__Significant Figures__

·
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practice worksheets**

One
of the most baffling subjects for students is frequently significant
figures. The reason for this is
simple: Nobody ever seems to know what
they’re supposed to be used for. Why
should I care if “100” has one significant figure or “100.0” has four?

Fortunately,
your friend Mr. Guch is here to help.
Let’s take a look at the joyous excitement produced by significant
figures:

__Why do we need
significant figures?__

For
some reason, teachers never really tell students why significant figures are
important (note to any teachers who are reading this: I’m not talking about *you*. I’m talking about those
other teachers).

Significant
figures are important because they tell us how good the data we are using
are. (Incidentially, the word “data” is
plural for “datum”, so even though it seems weird saying that “data are
[something]”, it’s grammatically correct.)
For example, let’s consider the following three numbers:

**100
grams**

**100.
grams**

**100.00
grams**

- The first
number has only one significant figure (namely, the “1” in the
beginning). Because this digit is
in the “hundreds” place, this measurement is only accurate to the nearest
100 grams (i.e. the value of what we’re measuring is closer to 100 grams
than it is to 200 grams or 0 grams).
- The
second number has three significant figures (the decimal makes all three
digits significant, as we’ll discuss later). Because the last significant figure is
in the “ones” place, the measurement is accurate to the nearest gram (i.e.
the value of what we’re measuring is closer to 100 grams than it is to 101
grams or 99 grams).
- The third
number has five significant figures (as we’ll talk about later). Because the last significant figure is
in the “hundredths” place, the measurement can be considered to be
accurate to the nearest 0.01 grams (i.e. the value of what we’re measuring
is closer to 100.00 grams than it is to 100.01 or 99.99 grams).

In
short, when you plug these three numbers into your calculator, there’s no
difference in how the calculator will manipulate them – your calculator neither
knows nor cares about how good the numbers it’s working with are. However, to you, the taker of data, these
three numbers tell you whether or not your data is good enough to pay attention
to.

__How do we find
the correct number of significant figures?__

Right
now you’re thinking to yourself, “Mr. Guch, my teacher never mentioned anything
you talked about above, but for some reason just likes to ask me a bunch of
questions in which I need to figure out how many significant figures a number
has. What should I do?”

What
you *should* do is march
right on down to your teacher and tell them that they’ve been wasting your
time, teaching you something that’s totally irrelevant (because as I mentioned,
significant figures are completely irrelevant if you don’t understand *why* they’re important). Of course, your teacher will laugh at you,
call your parents, and make you stay after school, so this probably isn’t a
great idea.

What
you’ll probably end up doing is just learning how to figure out the rules for
measuring significant figures. However,
make sure that you tell your friends why significant figures are handy so they
know why they’re bothering with all of this.

Rule
1: Any number that isn’t zero is
significant. Any zero that’s between two
numbers that aren’t zeros is significant.

- All
this means is that if you have actual numbers written down
(or zeros between these numbers), they have actual meaning and give you
meaningful information.
- Example:
198, 101, and 987 all have three significant figures.

Rule
2: Any zero that’s before all of the
nonzero digits is insignificant, NO MATTER WHAT.

- Basically,
this applies to numbers that are very small decimals. For example, if you have the number
0.000054, there are only two significant figures (the 5 and the 4),
because the zeros in front are insignificant.
*But Mr. Guch, don’t those zeros tell me something?*Yes and no. The reason that you don’t count these numbers as significant is mainly because of rule 4, which we’ll talk about after…

Rule
3: Any zero that’s after all of the
nonzero digits is significant only if you see a
decimal point. If you don’t actually see
a little dot somewhere in the number, these digits are not significant.

- Let’s consider
the numbers “10,000 lbs” and “10,000
**.**lbs”. The first number is significant only to the nearest ten thousand pounds (only the first “1” is significant) and the second is significant to the nearest pound (all five digits are significant). What the addition of the decimal does is tell us how good our measuring equipment is. The first number, for example, was probably taken by a truck scale (which wouldn’t have much use for measuring things to the nearest pound) and the second was probably taken by a bathroom scale (which requires much greater precision).

Rule
4: When you write numbers in scientific
notation, only the part before the “x” is counted in the significant
figures. (Example, 2.39 x 10^{4}
has three significant figures because we only worry about the “2.39” part).

- Let’s go
back to rule #2, in which we said that “0.000054” had two significant
figures. The reason for this is
that if we convert it into scientific notation, we end up with the number
“5.4 x 10
^{-5}”. If we said that all the zeros in front were significant in rule 2, then we’d have the weird case where the same number had either two significant figures or seven significant figures, depending on how we write it. Since that kind of confusion isn’t too cool, we ignore them in significant figures to make our lives easier.

__How about some
practice problems?__

Knock
yourself out. Click **HERE** for a
practice worksheet.

© 2006 Ian Guch – All rights reserved.