Associated Topics || Dr. Math Home || Search Dr. Math

### Calculations Involving Significant Figures

```
Date: 02/22/2002 at 20:34:42
From: Cassandra Rideout
Subject: Calculations involving Significant Digits

Hi!  This is a great project you have here, but on with my question!

My physics class is having a fair amount of trouble with significant
digits. We understand what significant digits are and their purpose.
We have laboured over the rules for rounding and for multiplication
and addition, and so on. However, our BIG problem lies with lengthy
calculations that require several intermediate steps before the final
result is obtained. In other words, we don't know (even our teacher!)
what to do when we have to add and multiply in the same problem.

For example: 3.95 x 1.15 + 2.7503 / 8.49. Do you multiply the first
two numbers (3.95 x 1.15) and determine the correct number of
significant figures (4.59), which you then add to (0.324)? Or do we
not round or use significant figures (is there a difference between
these two terms?) until the very end? That is, add the raw data
(4.5425 and 0.3239458...) and then use the addition significant figure
rules.

Likewise, when determining slope m = (11.2cm - 10.2cm) / (200g - 0g)
do you say that 1.0 / 200 = .0050 cm/g, or do you look at your initial
numbers (least amount of significant figures would be 3 because masses
are constants and contain an infinite number of significant figures)
and say that the answer is 0.00500 cm/g?

I hope that you can help me (in fact our whole class). Thanks in
Cassie
```

```
Date: 02/22/2002 at 23:39:54
From: Doctor Peterson
Subject: Re: Calculations involving Significant Digits

Hi, Cassandra.

It sounds as if you are forgetting the fact that significant figures
discusses that:

Decimal Places and Significant Figures
http://mathforum.org/library/drmath/view/59014.html

So you really have to think about the precision at each step, since
each step has a different effect, some via significant figures, others
via decimal places. But you don't want to round to the appropriate
precision at each step, because then you would be introducing error.
Significant figures are only a rule of thumb, and rounding too early
can allow errors in the dropped digits to affect the result. In these
days of calculators you don't have to round anything until the end; in
the old days you would have been advised to keep at least a digit or
two extra until the end. (Calculators do that too - they have more
precision internally than they show, for exactly this reason.)

3.95 * 1.15 + 2.7503 / 8.49

You will do all the arithmetic with as much precision as your
calculator allows, giving 4.5425 + 0.3239..., and then add them to get
4.8664.... Now you look at the precision. The product has three
significant figures, giving you hundredths, and the quotient has three
as well, this time giving thousandths; the sum is accurate only to the
hundredths, so you use 4.87. This happens to have three significant
figures as well, but that is not necessarily going to happen.

(11.2 cm - 10.2 cm) / (200 g - 0 g)

Here you first subtract, giving tenths in the dividend and units in
the divisor, so you have 1.0 / 200. Since the dividend has only two
significant figures (and I'm going to assume the divisor has three,
though it could be taken as one), your answer should have two:
0.0050 cm/g.

I'm not sure why you said that masses are constants and have
infinitely many significant figures; if they are measured, they have
finite precision. Only _defined_ constants (like 2 when you are
doubling something) are treated as exact. But the 0 probably is exact.
I can't tell without knowing the source of these numbers.

I hope this helps!

- Doctor Peterson, The Math Forum
http://mathforum.org/dr.math/
```
Associated Topics:
Elementary Place Value
High School Physics/Chemistry

Search the Dr. Math Library:

 Find items containing (put spaces between keywords):   Click only once for faster results: [ Choose "whole words" when searching for a word like age.] all keywords, in any order at least one, that exact phrase parts of words whole words

Submit your own question to Dr. Math
Math Forum Home || Math Library || Quick Reference || Math Forum Search