J. Denker

# Introduction[1]

## How Many Digits Should Be Used?

Some Simple Rules That Apply Whenever You Write Down a Number

1. Use many enough digits to avoid unintended loss of information.
2. Use few enough digits to be reasonably convenient.

Important note: The previous two sentences tell you everything you need to know for most purposes, including real-life situations as well as academic situations at every level from primary school up to and including introductory college level. You can probably skip the rest of this document.

3. When using a calculator, it is good practice to leave intermediate results in the machine. This is simultaneously more accurate and more convenient than writing them down and then keying them in again.

Seriously: The primary rule is to use plenty of digits. You hardly even need to think about it. Too many is vastly better than too few. To say the same thing the other way: If you ever have more digits than you need and they are causing major inconvenience, then you can think about reducing the number of digits. If you want more-detailed guidance, some ultra-simple procedures are outlined below.

In many cases, when you write down a number, you need not and should not associate it with any notion of uncertainty.
One way this can happen is if you have a number with zero uncertainty. If you roll a pair of dice and observe five spots, the number of spots is 5. This is a raw data point, with no uncertainty whatsoever. So just write down the number. Similarly, the number of centimeters per inch is 2.54, by definition, with no uncertainty whatsoever. Again: just write down the number.

Another possibility is that there is a cooked data blob, which in principle must have “some” uncertainty, but the uncertainty is too small to be interesting. It is insignificant. It is unimportant. It is immaterial. There are plenty of situations a moderately rough approximation is sufficient. There are even some situations where an extremely rough approximation is called for, as in so-called “Fermi” problems. This point is discussed in reference 1.
Along the same lines, here is a less-extreme example that arises in the introductory chemistry class. Suppose the assignment is to balance the equation for the combustion of gasoline, namely

a C8H18 + b O2 → x CO2 + y H2O

by finding numerical values for the coefficients a, b, x, and y. The conventional answer is (a, b, x, y) = (2, 25, 16, 18). The outcome of the real reaction must have “some” uncertainty, because there will generally be some nonidealities, including the presence of other molecules such as CO or C60, not to mention NO2 or whatever. However, my point is that we don’t necessarily care about these nonidealities. We can perfectly well find the idealized solution to the idealized equation and postpone worrying about the nonidealities and uncertainties until much, much later.

As another example, suppose you use a digital stopwatch to measure some event, and the reading is 1.234 seconds. We call this number the indicated time, and we distinguish it from the true time of the event, as discussed in section 5.5. In principle, there is no chance that the indicated time will be exactly equal to the true time (since true time is a continuous variable, whereas the indicated time is quantized). However, in many cases you may decide that it is close enough, in which case you should just write down the indicated reading and not worry about the quantization error.
A few sections of this text are omitted here as they are more advanced that I think is relevant for this class.

Let us continue with the stopwatch example that was introduced in item 4. Suppose we make two observations. The first reading is 1.234 seconds, and the second reading is just the same, namely 1.234 seconds. Meanwhile, however, you may believe that if you repeated the experiment many times, the resulting set of readings would have some amount of scatter, namely ±0.01 seconds. The two observations that we actually have don’t show any scatter at all, so your estimate of the uncertainty remains hypothetical and theoretical. Theoretical information is still information, and should be written down in the lab book, plain and simple. For example, you might write a sentence that says “Intuition suggests the timing data is reproducible ±0.01 seconds.” It would be even better to include some explanation of why you think so. The principle is simple: Write down what you know. Say what you mean, and mean what you say.

The same principle applies to the indicated values. The recommend practice is to write down each indicated value, as-is, plain and simple.
You are not trying write down the true values. You don’t know the true values (except insofar as the indicated values represent them, indirectly), as discussed in section 5.5. You don’t need to know the true values, so don’t worry about it. The rule is: Write down what you know. So write down the indicated value.

Also: You are not obliged to attribute any uncertainty to the numbers you write down. Normal lab-book entries do not express an uncertainty using A±B notation or otherwise, and they do not “imply” an uncertainty using sig figs or otherwise. We are always uncertain about the true value, but we aren’t writing down the true value, so that’s not a concern.

Some people say there must be some uncertainty “associated” with the number you write down, and of course there is, indirectly, in the sense that the indicated value is “associated” with some range of true values. We are always uncertain about the true value, but that does not mean we are uncertain about the indicated value. These things are “associated” … but they are not the same thing.

In a well-designed experiment, things like readability and quantization error usually do not make a large contribution to the overall uncertainty anyway. Please do not confuse such things with “the” uncertainty.

It is usually a good practice to keep all the original data. When reading an instrument, read it as precisely as the instrument permits, and write down the reading “as is” … without any conversions, any roundoff, or anything else.

The key

No matter what you are trying to do, significant figures are the wrong way to do it.

People who care about their data don’t use sig figs.

When writing, do not use the number of digits to imply anything about the uncertainty. If you want to describe a distribution, describe it explicitly, perhaps using expressions such as 1.234±0.055.

When reading, do not assume the number of digits tells you anything about the overall uncertainty, accuracy, precision, tolerance, or anything else, unless you are absolutely sure that’s what the writer intended … and even then, beware that the meaning is very unclear.

Significant-digit dogma destroys your data and messes up your thinking in many ways, including:

• Given a distribution that can be described by an expression such as A±B, such as 1.234±0.055, converting it to sig figs gives you an excessively crude and erratic representation of the uncertainty, B.
• Converting to sig figs also causes excessive roundoff error in the nominal value, A. This is a big problem.
• Sig figs cause people to misunderstand the distinction between roundoff error and uncertainty.
• Sig figs cause people to misunderstand the distinction between uncertainty and significance.
• Sig figs cause people to misunderstand the distinction between the indicated value and the corresponding range of true values.
• Sig figs cause people to misunderstand the distinction between distributions and numbers. Distributions have width, whereas numbers don’t.
• Uncertainty is necessarily associated with some distribution, not with any particular point that might have been drawn from the distribution.

As a consequence, sig figs make people hesitate to write down numbers. They think they need to know the amount of supposedly “associated” uncertainty before they can write the number, when in fact they don’t. Very commonly, there simply isn’t any “associated” uncertainty anyway

• Sig figs weaken people’s understanding of the axioms of the decimal numeral system.
• Sig figs provide no guidance as to the appropriate decimal representation for repeating decimals such as 80 ÷ 81, or irrational numbers such as √2 or π.

# Pedagogical Digression – Extreme Simplifications

## Postponing Uncertainty

In an introductory chemistry (or physics) class, you should start with some useful chemistry ideas, such as atoms, molecules, bonds, energy, atomic number, nucleon number, etc. — without worrying about uncertainty in any form, and double-especially without introducing ideas (such as sig figs) that are mostly wrong and worse than useless.

Roundoff procedures are necessary, so learn that. Scientific notation is worthwhile, so learn that. The “sig figs” rules that you find in chemistry books are not necessary and are not worthwhile, so the less said about them, the better.

# In place of the “sig figs” rules, you can use the following guidelines:

In this class, the following “house rules” apply:

1. Basic 3-digit rule: For a number in scientific notation, the rule is simple: For present purposes, you are allowed to round it off to three digits (i.e. two decimal places). Example: 1.23456×108 may be rounded to 1.23×108.
2. For a number not in scientific notation, the rule is almost as simple: convert to scientific notation, then apply the aforementioned 3-digit rule. (Afterwards, you can convert back, or not, as you wish.).
3. All homework and exams work on a 5% rule: if you are within 5% you will be counted correct.

The point of these rules is to limit the amount of roundoff error. As a corollary, you are allowed to keep more than three digits if you wish, for any reason, or for no reason at all. This is makes sense because it introduces even less roundoff error. As another corollary, trailing zeros may always be rounded off, since that introduces no roundoff error at all.

Examples

1.80 may be rounded to 1.8, since that means the same thing. Conversely 1.8 can be represented as 1.80, 1.800, 1.8000000, et cetera.
These rules apply to intermediate steps as well as to final results.

These “house rules” apply unless/until you hear otherwise. They tell you what is considered significant at the moment. As such, they have zero portability outside the introductory class, and even within this class we will encounter some exceptions. Still, for now three digits is enough. There is method to this madness, but now is not the time to worry about it. We have more important things to worry about.

These rules differ in several ways from the “sig figs” rules that you often see in introductory chemistry textbooks.

1. First of all, these rules are much simpler.
2. Secondly, the conceptual basis is different. The “sig figs” rules in the textbooks are a crude attempt to keep track of uncertainty. Despite the name, those “sig figs” rules do not even attempt to express significance. (See section 14.3 for details on this.) The roundoff rules given here are actually based on significance, i.e. on the importance of the numbers and how they will be used downstream. They say nothing about the accuracy, precision, or uncertainty of the numbers.

This is important because of the following contrast:

1. Every time you write down a number, you have to write down a definite number of digits, and this almost always involves rounding off. Therefore you must have a roundoff rule or some similar guidance as to how many digits are needed. There are many cases when you want to write down a number without any indication of uncertainty.
2. A roundoff rule is necessary and harmless (unless abused). A “sig figs” rule that forces a connection between the number of digits and the uncertainty is unnecessary and harmful.
3. Thirdly, these rules (unlike the textbook “sig figs” rules) permit you to get rid of trailing zeros. This is important because it means these rules are consistent with the axioms of the decimal number system that we all learned in 3rd grade and reviewed every year since then: 1.80 is a rational number. It is by definition equal to 180/100, which when written in lowest terms is 9/5. Similarly 1.800 is by definition equal to 1800/1000, which is also exactly equal to 9/5.

Remember, these are roundoff rules. Do not confuse roundoff with uncertainty. Roundoff error is just one contribution to the overall uncertainty. Knowing how much roundoff has occurred gives you a lower bound on the overall uncertainty, but this lower bound is rarely the whole story. Looking at the number of digits in a numeral gives you an upper bound on how much roundoff has occurred. (This is not a tight upper bound, since the number might be exact, i.e. no roundoff at all.) At the end of the day, the number of digits tells you nothing about the overall uncertainty. Roundoff error is in the category of things that we generally do not need to know very precisely, so long as it is small enough. Uncertainty is not in this category, and will be explored in lab.

1. This is a selection of sections taken from: Denker, J. Uncertainty as Applied to Measurements and Calculations. Uncertainty as Applied to Measurments and Calculations (2011). Available at: http://www.av8n.com/physics/uncertainty.htm. (Accessed: 26th August 2016)