Leaving Answers in Terms of Pi

Date: 12/11/2006 at 14:38:07
From: Alex
Subject: Why not solve for pi?

I am a 12-year-old homeschooler and I cannot understand why it is not
important to always solve an equation that contains pi to the full number.

For example, the area of a circular cylinder is to be found.  The 
correct answer is 1884 square meters, not 600pi square meters.  I
ALWAYS use pi immediately in the first step, calculate, and take the
larger number all the way through the entire problem when my 
teacher insists that it isn't necessary.  She says that it is ok to
write it as 600pi square meters, but that isn't finishing the problem!  

SA = 2(area of a base) + (lateral surface area)
SA = 2(144pi) + [(24pi)13]
SA = 2(452.16) + (75.36)13
SA = 904.32+ 979.68
SA = 1884

Date: 12/11/2006 at 15:09:02
From: Doctor Peterson
Subject: Re: Why not solve for pi?

Hi, Alex.

In real life, of course, you would want an actual number, which you 
would then use to buy your paint or whatever.  So you're right that 
the problem is not REALLY finished if you leave it as 600pi.

In class, a teacher likes you to leave the answer in terms of pi, 
because it's easier to check: the number you get won't depend on 
whether you used 22/7, 3.14, or 3.14159 for pi, or on how you rounded.  
And she knows you can do the last step of multiplying; all the 
important part of the problem has been done, so why waste your or her 
time?

To a mathematician, the answer in terms of pi is nicer, because it is 
exact--there's no rounding involved.

There's something else that ties all these viewpoints together: the 
safest way to be sure your answer is accurate is to work with exact 
values right up to the very end, and only THEN use the actual value 
of pi.

In your example, by replacing pi with its value (evidently 3.14, using 
three significant digits) early on, you introduced an approximation.  
If there had been a lot more arithmetic following that, each step 
might have brought in more inaccuracy (called "rounding error").  In 
this case, plugging in pi=3.14 at the end would still give 600*3.14 = 
1884, so there is no difference in the result; but sometimes that is 
not true!  Furthermore, by using 3.14, you committed yourself to 
needing only three digits of precision.  If you had left it for later, 
at the end you could have used a more accurate value and obtained a 
more precise answer: 600*3.14159 = 1884.954.  And you would not have 
had to carry around lots of decimal places in your work in order to 
get this precision:

  SA = 2(area of a base) + (lateral surface area)
     = 2(144pi) + (24pi)(13)
     = 288pi + 312pi
     = 600pi
     = 600*3.14159
     = 1884.954

So I would recommend making a habit of retaining pi (or variables, in 
other problems) until the last step, just simplifying the expression 
rather than evaluating it numerically; then, if you need a usable 
value, put in an appropriate approximation for pi at the end.  That 
will satisfy everyone, including you when you need an exact value and 
can get it with less chance of error.

If you have any further questions, feel free to write back.


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

Date: 12/11/2006 at 19:39:21
From: Alex
Subject: Thank you (Why not solve for pi?)

Thank you for helping me with my math problem.  It answered all of my 
questions.