Significant (and insignificant) figures

When we use numbers and symbols in science, we are not just doing math; we are representing things in the physical world. An important issue is: How specific should we be in reporting a number? For example, consider the following problem in arithmetic:

1.843 x 3.686 = ?

If you were doing this in your math class, it would be easy. You would plug the numbers into your calculator and the result would come out.

1.843 x 3.686 = 6.793298

This is an exact result and you would get full credit in your math class. Indeed, if you gave anything else, you would probably get points taken off!

The problem is, that's not what you should do in a science class.

Let's see how this works in a problem in which the math is representing something in the real world. Suppose I am making a rectangular box to exactly fit two pennies into it side by side, how much area do I need? This is shown in the figure below.

(Source: adapted from Wikimedia Commons: Measurement.)

Reading off the width of a penny in centimeters, I decided it was more than 1.84 cm, but not 1.85 cm, so I called it 1.843. The width of two pennies would then be twice that, and the area would be the product in the multiplication given earlier on this page.

But it "really" could be 1.841 cm or even 1.839 cm. I wouldn't bet against that. If it were 1.841 cm, then my multiplication would actually be

1.841 x 3.682 =6.778562.

This is not the same as what I got in my first calculation — 6.793298.  This tells me that the last four digits — the "3298" — are totally bogus. I have no idea whether they are correct or not. They could be anything.  Even the third place is a bit uncertain — I got "9" the first time and "7" the next.

So I really shouldn't cite the result with more than three digits. Those are the only significant figures. The rest are insignificant figures.

You will sometimes find "automatic" rules for significant figures — how many to keep. They can be convenient for those times when your brain has stopped working. For other times, use this rule of thumb:

          Only quote figures that you are pretty sure of — perhaps plus one more.

There are (at least) four reasons to understand and use significant figures in this class and in science in general:

• Quoting a number is a communication.  When you cite a number, you are telling your listener that you really believe those numbers. If you don't, you are misleading your listener — a serious error.
• Much of what we do in science relies on careful measurement. No measurement is perfect, so citing a correct number of significant figures (plus an uncertainty range — usually called an "error bar" even though it's not about mistakes) tells you how much to trust the measurement.
• It's extremely important to understand how accurate results are. Sometimes results reported in public forums (the web, news media, etc.) are given inappropriate significance (for example., political polls, results of medical studies, etc.). And you can even misinterpret results in scientific journals if you don't pay attention to how reliable a measurement is! 
• And of course the fourth reason is that you will lose points on your work on exams, homework, and lab reports for giving inappropriate numbers of sig figs!

Joe Redish 7/6/11