Friday, January 2, 2015

The math problem inside every math problem

A great deal of high school math is about solving problems. Yet for mathematicians throughout history and right up to the present, math is about proofs. Are students doing one kind of math, while mathematicians are doing something quite different? Not really. The kind of math that professional mathematicians do is much more challenging and abstract than a high school math problem, but in its essence the math is the same no matter who does it.

How is this possible? The steps in solving a math problem feel like the steps in a proof, but in some ways the process feels different. What is the theorem to be proved, and where is it? It turns out that theorem and its proof are both hiding just a bit inside the math problem.

Let’s look first at solving equations, perhaps the most classic form of algebra problem. A very simple equation to solve could be something like:

4 ⋅ x + 3 = 15

which has the solution:

x = 3

We could solve it by subtracting 3 from both sides and then dividing both sides by 4. The steps would look something like this:

4 ⋅ x + 3 - 3 = 15 − 3
4 ⋅ x = 12
(4 ⋅ x) / 4 = 12 / 4
x = 3

There is indeed a proof lurking inside the solution to this problem! How can this be? None of the statements in the proof is a theorem. A theorem should always be true, like the commutative law is true for all real numbers. None of these statements is true unless x is equal to 3, though each is true if x is indeed equal to 3.

We could say there are two theorems here. Stated in words, one is that for all real numbers x, if 4 ⋅ x + 3 = 15 then x = 3. Mathematical notation for this is:

4 ⋅ x + 3 = 15 ⇒ x = 3

(You can read the arrow as “implies”.)

When we solve equations, we usually work in this manner, starting with the problem statement and applying rules to get a sequence of statements, each true based on earlier steps.

The problem statement is assumed at each step. So the statements being proved along the way technically are like this:

4 ⋅ x + 3 = 15 ⇒ 4 ⋅ x + 3 - 3 = 15 − 3
4 ⋅ x + 3 = 15 ⇒ 4 ⋅ x = 12
4 ⋅ x + 3 = 15 ⇒ (4 ⋅ x) / 4 = 12 / 4
4 ⋅ x + 3 = 15 ⇒ x = 3

And the final step is the first theorem being proved. Repeating the assumption every time is tedious, so you can see why the assumptions are generally not written at each step in solving a problem!

The other theorem is that for all real numbers x, if x = 3 then 4 ⋅ x + 3 = 15. In mathematical notation this is:

x = 3 ⇒ 4 ⋅ x + 3 = 15

For many problems like this one, we can confirm this theorem just by “plugging in” the value 3 for x in the equation 4 ⋅ x + 3 = 15. In math textbooks this is sometimes called “checking your work”. It can help you avoid mistakes, and in some problems is necessary even if all of your steps are done correctly! We will say more about that in another posting.

Mathematical notation also lets us combine the two statements into a single theorem. In mathematical notation it looks like this:

4 ⋅ x + 3 = 15 ⇔ x = 3

This has an arrow pointing in both directions, and means that 4 ⋅ x + 3 = 15 ⇒ x = 3 and also that x = 3 ⇒ 4 ⋅ x + 3 = 15. That is why the arrow in this one points in both directions. In mathematics, a statement of this kind is called an equivalence.

Solving equations is mathematically the same as finding an equivalence between the given equation (or equations) and the "solution". In the solution, each variable should be shown equal to some expression, and the expression should be reasonably simple — the simpler the better.

There are other kinds of math problems too, including simplification of expressions. More on this later.