The more you understand, the less you have to memorize.

A good example is trigonometric identities, of which there are quite a number. Should a student memorize trigonometric identities? Well, at first, it is probably wise to memorize a few of them. Part of a teacher’s job is to help students identify what is essential to memorize, and what is more peripheral. In the case of trig identities, the most important ones are

$(1) \quad \sin^2 \theta + \cos^2 \theta = 1$

$(2) \quad \sin (A \pm B) = \sin A \cos B \pm \cos A \sin B$

$(3) \quad \cos (A \pm B) = \cos A \cos B \mp \sin A \sin B$

Even the $\pm$ signs are superfluous in equations (2) and (3); one can remember the top signs only, and make use of the symmetry properties of the sine and cosine functions (i.e, sine is odd and cosine is even). That is, to derive the bottom signs from the top signs, just replace $B$ by $-B$, and use the facts that $\sin (-B) = -\sin B$ and $\cos (-B) = \cos B$ on the right sides of equations (2) and (3).

Now divide each term of equation (1) by $\cos^2 \theta$ and you will get another standard identity; similarly if you divide each term of equation (1) by $\sin^2 \theta$.

Let $B = A$ in equation (2) (with the plus signs), and you will instantly derive another important identity, $\sin 2A = 2\sin A \cos A$. Similarly for equation (3); then combine the result with equation (1) to derive the three versions:

$\cos 2A = \cos^2 A – \sin^2A = 2\cos^2 A – 1 = 1 – 2\sin^2 A$

Rearranging the last two equations, one can derive the following identities (important for integrating even powers of sine and cosine):

$\cos^2 A = \dfrac{1 + \cos 2A}{2}$

$\sin^2 A = \dfrac{1 – \cos 2A}{2}$

And so on; but you get the idea: Knowing a few basic identities, you can rapidly derive many more. If you practice these kinds of calculations, you will get good at them, and will be able to effect them very quickly, even on exams.

But is even the three basic identities too much for you to memorize? Well, once you learn a little bit about complex numbers, including Euler’s formula

$e^{i\theta} = \cos \theta + i \sin \theta$

then you don’t even have to remember them! (Of course, you do have to remember Euler’s formula, but there is a nice picture to help you!)

For example, consider

$e^{i(A + B)} = e^{iA}\cdot e^{iB}$

which follows from the properties of exponential functions. Apply Euler’s formula to each side of the previous equation to get

$\cos (A + B) + i \sin (A + B) = (\cos A + i \sin A)\cdot (\cos B + i\sin B)$

Expanding the right side, we get:

$\cos (A + B) + i \sin (A + B) = \cos A \cos B + i \sin A \cos B + i\cos A \sin B – \sin A \sin B$

Equating real and imaginary parts on each side of the previous equation, we end up with the trigonometric identities in equations (2) and (3). Voila!

To derive equation (1) in a similar way, use

$e^{i\theta – i\theta} = e^0 = 1$

Then write the left side of the previous equation as

$e^{i\theta} \cdot e^{-i\theta}$

and use Euler’s formula on each factor; within a few lines of algebra you will have it.

For more along these lines, see this page at John Baez’s site.

Returning to a general discussion, I have always advised my students to memorize the absolute minimum necessary, but make sure to know that bit cold. And the best way to do this is to practice using what you wish to memorize in solving exercises and problems. In this way, you naturally memorize just by repetition of use. Just staring at a formula and trying to will it into memory never worked very well for me, and I don’t think it typically works well for others either. And it wastes precious time that would be better spent solving problems or reading around the subject so as to open the mind to new concepts.

Another excellent way to memorize things is to represent the thing to be memorized by a picture, which is much more memorable than a collection of words. A good example is the mean value theorem. As a student, I found it very easy to remember this theorem just by remembering the picture. One can remember such pictures for a lifetime, whereas theorems remembered merely as collections of words tend to fade a lot sooner.

The best way to remember something is to understand it thoroughly, from many different perspectives. This takes time and work, but ultimately it is the most satisfying. For example, one can memorize the multiplication table by rote (I did, but then I grew up at a time when this was extremely common). But once you’ve memorized it, that’s all you know. But if you examine the table carefully, noting the many patterns that are present in it, then you will know a lot more.

For instance, you will notice that every number in the multiplication table that is one step NE or SW from a number on a main diagonal is *one less* than the nearest number on the main diagonal. For example, $4 \times 6 = 24$, which is one less than $5 \times 5$, $8 \times 10 = 80$, which is one less than $9 \times 9$, and so on. You might speculate that this is always true, no matter which whole numbers you use, as long as you follow the pattern. And you might convince yourself that this is true using an area model: Take a square “floor” that has $n$ tiles on each side. If you take a row of tiles off the North end of the floor and move the tiles so that they are all on the East side of the floor, you will end up with a rectangular floor of dimension $(n – 1) \times (n + 1)$, but with an extra tile sticking out at the East end of the North side of the floor. Algebraically, the fact that the number of tiles is the same in each figure can be represented by $n^2 = (n – 1) \times (n + 1) + 1$, which can be rearranged to get a common factoring pattern learned in high school: $n^2 – 1 = (n – 1) \times (n + 1)$.

The moral is that by understanding the multiplication table, not just memorizing it, your knowledge can extend far beyond the table (for instance, you can immediately determine that $19 \times 21 = 20^2 – 1 = 399$), but you also prepare your mind to understand more advanced material. In our example, noticing a simple pattern in the multiplication table prepares you to understand a factoring pattern when you tackle algebra in high school.

To continue the story about patterns in the multiplication table, you can go two steps away from the main diagonal; this corresponds to taking two rows of tiles from the North end of the floor and placing them on the East end of the floor (except now a $2 \times 2$ block of tiles will stick out). Then continue with three rows, four rows, and so on. This gives a concrete model for the factoring pattern $n^2 – m^2 = (n – m) \times (n + m)$.

But, of course, to each his (or her) own. So experiment, play with mathematics, and then do what works best for you.

(This post first appeared at my other (now deleted) blog, and was transferred to this blog on 25 January 2021.)