Does Pi Equal 3.14?

Date: 01/27/98 at 15:39:39
From: Adam Glover
Subject: We use pi as 3.14. How can this be right?

In my math class pi has always been 3.14. We use this to find area, 
volume, and any other equations using pi. If pi has an infinite number 
of digits, how can 3.14 be exact? I am very concerned about our 
society using 3.14 as pi when it has an infinite number of digits. 
Anyway, I was just wondering about this.

Adam Glover

Date: 01/27/98 at 16:08:32
From: Doctor Rob
Subject: Re: We use pi as 3.14. How can this be right?

You only need to use a value of Pi with as many significant figures as 
the other measurements you used for other values in the same equation.  

For example, if you wanted to measure the area of a circle, and the 
radius was measured as 36.2 feet (to three significant figures), the 
actual radius might be anywhere in the interval 36.15 < r < 36.25.  
Then the area would be Pi*1306.8225 < A < Pi*1314.0625. Even if we 
used 15 decimal digits for Pi, all we would know about the area is 
that

   4105.50396554585 < A < 4128.24909635784.

If we use just 3.14 for Pi, we get 

   4103.42265 < A < 4126.15625.

In either case, we can't tell what the fourth significant digit of A 
is, and the third digit is probably a 1, but maybe a 0, 2, or 3. The 
extra digits of Pi don't help unless you make more accurate 
measurements.  

The rule of thumb is to use the same accuracy for your constants as 
for your least accurate measurement, and the answer will have the same 
accuracy. In this example, 3.1400000*36.200000^2 = 4114.7816 
(exactly), but if we round off to three significant figures, we get 
3.14*36.2^2 = 4110 as the best thing we can write for the area.

Don't worry about society - engineers and mathematicians are well 
aware of this situation! 

If you want to use more decimals, you can do so, but your answers will 
still only have the accuracy of the least accurate quantity in the 
formula. We usually used 22/7 when I was in school, which is about the 
same accuracy as 3.14. If you want a better value, you can use 3.1416, 
or 3.14159, or 355/113, or 3.14159265, depending on the accuracy of 
your inputs.

If the measurements are ideal, and hence exact to infinitely many 
decimal places, I would leave the answer in the form 100*Pi 
(if r = 10, say).  The context of the problem should make it clear 
whether the quantities are exact or only approximate.

-Doctor Rob,  The Math Forum
 Check out our web site!  http://mathforum.org/dr.math/