The Context Around How and Why to Use Significant Digits (a.k.a. Significant Figures)
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A truth about me: I struggled with significant figures until I taught them in graduate school…
First, a Story and a Truth
There is a dirty little secret in physics and chemistry. Anyone who has taught general chemistry — in my experience — also knows what it is. It’s significant figures!
More specificially, it’s how we discuss them, how we contextualize them, and how even most graduate students will admit in the strictest confidence that we all struggled through learning them in undergrad, promptly forgot about them, and only really figured them out by teaching it as Teaching Assistants.
I’ll admit it: significant figures were the bane of my existence in my undergraduate chemistry courses.
Anyone who knows anything about how my mind works will tell you that I struggle greatly with memorizing seemingly-arbitrary rules that have no real motivation behind their use. I struggled mightily against the subject, finally forgetting I had tried to learn it at all and picking up the rules for the arithmetic of significant figures through scientific practice and observation — indeed giving very little thought to the why and the how of choosing precisions while measuring and recording values in the laboratory. In a sense, the way I learned significant figures was much like the process of learning a language as a child. Great for me in my scientific practice, but not so great for my students when it came time for me to teach the basics of significant digits.
To put this in a little bit of perspective for you: do you remember what a participle is in English? Without looking it up?
Well you’re a better student than I was, I suppose!
At any rate, I find my experience is not an isolated one. Many students in introductory chemistry and physics courses struggle with the why and the how of significant figures.
Unfortunately, by the time we discuss it I find most of my students already frustrated and confused about what significant figures are, why we care about them, and how to use them. From a purely mathematical point of view, it makes no sense why we would apply this set of rules to our arithmetic!
That’s probably because the notion of significance is more a statistical and scientific one than a mathematical problem.
However, the proper application of significant figures is critical to science and technology. These are of course realms where numbers are used along with arithmetic and calculus in order to construct or analyze something. In engineering and technology, significant digits can even make the difference between life and death!
Another unfortunate thing that I have observed is that many students will have a complete disconnect in their minds between mathematics, science classes, and the actual practice of observation and critical thought. It seems the main approach my students often take to get through high-school chemistry and physics is to memorize formulas and regurgitate them with some numbers plugged-in from the question prompts. Naturally, when this approach is used exclusively, the connection between the numbers used in the formulas and the numbers that one obtains from observing nature or deriving physical relationships is never solidified.
Of course you’re frustrated! I would be too (and I was)!
So here I will begin by discussing some background on the nature of measurement itself, precision, accuracy, and what we might find ourselves doing with these values we’ve collected from nature.
A student might ask, “Why don’t you just get on with showing me the arithmetic rules?!?”
I am explaining it this way because the rules make no sense out of context. So first I will provide context and expound a bit on why we need these wonky “sig figs” in the first place.
Then I will tell you how to do the arithmetic.
You are of course free to skip ahead, but I will say that the vast majority of the value in my approach will be lost on you if you just “plug and chug” the formulas and memorize the rules.
So let’s go!
What are Significant Figures (SigFigs) and Why Should You Care?
Science and technology often involve measurements. In fact, so many things in life involve measurement. An example where my dad and I were tiling a bathroom comes to mind. When tiling a room, you have to measure carefully to account for the area of each tile as well as the gaps between them for grout.
If you don’t measure the tiles correctly, the combination of many small errors while measuring and cutting each tile will add up to form a giant, horrible, awkward mess of tiles that are partially cut at weird angles — as my dad and I found out the hard way!
The point is — your measurements are only as good as the precision of the instrument you’re using to take them.
What does this have to do with significant figures?
Significant figures provide a mechanism for minimizing the error in a value derived from multiple measured values of varying precision.
What do I mean by precision?
The number of decimals you measure out to confidently is your degree of precision. For example, a balance that is guaranteed to measure out to five decimal places, or 0.00001 = 1\times 10^{-5}, would give us confidence to the fifth decimal place.
In other words — if you have a tape measure and you measure a square tile to have an edge length 0.47” and my dad (I don’t know why he’s here too… just go with it…) measures the square tile to have a length of 0.5” per side, which measurement do you choose? If you have to calculate the amount of tile to cover the space and leave enough room for grout between them, what do you do?
Your measurement has a greater precision than my dad’s measurement!
Wait, isn’t that good?
Not necessarily!
What would cause probably the least headache in this example might be to err on the side of caution and calculate the number of tiles covering the bathroom using the combination of measurements, but then reduce the precision of the resulting value to match that of my father’s original measurement.
This would have made that bathroom renovation much easier…
Having said all this, significant figures allow us to be confident in the accuracy of our calculations within a given degree of precision. The degree of precision — and thus the number of significant figures — is set by the degree of precision in the measurements themselves.
Why Do We Care About Them?
If you did not collect the data you are working with, you only have as much confidence in the value’s precision as you are given by the figure itself. For example, you have the number 50,000 — was it rounded from 49,999.99? 50,000.001? 50,000.4? Was it rounded from 49,999? Or from 49,000? We don’t know unless we ask the person who took the measurement, “Hey dude, did you round this? If so, from what?”
Rather than scratch our heads and drive ourselves nuts trying to find our Mystery Technician — who is probably on lunch for the next four months anyway — we just look to see how many digits in it are significant for our arithmetic. If it turns out that it was 50,000. (notice the decimal point at the end) then we can be sure it was an even 50,000. where we have five significant digits to play with instead of just the one.
This is also important when it comes to the increasingly common applications of numerical algorithms for solving both everyday and scientific problems. One might assume that, because a small roundoff error in one calculation makes a miniscule difference, it isn’t a big deal to worry about the significance or about so-called ``roundoff errors’’. However, this person might also write an algorithm to perform millions of calculations per second repetitively — if they paid no attention to the precision and significance of the input data and the arithmetical manipulations of those values, you would get utterly useless, flaming piles of garbage for output data in some cases because of repetitive, cumulative roundoff errors!
Essentially, the reason we do this is so we don’t end up inserting a bunch of meaningless numerical hot-garbage into our derived values — that is, the values we calculated from the raw data. Typically in science, engineering, analysis, forensics, and various other technical disciplines, one will have to deal with the confidence one has in raw data collected by someone else.
Hell, my memory is so bad that half the time I don’t even remember if I rounded to a specific decimal place. At least if I didn’t put that handy decimal point at the end of a round number.
How to Apply the Arithmetic of Significant Figures
There is an algorithmic set of rules provided on Wikipedia (among other locations on the Interwebs like this one!) that boils down to:
Definition — Identifying Significant Digits
Nonzero digits (like 123) are significant in raw measured data or properly derived values calculated previously!
Zeros between two significant non-zero digits are significant: 101.1203 has seven significant digits!
In figures containing decimal points and zeros, things get a bit more complex, but still not too scary:
A. If the figure has leading zeros before nonzero digits (e.g. 0.000052) those leading zeros are not significant. Phrased another way, “If you can express it with scientific notation without losing any nonzero values, the leading zeroes are not significant.” So using this reasoning, 0.000052 = 5.2
\times 10^{-5}, containing only two significant digits.B. If the figure has trailing zeros written explicitly (e.g. 12.2300), that means whoever wrote it down for you is implying that those zeros are significant and thus 12.2300 has six (6) significant digits! This is one of the major sticking points I along with my students had with significant digits!
C. Finally, values terminated with a decimal point, such as 1,230. are written that way to indicate that the final zero is significant and that we are confident in the precision of 1,230. being exactly 1,230. without any rounding!
If we are very lucky, our Mystery Technician underlined the most precise significant digit they have confidence in given their instrumentation. Most often, they will round it for us before we see it, but sometimes they’re very precise and thoughtful! That would look something like: 50,000.012 which has seven significant digits. The 2 at the end is in the wibbly-wobbly part of the measurement that might fluctuate, but the Tech included it anyway for us.
Arithmetic — Manipulating Significant Digits
The algebra of significant digits is straightforward. You only really have to remember two sets of rules once you’ve ascertained the nature of the value, and its significant digits. A handy reference I used (and paraphrased) for this blog post can be found on Wikipedia if you want to go into a deeper dive.
Since the algebra of real numbers is closed under addition and multiplication and also has inverse laws for both operations — which I personally find convenient! — subtraction is adding the inverse of the value being subtracted; division is multiplication by the reciprocal value of the divisor. If this seems confusing, it will become clear shortly, so don’t fret!
Another consideration alluded to previously is whether or not the value is directly measured or if it has been calculated (or derived) as this will affect the rules for the arithmetic. Constants that have been measured exactly for things like unit conversions are the most common cases we deal with in the physical sciences where this distinction should be made.
Addition and Subtraction
The result of the calculation is rounded to the position of the least significant digit in the most uncertain of the numbers added or subtracted. In other words, the result is rounded to the last digit that is significant in each of the numbers.
Some examples of this include:
1 + 1.1 \approx 2where the value with fewest significant digits and most uncertaity is integer 1, so the significance of the result is integer 2 — consistent with the position of the significant value in 1 — rather than decimal 2.0 as one would find in:
1.0 + 1.1 \approx 2.0See the distinction? If we were to include the higher precision here, we could introduce all sorts of insignificant junk and create an ``artificial’’ precision that we have no confidence in. It would all be rather confusing without this set of rules and conventions, don’t you think?
Multiplication and Division
Oddly enough, this one is, I think, a bit more intuitive. When performing multiplication in this way, one only has to round to the number of significant figures in the factor with the fewest significant figures. That old, familiar “Just round to the least precise“ rule. Note that I don’t say the position of the most uncertain value (as with addition). Rather, it is about the precision when it comes to significance multiplication.
Some examples:
8 \times 8\approx 6\times 10^{1}With addition, the following
8 \times 8.0 \approx 6\times 10^{1}would have a different value. Not so with significance multiplication, as the value of least precision is not different between the two examples.
However, if we had
8.0 \times 8.0 \approx 64note that the precision has changed, and we include one more significant digit (in this case, the number 4).
Going a step further, we see that
8.02 \times 8.02 \approx 64.3because now we have rounded to the number of significant digits of the least precise factor (e.g. three significant digits).
The same rule holds for division, thus
8 / 2.0 \approx 4and
8.6 / 2.0012 \approx 4.3and
2\times0.8\approx2based on these rules.
OK great… so that seems a bit more intuitive! But what about the really weird stuff like…
Transcedental Functions
Ahh, good old e^{x}, \ln x, \log x, and the trig functions…. surely those can’t be too bad right?
Right?
Well.. these are somewhat complicated, so I’d direct you to Wikipedia for a set of examples and explanation. Perhaps another blogpost could be devoted just to this subject were one so inclined to write it (or read it for that matter!)
Directly Measured vs. Derived Values and Why This Matters
Some values are directly measured. Meanwhile, for others that are measured elsewhere or derived from calculations, you can only trust the significance value you are given by your data. Let’s say you have a formula cheatsheet from NIST or from somewhere else that has constants like \pi or various unit conversions on it, but you are working on your own data and deriving values from a combination of those tabulated values and your own data.
What do you do?
Generally, the exactness of a constant or other pre-tabulated value will be implied by how many digits are printed on your source. Sometimes they’ll use the aforementioned underlines. Sometimes it’s parentheses on the first value for which the tabulator is not confident about.
Sometimes you get lucky and you have a value that is exact and is typically listed as such on your pre-tabulated conversion factors.
Finally, you also have the numbers 2, 3, and \pi that pop up absolutely everywhere! Those values should be used without applying the significance to them as they are exact integers in the case of 2 and 3, but \pi written as a symbol is exact.
It is also infinitely long and irrational, so there’s no point in worrying about it because you’ll just end up rounding to the position of most uncertainty if you add it, and the position of least precision for multiplication.
Some Remarks
Hopefully by now the concept of significant figures seems less frustrating and inconsistent. As I mentioned, I distinctly remember struggling with how to apply them properly up until I began teaching one of the most introductory chemistry courses at the University of Washington. My students in that class asked some of the most insightful questions I have ever encountered as an instructor (shout out to CHEM 110 students from Fall 2018! Thanks guys!)
Strangely, in retrospect, as a practicing professional lab technician and later on as a scientist I have definitely been using significant figures more intuitively than algorithmically.
In a sense, if you were to ask a practicing professional chemist or physicist to explain significant figures to you might sound a lot like asking them to explain how they make sure they use the right form of a verb in the English language — they just do it.
On the other hand, the explanations I have found online vary between helpful and ludicrous. Most textbooks have very precise definitions, but in the context of the knowledge base already possessed by most students at the introductory level, these definitions are not helpful nor do they provide a motivating context for obtaining mastery of significant figures.
Please feel free to add your own comments to the discussion. If you are an instructor or scientist and have something to add, I could always use more approaches for teaching a seemingly basic and simple but rather confusing topic to my students. If you are a student, please let me know what you are confused about and I can try to clarify — you may end up helping me understand the topic even better with your question!
Cheers!
-J.J.R.
