Using Pi in Calculations and the Significance of the Decimals

Date: 12/18/2007 at 11:01:10
From: Lindsay
Subject: Trying to understand why Pi x 30 squared gives you 2826

30 squared times Pi versus 30 squared times 3.14

I don't understand why if you manually calculate 30 squared times 
3.14, you get 2826, but if you type in 30 squared times pi in your
calculator you get 2827.433388...etc.  I don't get it!

I can't understand why there is a whole number difference.  I'm sure 
it has something to do with the numbers after the decimal, but then,
wouldn't your answer continue to change every time you try to
calculate the answer using more of pi's numbers after the decimal? 
How can you ever get the question "right"?

Date: 12/18/2007 at 11:46:59
From: Doctor Ian
Subject: Re: Trying to understand why Pi x 30 squared gives you 2826

Hi Lindsay,

>30 squared times Pi versus 30 squared times 3.14

They're different, because 3.14 is only an approximation to pi.  That
is, pi and 3.14 are not the same values.  Pi is a little larger. 

>I don't understand why if you manually calculate 30 squared times
>3.14, you get 2826, but if you type in 30 squared times pi in your
>calculator you get 2827.433388...etc.  I don't get it!

Suppose there is something that sells for $1.98 at a local store, and
you want to figure out how much it would cost to buy 20 of them. 
You'd approximate $1.98 as $2.00, and multiply by that to get 

  20 * 2.00 = 40.00

which would be a little higher than what you get if you do the actual
multiplication,

  20 * 1.98 = 39.60

right?  

It's the same kind of thing with pi and 3.14.  When you use the 
latter, you're using a number that's a little different than the 
actual value, so you get an answer that's a little different than the
actual answer. 

Note that when you hit the Pi button on your calculator, you don't get
3.14, but rather something like 3.14159265358979, which is itself just
an approximation, although a somewhat more precise one.  

>I can't understand why there is a whole number difference.  I'm sure
>it has something to do with the numbers after the decimal, but then,
>wouldn't your answer continue to change every time you try to
>calculate the answer using more of pi's numbers after the decimal? 
>How can you ever get the question "right"?

You never can, except to just write 'pi' for the value of pi.  It's
the same sort of situation you're in when you find that something is,
say, 12 times the square root of 2.  You can approximate the square
root of 2 as 1.4, or 1.414, or 1.41421356, and so on; but it's always
just an approximation.  

An easy way to see this is to multiply 1.414 by itself.  When you do
that, you get 1.9881, which is a little less than 2, because 1.414 is
a little less than the actual square root.  On the other hand, if you
do this, 

  sqrt(2) * sqrt(2) = 4

you get the exact answer... but at the cost of having to keep the
square root around.  It's just a trade-off that we have to make with
some numbers, like pi and e and the square roots of non-squares. 

Here's another way to think about it.  We can represent pi by 

  pi = 3.14159265358979...

     = 3.14 + 0.00159265358979...

right?  Now, suppose you multiply that by 30^2:

    30^2 * pi 

  = 30^2 * (3.14 + 0.00159265358979...)

  = (30^2 * 3.14) + (30^2 * 0.00159265358979...)

That second part is the error that you introduce by using 3.14 instead
of pi, right?  How much does that work out to?  

  30^2 * 0.00159265358979... = 1.43338823082...

That is, you're taking a pretty small error (about 1.5 over 1000), but
multiplying it by a big number (900, or nearly 1000), which means you
should get an error of about 1.5... which is just what happens. 

Does that make sense?  Let me know if you need more help.

- Doctor Ian, The Math Forum
  http://mathforum.org/dr.math/ 

Date: 12/18/2007 at 11:52:40
From: Lindsay
Subject: Thank you (Trying to understand why Pi x 30 squared gives you
2826)

THANK YOU THANK YOU THANK YOU!!!!  That makes total sense!  Silly
decimals!!! 

Cheers,
Linz