equal to pi. But it is close to it up to couple of digits so that it is enough to use it as a reasonable approximation to pi.

Answer 9

its 'pi'...and it is an irrational #. 22/7 ends at 3.142857143. its an estimatin of pi. the actual decimal of pi goes on forever..its a neverending number.

Answer 10

pi is not exactly 22/7

pi is the ratio of circumference of a circle and its diameter. It is just a characteristic of a circle.