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### Rules for Significant Figures and Decimal Places

Date: 03/23/99 at 21:29:42
From: Ashley Seither
Subject: Definition and rules

I need to know what significant digits are, and what rules go with
them.

Date: 03/24/99 at 12:00:26
From: Doctor Peterson
Subject: Re: Definition and rules

When we work with numbers that come from the real world (such as measurements
from a ruler), the numbers are not exact, but carry some amount of inaccuracy
with them (because, for example, no ruler is absolutely, perfectly straight).
There are two main ways we can describe the accuracy of a measurement:

If it is accurate to N *decimal places*, this means that there are N
digits to the right of the decimal point that you can trust. For
example, if I measure a length with a ruler marked off with millimeters, then
the measurement will be accurate to the nearest millimeter. (If I write it in
meters, to three decimal places: 0.001m.) If I claim to have measured it as
1.1293m, you know I was guessing about the 3 ten-thousandths, and you would
round it off to the nearest thousandth: 1.129. If I say it was 1.100 m to three
decimal places, you know that the two zeroes are not just guesses, but what
I actually read from the ruler. The ruler will always produce the same number
of decimal places, since there is a certain minimum size it can measure.

If a number is accurate to N *significant digits* (or figures), this
means there are N meaningful digits that you can trust. For example,
in my 1.129m, there are four digits I consider dependable, based on
how I measured. If I had measured 0.024m with the same ruler, there would
be only two significant digits. (The zeroes are there only to show the place
value of the other digits, and are not 'significant'.) The ruler does not
always produce the same number of significant digits, because it is better at
measuring larger things. If I tried to measure something smaller than a
millimeter, it would be useless. It would not give me any significant digits
at all!

Incidentally, be careful about zeroes in a number. If I told you a road was
12300m long, according to my car's odometer which shows tenths of a kilometer,
you would know that only three digits are significant, because I read "12.3."
The two zeroes, like the zeroes in 0.024, are there only to give the other
digits the right meaning. But if I used a more accurate instrument, I might
have read all five digits exactly. You do not know unless I tell you how I
measured it or how many digits are significant.

Now, what happens to the accuracy of a number when I use it in a
calculation? Or rather, how does the accuracy of the 'inputs' to a
calculation affect the accuracy of the 'output'?

When you are adding numbers, you want to look at the number of decimal
places. For example, if I add 1.2 and 3.45, with different numbers of
decimal places, I do not know what the hundredths place of 1.2 is, or
what the thousandths place of 3.45 is. I can put an X for the unknown
digits and see what happens:

actual    with X's

1.2       1.2XX
+ 3.45    + 3.45X
------    -------
4.65      4.6XX

You see, since I do not know all the hundredths I am adding, I really
have no idea what the hundredths place of the result is (an 'unknown'
plus 5 is still 'unknown'). So, to be honest, I have to drop the 5 and
call the answer 4.6 (or else round it up to 4.7), showing that my
answer is accurate to only one decimal place. (Even the tenths might
be wrong because of a carry, but it would not be too far off.) So, when
I add numbers, the result is only accurate to the smallest number of
decimal places I am adding. In this case, since 1.2 has only one
decimal place, that is all I can keep in my sum.

On the other hand, if I multiply numbers, what counts is the number of
significant digits. Suppose I run for 1.45 hours at 6.1 miles per hour.
Then I have gone 1.45 x 6.1 miles. How accurate is that? Again, I will
put an X for the unknown places and see what happens:

actual       with X's

1.4 5        1.4 5 X
x     6.1    x     6.1 X
---------    -----------
1 4 5        X X X X
8 7 0        1 4 5 X
---------    8 7 0 X
8.8 4 5    -----------
8.8 X X X X

You can see that the number of significant digits in the result (two)
is the smaller of the significant digits for the two multiplicands (three
and two respectively), so I have to write my product as 8.8, rounding it
to two significant digits and dropping two digits that I worked hard for
and would otherwise have thought were good. Since 6.1 has only two
significant digits, I cannot keep more than that in my product.

So those are the rules:

When you add (or subtract), you keep as many *decimal places* as
there are in the least accurate number.

When you multiply (or divide), you keep as many *significant
digits* as there are in the least accurate number.

I should mention that this is only a 'rule of thumb', and it sometimes
multiplication with a larger factor in place of the 1.45, you would
have seen an extra significant digit because of a carry. There are
more careful rules for measuring the accuracy of a result, when you
really need to know just how accurate a number is, but significant
digits work well as a general rule.

In this age of calculators, when you can get many digits in any
calculation with no trouble, it is important not to keep all those
digits and thus get a false sense of the precision of your results.
We do not want to pretend we know seven digits when we really only know
two or three.

Here are some answers in the Dr. Math archives that may be of interest
to you:

Rounding to One Digit Accuracy
http://mathforum.org/dr.math/problems/lori4.16.98.html

Significant Digits
http://mathforum.org/dr.math/problems/casey9.7.98.html

Significant Figures and Scientific Notation
http://mathforum.org/dr.math/problems/smith5.21.97.html

I hope this does not overwhelm you! I wanted to give you not just the
basic definition, but some background so you could see why it makes
sense.

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

Date: 03/09/2001 at 08:44:08
From: Solomon JHS Math Club
Subject: Significant Digits Dividing

We reviewed this answer and we still have trouble understanding the rule
for dividing. For example, the exact answer for 366/2 = 183. Since 2 has
only one significant digit, does that mean that the most accurate answer
is 200?

Date: 03/09/2001 at 09:22:32
From: Doctor Peterson
Subject: Re: Significant Digits Dividing

Hi, Solomon.

It depends on whether 2 is a measurement, or a known exact value, as, for
instance, if you are calculating a radius from a diameter, so you know you
have to divide by exactly 2. In the latter case, you can think of the 2 as
having infinitely many significant digits when you apply the rule, since
EVERY possible digit is known exactly. You have three significant digits
in 366, so you can keep three in the answer. But if 2 represents, say,
the number of hours it took to go 366 miles, then the answer should be
200 mph, since you have only one digit of precision in the time measurement.

We usually talk of significant digits only when there are decimal points
present to show the assumed precision of the numbers; in fact, we should
only do this if we know something about the actual measurements, rather
than just assuming the significant digits from the way a number was written.
Also, it's best to write numbers in scientific notation if we want to be
completely clear about accuracy. (Does 23000 have 2 or 5 significant digits?
If we write it as 2.30 *10^4, we can see that it has 3.) In such a context,
writing "2" with no decimal place would make it clear that it is an exact
number.

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

Associated Topics:
Elementary Definitions
Elementary Place Value
Elementary Terms & Units of Measurement
Middle School Definitions