Significant Figures

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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 104 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.