Symbols in science

In science, we use symbols in ways that can be very different from the way you've seen them used in your math classes. This can be quite confusing when you first encounter it. Let's consider some examples to make these differences more concrete. 

Mapping physical meaning onto symbols

As discussed in our page, How math in science is different from math in math, in science our symbols don't just stand for numbers, they stand for physical quantities.

Here's a "biologically relevant" problem from a calculus class that is trying to be "biologically relevant" and missing some critical issues. Despite on the surface using measurement terms, it treats the quantities in the problem as if they were numbers, not measurements in the physical world.

The population density of trout in a stream is

$f(x) = 20\frac{1+x}{x^2+1} $

where $f$ is measured in trout per mile and $x$ is measured in miles. $x$ runs from 0 to 10. Write an expression for the total number of trout in the stream. Do not compute it.

Although this is a perfectly reasonable problem for a math class (to see if you know how to write an integral), in a physics class, any student writing an equation like this would lose significant credit for failing to be consistent about dimensions (and we've shown this with an "unhappy" emoji). If $x$ is measured in miles, this means that the "1" in the numerator has to mean "1 mile", not just "1", and the "1" in the denominator has to mean "1 mi²", not just "1", So in this equation "1" is used twice, but means two different things, neither of which is "1".

We can fix this for a science class by using parameters — symbols that stand for a measured value with unit. So we might rewrite the equation by doing the following:

• Instead of a generic "$f$", choose the symbol "$t$" to remind us that it is the "density of trout" and has units of "trout/mile".
• Instead of the number "20", write the symbol λ to stand for "trout density parameter" — a constant with value of "20 trout" (since we will see that the "per mile" needed is introduced by the distance function following it).
• Replace the "1" in the numerator by a parameter "$a$" that has the value "1 mile"
• Replace the "1" in the denominator by a parameter "$b^2$ that has the value  "1 mile²". (We make it "$b^2$ rather than just "$b$ since it is added to a square distance and that makes "$b$" the same kind of quantity as "$a$" and "$x$.)

Since in this case, $a=b=$ 1 mile, we could only have used "$a$", but we have chosen to include 2 separate parameters so we could vary them independently if we chose. This is why equations in physics have so many symbols! Our resulting equation is this:

$t(x) = \lambda\frac{a+x}{x^2+b^2} $

a much nicer equation for a physics class!

The smiley face form of our equation has the advantage of having correct dimensions. We could easily change units if we wished (say to kilometers), but we could also imagine varying the parameters a, b, and  λ to see how our results changed.

Since a lot of what happens in biology is about changing parameters (coming from evolution, mutation, or diseases messing with values of chemical concentrations), this is particularly useful.

In making the transition from math in math to math in science, you need to learn to be as comfortable with the second form of the equation as you are with the first.

Annotating symbols: subscripts, superscripts, and vector markers

Assigning physical meaning to symbols is just the beginning. We like to develop a kind of "code language" of symbols to remind ourselves just what kind of physical quantity a symbol stands for.

Thus, in many examples we use the symbol "$t$" to stand for time. We borrow, "$x, y, z$" from math practice to stand for positions = distances from an origin (now having units). We use "$F$" to represent a force, "$T$" to represent a temperature, and "$E$" to represent an energy. (Not all of our canonical assignments match common speech like this, unfortunately.)

That sounds nice in principle, but in practice we deal with complex problems. We often have many different quantities of the same type even in a single problem.

Let's show how this works with some examples that will appear later in the course. You don't have to understand the details now, only get the point that you have to pay careful attention to the annotating markers on symbols.

Subscripts and superscripts

We might have to consider a starting time and an ending time. We might have to consider different kinds of forces acting on many different objects. You can get into real trouble if you are considering the weight of three different objects, and you write the weight for each object as "mg" despite the fact that there are three different masses. (You will learn that an object's weight is mg -- its mass times the gravitational field, g ~9.81 N/kg on the earth's surface.)

We get around this by adding notation to our symbols. This makes them look more complicated, but it lets us keep straight what they mean physically.

For example, we might have three different forces to talk about when thinking of the motion of an object A:

$F^N_{B\rightarrow A}$ = normal (contact) force object B exerts on object A

$F^W_{earth\rightarrow A}$ =weight (gravity) of object A produced by the pull of the earth

$F^{net}_A$ = net (total) force acting on object A.

These mean three different things and they play different roles. We can't just write "$F$" for all the forces. It helps to put enough markers on the symbol so that you can easily remember what physical quantity it stands for and distinguish it from other similar kinds of quantities in the problem. (Of course, if there is only one force in the problem, it's OK to just call it "$F$".)

Vector markers

Subscripts and superscripts are just a start. We don't only have different units to represent different kinds of quantities. Some of our quantities represent not just a measurement but a measurement with an orientation or direction. We mark these quantities by calling them "vectors" and putting a little arrow above them. Sometimes we are interested only in the direction of a vector, not its magnitude. Then we might write a "unit vector". We would annotate them something like this:

$\overrightarrow{R}_{12}$ = displacement from point 1 to point 2 (magnitude and direction)

$R_{12}$ = displacement from point 1 to point 2 (magnitude only)

$\hat{R}_{12}$ = displacement from point 1 to point 2 (direction only)

Again, our point here is not to teach you these now, but just to warn you that it is important to pay attention to the little markers on the symbol.

The "change" operator

A third kind of marker that is sometimes added to symbols is a "change" operator. Since a critical difference in our discussions will be separating values of a quantity, changes in a quantity, and rates of change of a quantity (See Values, change, and rates of change.) we mark that we want to stress a "change" by putting a symbol in front of the quantity we want to change. We use the "$\Delta$" and the "$d$" symbols this way:

$\Delta f = $ the change in the quantity $f$

$df = $ a (very small) change in the quantity $f$.

You should read "$Δf$" or "$df$" as a single symbol. (The symbol "$df$ here is NOT a product of a delta or a $d$ and an $f$.)

Symbolic homonyms

Although symbols allow us to express complex relationship more clearly and precisely, and to carry out complex chains of reasoning, they aren't perfect. The same symbol may sometimes be used to express different things in different contexts.

Math symbols are a kind of language and as such share many of the problems all languages have. One problem that you will have to deal with is symbolic homonyms. A homonym is where the same sound represents two (or more) different words. For example, in English, "their", "there", and "they're" all mean different things but sound the same. You just have to tell by context — the situation in which it's used. A symbolic homonym is where the same symbol can stand for different things in different contexts. For example, under an integral or in a derivative, "$df$" means a small change in "$f$". In another context (say an equation for work) it could mean a distance "$d$" times a friction force "$f$".

We have LOTS of physical quantities in the world and only a small number of symbols (26 English + 24 Greek), so we make many of them do double duty. In this class you will see the symbol $k$ standing for a spring constant (in mechanics), Boltzmann's constant (in thermodynamics), and a wave number (in wave mechanics). 

To keep these apart and write correct equations, you need to learn to think of equations in science NOT as mathematically abstract symbols standing for a number, BUT rather for a physical measurement — a quantity that has a meaning in the real world.

Joe Redish and Mark Eichenlaub 8/10/15