# Infinity times zero equals pi

## Main Question or Discussion Point

I'm 16 and in high school. I've never taken physics or anything before.

I have this equation that shows that pi = infinity times zero

Ok.

So if u take a circle with a diameter of 1 and draw equal spaced radius' and connect them you get a polygon with all equal sides.

the more radius' you draw the more sides the polygon has and the closer the perimeter gets to pi

the formula for the perimeters of these polygons is:

n(-.5(cos(360/n)-1))^.5

where n is the number of sides of the polygon

as n gets larger and larger in this equation it approaches pi. Like if you make n = 1000 you get the first 5 digits of pi correct.

we can say then that:

infinity(-.5(cos(360/infinity)-1))^.5 = pi

360/infinty = 0 and cos(0) = 1

so this simplifies to...

infinity(-.5(1-1))^.5 = infinity(-.5(0)^.5) = infinity(0)

so therefore

infinity(0) = pi

my calculater says infinity(0) is undefined so i don't get it.

How am i wrong?

cepheid
Staff Emeritus
Gold Member
Your calculator is right. Infinity multiplied by zero is NOT defined. Infinity is not a number. You can't just plug it into the equation. If you took the LIMIT of the equation as n approached infinity (which is a statement that should be taken to mean, "as n increases without bound*"), then perhaps you might be able to show that the result equalled pi. Do you know what limits are? If not, you will learn about them in your first calculus course.

EDIT: to summarize: if you carry out meaningless operations, you will get meaningless results.

EDIT 2:

* For clarity, this does not mean that n will eventually reach some destination (final value). What it means is that you can make the expression *arbitrarily* close to pi by making n arbitrarily large. Here, "arbitrarily close to pi" should be taken to mean "as close to pi as you like." This is what we mean when we say that the expression has a "value", in the "limit as n goes to infinity."

Last edited:
There are lots of things wrong with your calculation. You need to remember that infinity is not a number, and anything times infinity is not well defined. Infinity only exists in limits, eg you can take the limit of x to infinity.

It is true that your formula converges to pi in the limit that n goes to infinity. However, taking a limit is not the same as simply substituting "n = infinity".

360/infinity is NOT zero. 360/n is zero in the limit that n goes to infinity. There's a difference!

ok thank you.

so you cant use infinity like a number that makes sense i guess. Haven't learned about limits, but yea i think i get the idea...

diazona
Homework Helper
Well, I guess you just discovered limits for yourself ;-) Time to feel smart! Seriously though, when you do learn about limits in class, some of what you hear will sound pretty familiar.

HallsofIvy
Homework Helper
One reason why we don't just say 'infinity times 0 equals..." is that it is possible to find other functions f(x)*g(x) where f(x) goes to infinity, g(x) goes to 0, but f(x)*g(x) goes to some other number. By choosing f and g carefully, we can make the limit of that product anything.

What is wrong with this calculation for the value of e?

e = Limn->infinity(1 + 1/n)n

so e = (1+ 1/infinity)infinity = 1infinity

Hurkyl
Staff Emeritus
Gold Member
What is wrong with this calculation for the value of e?

e = Limn->infinity(1 + 1/n)n

so e = (1+ 1/infinity)infinity = 1infinity
Because the conclusion of the limit theorems only apply when you don't have an indeterminate form.

Please note, as I've said before, that
$$\displaystyle \lim_{n \rightarrow \infty} x^n$$

is NOT the same as

$$x^{\infty}$$