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### Why Pi?

```Date: 07/19/2002 at 04:51:41
From: Maxx
Subject: Pi ratio

Hi, I'm Maxx and I'll be taking Analysis 2 for math when I start
10th grade next year. I know that if a diameter of a circle is an
integer then then the circumference cannot be an integer and vice
versa, and that is why although pi is a ratio, it is an irrational.
But then, why don't we just use the ratio (for example 2.4/4 - not
really pi, just an example) when we calculate the area of a circle
to get more accurate answers? Is it impossible to accurately measure
circumferences?
```

```
Date: 07/19/2002 at 13:35:35
From: Doctor Ian
Subject: Re: Pi ratio

Hi Maxx,

A subtle point, which a lot of people fail to appreciate, is that
mathematical shapes are _idealizations_ of real shapes. Or to put that
another way, real shapes are only _approximations_ to mathematical
shapes. You can look at it either way (although philosophers would
probably make a big deal out of which way you choose to phrase it).

For example, you can't construct a perfect circle in the real world,
if for no other reason than that a perfect circle contains an infinite
number of points, and a real object contains a finite number of
pieces.

The formula

circumference    area
pi = ------------- = --------

holds exactly _only_ for mathematical circles, not for real objects.
If you try to apply it to real object, you run into several problems,
which you're probably already aware of.  One is that you can't make
exact measurements! Another is that if the object isn't a perfect
circle, the formula doesn't exactly describe the shape of the object.

A third problem is that in many cases, you can't really measure the
quantities you want to measure, because they don't really exist. For
example, in a mathematical circle, every point on the circle is the
same distance from the radius. That is, there is just the one radius,
and it's the same everywhere.

But now suppose you have an object that is approximately circular, and
you want to find the 'radius'.  Where should you measure it? Depending
on how precisely you measure things, you'll get a slightly different
value everywhere you look! So what _is_ the 'radius' of the object?

(It might seem at first that 'circumference' would be immune to this
problem, but it's not. Think about measuring the circumference of an
island. If you use a yardstick, and measure points a yard apart, you
get one value. If you use a foot-long ruler, you get another value.
If you use an inch-long ruler, you get a third value. And so on. By
determining the level of detail that you're willing to ignore, the
length of your ruler determines the circumference of the object!)

It's to get past problems like these that we agree to treat objects as
if they corresponded to exact shapes like circles and squares. Then we
can forget about the objects and just deal with the shapes to do our
calculations. We give up a little accuracy this way, but we more than
make up for it in convenience.

Does this answer your question?  Write back if you'd like to talk

- Doctor Ian, The Math Forum
http://mathforum.org/dr.math/
```
Associated Topics:
High School Conic Sections/Circles
Middle School Conic Sections/Circles
Middle School Measurement

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