The idea of algebra: Unknowns and relationships

Most of you will already have studied a lot of math — algebra, geometry, trig, and calculus — and most of you have been successful in these studies. Some of you will handle the transition to math in science just fine. Yet we find many students get stuck in applying what they have learned in math to a science class. Part of the reason is that in science classes we have a different goals than we do in math (as discussed on the page How math in science is different from math in math). 

Most of the math we use in this class will be straightforward algebra. Most of you can do this. But sometimes, we find that students have learned to manipulate symbols without getting the conceptual point of algebra. In this class, we rely heavily on the conceptual underpinnings of algebra.

Using algebra in the context of science

Three critical concepts you need to understand about algebra to use algebra successfully in science are

• An algebraic symbol is a container for holding a value, which may or may not be known.
• An algebraic equation is a relationship among variables, constants, and parameters. 
• An algebraic equation says that two things expressed in different ways are actually the same. 

An algebraic symbol is a container for holding a value, which may or may not be known.

Sometimes algebra uses abstract symbols to emphasize that something is not known and needs to be solved for. If you have an equation for an unknown and you want to find its value, you have learned procedures in math to solve for it. In this case, it's useful to think of an abstract symbol as a container — something holding a value that you may not know.  

But in this class, you will not only solve given equations, but you will also be considering situations where our container doesn't just hold one unknown value but we are interested in what happens if the value in the container changes.

For example, we might be considering the viscous force acting on a blood cell as it moves through the blood. This force depends on a variety of parameters in the physical situation: the radius of the cell, $R$, the viscosity of the blood, $\mu$, and the speed of the cell through the fluid, $v$. We have an equation for this force that looks like this:

$$F_{viscous} = 6\pi \mu Rv$$

In this case, we are not necessarily calculating anything at first. We may be using this force in the context of other forces and we could be interested in how the balance of force changes when the viscosity of the blood increased (for instance due to smoking tobacco).

This approach is also useful when we are trying to solve a physical problem with multiple unknown quantities. When asked to do this, the way to move forward is to give a name to all the things that you might need to know to solve the problem — create a container for the value and work with it whether you know what's inside it or not. (And be sure to give symbols that represent different quantities different names!) Then write down all the relationships that you can find to describe the physical situation. You might wind up with a set of equations for multiple unknowns — but a set that you can solve. 

Don't be afraid to give a name to some quantity that you might not know. Remember that an algebraic equation expresses a relationship among algebraic symbols. That relationship holds whether we know the values or not and may be useful even when we don't.

An algebraic equation is a relationship among variables, constants, and parameters.

Sometimes algebra uses abstract symbols to emphasize relationships among different quantities even when they are known. Often we will use abstract symbols to represent parameters — symbols that stand for something in the physical world that in a given situation we may want to consider to be a constant, but where we might think about it taking on different values.

The value of a mathematical equation is often that it expresses the relationships among physical observables, not only as a way to calculate an unknown.

Consider the example of Fick's law -- the equation that tells how a chemical diffuses through a fluid. It says that if we have a small bit of chemical emitted into a fluid, if it can diffuse out in three dimensions, then the size of the blob of chemical will spread according to the rule

$$\langle r^2 \rangle = 6Dt $$

where r stands for the size of the blob, t stands for the time it's been diffusing, and D is a constant known as the diffusion constant.

This is a relationship among three things: two variables (the size of the blob and the time) and a constant. The form of the relationship tells us something important: that as the time grows, the size doesn't grow as quickly as the time does. If t doubles, then the average value of r² doubles, which means that r only grows by a factor of the square root of 2 (that is, by a factor of 1.4...). This has important consequences for the design of biological structures (and we'll be doing a lot more with it during this course). The critical point to be made here is the following:

A mathematical equation is not just a way to calculate something. It tells you how various quantities depend on each other -- their functional dependence.

In the example above we looked at r and t as variables and D as a constant. But working through the physics of diffusion tells us how D depends on other properties of the system (the Stokes-Einstein equation):

$$ D = \frac{k_BT}{6 \pi \eta R} $$

Here, the diffusion constant is not just a constant, but is expressed in terms of other parameters of the system — the temperature, T, the radius of the diffusing particle, R, the viscosity of the fluid it's diffusing in, η, and some constants. (k_B is called Boltzmann's constant.) This relation tells us that the diffusion constant can be made bigger (so things will diffuse faster) by increasing the temperature, decreasing the size of the particle that is diffusing, or decreasing the viscosity of the fluid it is diffusing in. This kind of analysis can be critical in understanding how to design experiments or how evolution works on the parameters of a system to select a more effective result.

An algebraic equation says that two things expressed in different ways are actually the same.

The third important concept about algebra is that an equation represents an equality. An equation says that two expressions — that may look very different or that we think of in different ways — are actually the same. This is particularly important in science.

We think of the diffusion constant as something that controls how fast diffusion takes place. The Stokes-Einstein equation above says that the messy combination of parameters on the right actually is the same thing as the diffusion constant, even though it looks different. This tells us that anywhere we have an expression with the diffusion constant, we could actually use the expression on the right. (Why we might want to do something like that will become clearer as we move through the course.)

Seeing the two sides of an equation as the same has a very practical consequence: It tells you what you can do with an equation. Often in math you are taught procedural rules for solving equations that may seem to make no sense, be arbitrary, or even bizarre. But if you look at these rules as saying that both sides of an equation say the same thing, they become not just sensible, but even obvious. 

Since both sides of an equation represent the same thing, you can only perform manipulations on an equation that maintain the balance between the two sides, that is, the same manipulation on each side.

You can:

• do the same things to both sides of the equation (add, subtract, multiply by, or divide by something)
• multiply one side of the equation by an expression that you know to be equal to 1 (for example, by a/a)
• add an expression to one side of the equation that you know to have the value 0 (for example, a - a)

These all make sense if you are thinking about maintaining the balance represented by the equals sign.

One example of keeping the equality is the heuristic rule, "you can move something from one side of the equation to the other if you change the sign." This sounds pretty bizarre. Why should we be allowed to do that? But if you think about it in terms of doing the same thing to both sides of the equation it makes sense.

Consider the equation from the beginning of this webpage: ½ x + 7 = 5. To solve this, we might use the heuristic and bring the 7 to the other side and change its sign. But a better way to see what we are doing is to subtract 7 from both sides, maintaining the balance. Then we get:

$$½ x + 7 = 5$$

$$½ x + 7 - 7 = 5 - 7$$

$$½ x = -2$$

$$x = -4$$

and we can easily see where the rule comes from.

A second example is the heuristic rule: "to clear two equal fractions, cross multiply."  That is, if you have $a/b = c/d$ you can get rid of the fractions as follows:

This seems rather strange. But if you looks at the first equation and say, "I want to get rid of the denominators. Since the opposite of dividing is multiplying, let's multiply both sides of the equation by bd," then the result becomes obvious.

The $b$ cancels on the left and the $d$ cancels on the right and you are left with the same result — but know you can see why you do it.

Heuristic rules work if you know when to use them, but for more complicated situations where you have no rule, if you understand the processes that led to the heuristics you will still be able to figure out what to do.

Be careful! = must mean "equal", not "goes to"!

Some students get into a lot of trouble because of a bad habit they picked up in earlier math classes. This is to use the equal sign as a sort of "goes to" marker rather than maintaining it as a true equality.

Treat equals signs as sacred. NEVER write an equals sign between two quantities that are not in fact equal. 

Here's what I mean. Consider our equation again:

$$½x + 7 = 5$$

It's fairly obvious what you have to do. Subtract 7 and multiply by two and to get the $x$ by itself on the left. I have seen students write something like:

$$½x + 7 = 5 = 5 - 7 = -2 = 2 * (-2) = -4$$   
(RIGHT ANSWER BUT WRONG SINCE 5 ≠ 2 ≠ -4 )

saying, "Let's see. I first have to move 7 over to the other side and subtract it. That makes the 5 into 5-7 or -2. Now I have to multiply by 2 to get the x by itself, this make 2 * (-2) or -4. The answer is -4."

Although the answer (x = -4) is correct, a lot of the equations do not have the same value on both sides. For example, 5 is not equal to 5 -7. The student has used the equal sign as a "next step" marker rather than a true balance.

Doing the steps in the opposite order they would yield the wrong result. "Let's see. I have to multiply by 2 to get x by itself and then I have to subtract 7 in order to get x by itself so: 

$$½x + 7 = 5 = 2 * 5 = 10 = 10 - 7 = 3$$  
(WRONG ANSWER AS WELL AS WRONG PROCESS)

Even if you would never make that mistake, it is a very bad habit to write an equal sign that connects two things that aren't equal. You can't go back to check your work later and, on an exam, you might look back and assume the equality holds.

We will be doing math that is much more complex than this. Setting yourself up with wrong statements in the middle of your calculation strongly increases the probability you will make a mistake. Don't do it! (And your grader may well deduct points if you write any equation that is not true.) 

Joe Redish 9/22/14