Elizabeth Drellich

# There’s an equation in this paper! Now what?

Just like when you come across a bear in the woods, the first rule of math is “don’t panic!” But just telling people not to panic is never a great way to deal with an issue, so here are some more concrete ways to approach mathematical equations that appear in scientific papers.

Ideally when you are reading a scientific paper and an equation is included, you would be able to understand the equation and continue reading. Unfortunately almost none of us live in that ideal world. I know a LOT of math and my first reaction to a collection of symbols can still be to panic and assume I will never understand what they mean. Anyone reading scientific papers will eventually, if not frequently, come to a line of mathematical symbols that do not make sense.

Three common approaches to unfamiliar equations in a paper are to

1. skip the math and continue reading the paper, or
2. assume that the rest of the paper cannot be read without understanding the math and stop reading entirely, or
3. assume that the rest of the paper cannot be read without understanding the math and spend a lot of time and energy trying to understand every part of the equation before continuing.

With the first approach, you might miss out on a part of what the authors were trying to communicate, but with the second, you miss out on the rest of the paper. In neither case are you getting everything you can out of the paper. The third approach might be a way to extract all of the information from the paper, but learning or relearning math concepts is not necessarily the best use of your time or brainpower. This article is about what to do instead.

# Assess the situation

The first thing to do is read the next few sentences after the math equation. They, along with the few sentences before it, may explain what the authors are trying to say with their equation. It is always a bummer to spend a lot of time and frustration figuring out an equation only to see it clearly explained in the next paragraph, so make sure you read on a bit before even thinking about the equation.

You can practice this by continuing to read the words after this equation:

Good job! Because you kept reading, I am happy to tell you that the equation you just skipped over doesn’t have any meaning whatsoever, at least not yet. Now let’s see that equation in context in the paper I made up about bears:

We are using the following model for a bear’s daily intake of salmon, $s_t$ on day $t$ based on the work of Yogi (1961) and Pooh, et. al. (1926):

The parameters $\alpha_0=1.432, \alpha_1 = 1.243$ and $\beta = .0076$ are taken from Goldilocks (1837). The bear’s mass in kg ($m$) and age in years ($a$) have large impacts, as do the annual population of salmon in the river studied ($x$) and the number of bears in the area ($n$).

Okay, now that you have read some sentences surrounding the equation, it is time to figure out whether you need to dig more deeply into it.

If you followed the flow chart and it told you to continue reading with confidence, you have my full support in doing just that! If it turns out later that you need to understand more about this equation - maybe it is referred to later in the paper or you want to use it in your own research - the paper will still be there and the next time you will have a different path through the flow chart.

The rest of this article is about what to do when you need to learn more about the equation. The first thing to do is to identify what the variables are. This can be particularly tricky because there may be a lot of symbols that are not variables. Again, the sentences just before and after the equation are going to be a big help.

# Initial investigation

Start by making a list of what each variable represents. If you are taking notes on the paper, highlight each of the variables in the equation. Each variable should be described in the text surrounding the equation. Usually one of the variables, the output variable, will be alone on one side of an equals sign.

In our made-up paper about bears, the sentences before and after the equation describe these variables:

$s_t$ number salmon a bear eats on day $t$
$m$ mass of the bear in kg
$t$ date
$a$ age of the bear in years
$n$ number of other bears within a 5 mile radius
$x$ river’s annual salmon population

They also describe three other symbols, $\alpha_0$ (pronounced “alpha naught” or “alpha zero”), $\alpha_1$ (“alpha one”) and $\beta$ (pronounced “beta”). These symbols are given specific numerical values, each one is simply a number. It may seem confusing to use symbols instead of numbers, but this is a way to simplify what is written down so that the equation isn’t full of numbers like 1.432 (which can be pretty hard for a reader to distinguish from, say 1.243 or .1243).

There could also be a lot of non-variable symbols. At this point you can completely ignore anything that is NOT a variable. To emphasize that we are ignoring all of the non-variable symbols, replace the equation with the sentence “The output variable is a function of input1, input2, and input3” or however many input variables there are. You can also write that sentence symbolically using only the variables and a function name (“f” is a very common symbolic name for a function, but you can name your function whatever you want).

In our example about bears, we would say “The number of salmon a bear eats each day is a function of the bear’s mass, the date, the bear’s age, the number of other bears nearby, and the number of salmon that pass each year.” We could write that sentence symbolically as:

Hopefully this equation looks much less scary than the original. And it might be tamed enough that you can continue reading the paper and get a whole lot more information out of it than if you had skipped the math entirely. But maybe you need a deeper understanding of this paper’s equations than this. How can you figure out how much time and energy you should be spending on this particular equation? This is going to be an iterative process. If a paper turns out to be extremely relevant to you, you will likely read it several times, and each time you might need to know more about the mathematics in it.

Depending on your work, you may have some equations you need to learn a whole lot more about. Learning iteratively (by learning a little more each time) is going to help you have an idea of what the equation is saying at each step.

It is ALWAYS okay to say “I don’t fully understand this math, but I have enough of an idea to keep reading the paper for the moment.” Don’t let imperfect understanding of some math keep you from a good enough understanding of the science!

# Learning more

Each time you want to learn more about an equation, try asking a different type of question about it. This will eventually be its own article, but if you are looking to get started right now, here are some categories you could look at next:

• Can one side of the equation be broken up into the sum of a few smaller expressions? Which of the variables are multiplied together or inside the same function? Which variables don’t interact? In our example, notice that everything inside the parentheses multiplied by $\frac{x}{365n}$. This is saying something about the number of salmon per bear.

• Look at the constants or numbers in the equation. Are they positive or negative? How does the sign impact the output variable? Are any of the constants related to each other? In our example $\alpha_0$ and $\alpha_1$ are likely related, while $\beta$ may not be related to either. The number 365 shows up twice, what is it doing in this equation?

• Look at the functions inside of the equation. Are there trigonometric functions like sine and cosine? Logarithmic functions? Square roots or powers? Are variables appearing in denominators or exponents? Each of these types of functions has different implications for how the input variables inside of them impact the output. Our example has $\sin (\frac{2\pi t}{365})$. What sort of oscillation does that imply?

• Are there summations, integrals, or derivatives? If there are derivatives, what is changing with respect to something else? If there are integrals or summations, what are you totaling up?

• Are there matrices or vectors? What role are these playing in the equation? They could be variables, constants, or functions. (oh my!)