Significant Figures

How much Water in the Cup?

Check out the image of the cup of water to the right. What is the volume of water? At first glance you might say it's 350 mL. Look again... closer. Click the image and get a full screen view - zoom in if you want. Want to adjust that initial number now? The level of the water is definitely below the 350 mark and well above the 300 mark. This is where you have to estimate (or some say guesstimate). To me, Dr. McCord - the guy writing all this, it looks like about 4/5ths of the distance between 300 and 350, which means I'd write down that it is 340 mL. Now here's the thing... How accurate is that 340 mL I just "read"? To be truthful, I'd need to stick an error range on that number. Just how far off could I be? 1 mL? 2? 3? 5? 10? Hmmmm. Hard to say, but I'm absolutely sure I'm not off by 10, that is a full 1/5 of the rather wide markings. Maybe I'm good enough to guesstimate to 1/10th of the markings which is 5 mL. So I'd adjust my measurement to be 340 mL ±5 mL. There, I've done the best I could with the data before me.

How accurate is that measured number?

The fact is that any measurement that one makes will carry with it a certain degree of uncertainty. The uncertainty has to do with how accurate the method is for the measurement (and let's say the competency of the one doing the measuring). Chemists will often communicate this uncertainty via the use of significant figures (ie "sig figs") in a recorded measurement. The number of significant figures implies an accuracy of no more than ±1 in the last digit recorded. It is important to be able to propagate this implied uncertainty when combining measured numbers so that we do not ultimately convey an accuracy that is really not there. And yes, I know, and you should know that using significant figures is no substitute for a full blown error analysis and statistical treatment. However, all of that could require multiple measurements and data sets which is a bit of overkill for common measurements. Using significant figures is a relatively simple way to convey accuracy in a number by just writing the number with the correct number of digits. To be clear here, go back up and look at that measuring cup again. What if I had told you that I recorded the volume as being 342.352 mL of water. Wow! that is an accuracy of better than ±1 drop! Crazy, right? No way! There is no point to imply false accuracy with all those digits. In general, we write out all the digits we are certain of, and then one more. That "one more" digit? It is called the uncertain digit and we definitely need to include it. That's how we chemists do it and you need to learn how to do it as well.

The Quick Way to know the Number of Significant Figures

Let's get you started by teaching you to properly count the number of significant figures in a recorded numerical value (a measurement). Let's start with any ol' plain number that actually has a decimal point in it.

Assuming there is a placed decimal: Starting from the left on the number, start counting on the first non-zero digit. Now count all the way to the end of the number - all digits count at this point. Here are a few examples.

13.52 has 4 sig figs

0.012 has 2 sig figs

12.0015 has 6 sig figs

No decimal numbers (integers) with no trailing zeros: In general, the same rule above applies if it is an integer with a non-zero last digit such as 135, 34, 1254, etc... each and every digit is significant here. We assume that there was a good reason to include all those digits. We assume that someone did do their best to accurately measure something and they recorded 1254 as the measurement. We will tend to believe that the number reported could be ±1 in that last digit (the uncertain digit). So count all the digits as significant for all integers with non-zero last digits. All a caveat here - some numbers like this are "counting" numbers and they are infinitely precise. Be sure and keep reading to find out about them.

Things get a little Complicated with Trailing Zeros

No decimal numbers (integers) with trailing zeros: These are the tricky ones. When we are forced to use trailing zeros to get us into the right order of magnitude such as 1000, 250, 365000, or even 63000000. Are all those zeros significant? Probably not, especially if these numbers are some sort of measurement. The first number is one thousand. But is it exactly one thousand? You know, like how many grams are in a kilogram? Or, is it a "about" 1000 as in we poured a sample of about 1000 mL into a drinking glass. See? One of those 1000s is infinitely accurate: 1000 g per kg. The other is most likely a "ballpark" figure or an estimate. A drinking glass typically has no measurement markings and you would be making an approximation of about 1000 mL? Context is everything here for trailing zeros. You must know a bit more information than just the number alone and be able to have a reasonable idea of how many significant figures there are. If that drinking glass has about a liter capacity, then pouring a "full" glass will be about 1000 mL give or take about 100 mL.

Be Careful: Other books and websites on significant figures will often tell you to not count any trailing zeros on numbers like this. This is an easy rule to follow but it isn't at all in keeping with the logic behind significant figures. If we follow that logic, our good ol' 1000 has been reduced to just one significant figure. That implies a ±1 in that digit which means 1000 ±1000... so you might have none, or you might have 2000. That is ridiculous if you have a glass of water sitting right in front of you in a liter glass and it is pretty much full. It is more of a ± 100 mL which means you'd need to rewrite your number to somehow communicate that. Bottomline on trailing zeros for integers - make a reasonable guess as to the accuracy of the number, then estimate the number of significant figures. Better yet, show the number in scientific notation which allows you to show only significant trailing zeros if need be.

Scientific Notation numbers: These numbers consist of a coefficient (a number between 1 and 10 with a placed decimal after the first digit) and an exponent (the number that is the power of 10). This allows one to only show significant figures in the coefficient and then allow the exponent to put the number into the right order of magnitude. Now you write 1.0 × 103 to show 2 significant figures and you interpret this as one thousand plus or minus about 100 (the last decimal place listed). If you're more accurate than that, then write 1.00 × 103 and you'll have 3 significant figures and your 1000 will now be ±10 which is a much tighter accuracy range - hence the 3 sig figs.

Counting Numbers are Infinitely Precise

I bought a box of one dozen golf balls. How many are there? Twelve (12). When I say or write that I have 12 golf balls there is no uncertainty at all that "12" is infinitely precise meaning is it like writing 12.0000... where there are an infinite number of zeros. Many numbers are exact because they are counting numbers like the golf balls or they are defined numbers like 2.54 cm is 1 in. The 2.54 here is infinitely precise. We now have more defined numbers in our measurement arsenal than we used to. 4.184 J is the definition of a calorie. It is exactly correct and has an infinite number of significant figures.

Avogadro's number (as of March 20, 2019) is now an exact number. How exact? Infinitely exact. Here it is written in all its infinite precision.

\[N_{\rm A} = 6.02214076 \times 10^{23}\]

Although it looks like 9 significant figures, it is actually infinitely precise because we have defined it that way. The speed of light is also infinitely precise now.

\[c = 2.99792458 \times 10^{8}\;{\rm m/s}\]

So if you can count individual items or pieces, you will have an exact number which will have an infinite number of significant figures. Also, if a number is labeled as defined then it too is infinitely precise. Counting numbers and defined numbers will NOT limit you on your precision/accuracy in a measurement. It is only those measured amounts where the technique and/or instrumentation have limitations where you will have a specified number of significant figures.

Remember: Just because your calculator display is showing 10 digits of precision doesn't mean that all those digits have any real world meaning. Keep track of significant figures and you'll have a better grasp of the precision of your measurement.

Sig Figs and Logarithms (pH of a solution)

Sometimes the numbers (measurement) we use can get a bit unwieldy. Like when we measure the hydrogen ion (H+) concentration in aqueous solutions. Those values span orders and orders of magnitude. Like a concentrated acid could be 1 M (molar) to 10 M. But neutral water has a concentration of 1.0 × 10-7 M. And a strong base solution can drive the H+ all the way down to 1.0 × 10–14 M. That is a lot of exponents of 10 to keep up with and it can get tediuous even writing it all out. And forget about ever plotting these vastly different values on the same linear graph. What to do?

Let's bring in our mathematical friend, the logarithm, or log(x). On your calculator, it is most likely just LOG. The logarithm gives you the power of 10 that will give you back the number. So the LOG of 10–7 is –7. The LOG of 10–14 is –14. Now the entire range of concentrations is on a new scale from 0 to –14. And it's real easy when you have perfect powers of 10.

Let's try something more like the real world that isn't a perfect power of 10. Let's imagine a water sample where the H+ concentration is measured to be 2.3 × 10–5 M. The LOG of that number on my calculator is –4.638272164. Although something seems off here. My original number only had 2 sig figs and there are way too many on my calculator. So what to do? You might think to say the answer is –4.6 and be done. Yeah, it looks like 2 sig figs, but it's not. It is only 1 sig fig because that number is a logarithm. You need another digit to be accurate to 2 sig figs. Let me prove it.

Let's UN-LOG that –4.6 we just talked about. UN-LOG is not the correct term either. It is actually called antilogarithm and mathematically it is ten to the x or 10x. So what is 10–4.6? My calculator says it is 2.511886432 × 10–5 which rounds to 2.5 × 10–5 to 2 sig figs. Yikes! That is NOT our original amount of 2.3 × 10–5 and it is actually 8.7% too high - bye bye scientific precision. Why didn't this work? Well, that is because... drum roll, please...

All the sig figs for logarithm numbers are AFTER the decimal.

So in our example, if we want to keep our two sig figs that we originally had, we need to record the LOG of 2.3 × 10–5 as –4.64. Only the 6 and the 4 are "significant". And note, I did have to round to get that trailing 4. Now lets try antilog again. 10–4.64 is equal to 2.290867653 × 10–5 and THAT rounds to 2.3 × 10–5 which is our original 2 sig fig number. No loss of precision!

p(x) is just -log(x)

Another thing you'll find about us chemistry guys... we really like the p(x) function. It is used anytime we want to get a large exponential range of tiny-ass numbers into a more user friendly scale. And you know what is not user friendly to us humans. Constantly having to say negative this and negative that. Every logarithm of a number less than 1 is a negative number. We want a positive scale. We want to drop the negative. That is what the p(x) function does. It is just the –log(x). And when x is the hydrogen ion (H+), the term becomes pH and it is a nice scale from about 0 to 14 to tell us the acidity of the solution. There is a whole chapter on it in this chembook (chapter 6).

So think of it like this for pH and sig figs. The number (digits) in front of the decimal are really just putting us in the right power of 10 range like the exponent of 10 in a scientific notation number. Those leading digits in front of the decimal are NOT significant. Now, the numbers AFTER the decimal - those are the sig figs. Most of our pH calculations have 2 sig figs which is why most all pH's are written with 2 digits after the decimal. Below is a graphic summary of all this.

Note: Although this discussion on logarithms pertains to LOG base-10, the same basic argument can be true for natural log (ln) as well. Admittedly, for natural log, the digits in front of the decimal can be somewhat significant, but not entirely. For this reason, it is still best to assume only the digits after the decimal are significant.

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