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### 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.
```
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