Why is pi an irrational number?

Answer 1

It is not just declared but was proven in the 18th century by the mathematician Johann Heinrich Lambert. Furthermore, it is not just irrational, but transcendental.

Irrational means that it cannot be written as a ratio of two integers, p/q (where q is not 0). Transcendental requires that it is not a solution to any algebraic equation. By way of contrast, sqrt(2) is irrational but it is not transcendental since it is the solution of x^2 - 2 = 0.

Answer 2

Numbers can be divied

up into Rational numbers and Irrational Numbers. A rational number will have an end point, for example, 3.14 has an end point of 4. On the other hand, the number pi is a forever continuing number. There is no end point, thus, it is considered an irrational number.

More precisely:
Rational numbers are those which can be expressed as a ratio of two real numbers such as 22/7 which is only a poor, inaccurate representation of pi since pi cannot be expressed as a ratio of integers. Since pi is itself a real number, pi can always be expressed as a ratio of real numbers - for example pi/1.

It has been proven that pi is irrational and, that definition is equivalent to the statement that pi cannot be expressed as a ratio. Any web site that claims that pi, or e etc can be expressed as rational numbers is using a non-standard

definition of rational.

But since the question is WHY is it irrational, it might be nice to offer an intuitive idea of that. A common way to prove that A, the area of a circle is (pi)r

2 is to arrange slices of the circle where we alternate the pointed part of the slice with the round part of the next slice. Sadly there is not way to give a picture of that here. As the slices get thinner, the shape becomes closer and closer to a rectangle with sides of r and c/2.Try to picture a rectangle with width r which comes from either slice of the circle on the ends and length c/2 because half the circumference is on top and half is on the bottom. We can substitute 2(Pi)r for c (since that is the definition of Pi). This gives us (2(Pi)r/2)x r which is Pi x r x r =

(Pi)r

2 . This helps us see why area, A=

(Pi)r

2. However, looking at the slice of the circle getting smaller and smaller helps us see there is always some error in estimating it as a rectangle. Even if the slice is very very thin, it is really not a rectangle. Fact is, we can never make it thin enough to be a perfect rectangle even if we keep making the slice smaller and smaller. Think of cutting the slices in half, then in half again and continue doing this FOREVER! This error in creating a rectangle helps us to see intuitively WHY Pi is irrational.
Have a look at the link to see a nice picture of this!

Answer 3

It was proven that if you assume that pi = a/b, with integers a and b, that you get some sort of contradiction. Therefore, it is irrational.There are several proofs that pi is irrational; for example, you can find several on YouTube.