Anecdotes, Mnemonics, and other Tricks
in Basic Mathematics

I like these for some reason. Most of them are borrowed....

All-Star Trig Class. This mnemonic says which of the three "standard" trigonometric functions are positive in a given quadrant. All three of them are positive in quadrant I, only the Sine function is positive in quadrant II, only the Tangent function is positive in quadrant III, and only the Cosine function is positive in quadrant IV. There are variants on this, such as "All students take calculus", "Awful stupid trig class", and so on.

Approximations of π. 355/113 is a really good rational approximation of π. Try it on your calculator! Note that one may also write this as π≈113\355.

Backwards Integration. Integrating backwards is the opposite of integrating forwards. Read:

Before and After. Everyone has seen those tiresome print advertisements for weight-loss, skin-care, remodeling, etc. There's a before picture which looks a certain way and an after picture which inevitably differs drastically from the before picture. This is very useful when trying think about transformations in 2 or more dimensions.

Continuity. The freshman definition of continuity says that a function is continuous if you can draw the graph without lifting your pencil off the page. You don't want to be a freshman your whole life, do you?

Elementary Row Operations. Performing a sequence of elementary row operations to put a matrix in reduced row echelon form is like solving Rubik's cube.

Equilibrium Solutions. A stable equilibrium solution sucks and an unstable equilibrium solution blows.

Euler rhymes with boiler. I'm not German, but my name is.

Exact Values for Trig Functions. There's something special about the angles 0°, 30°, 45°, 60°, and 90°. If one knows the exact values of the sine function for these five special angles, then one can in principle determine the exact values of every trig function everywhere, or something to that effect. This trick gives the exact values of the sine function:

sin(0°)=½√0
sin(30°)=½√1
sin(45°)=½√2
sin(60°)=½√3
sin(90°)=½√4

Notice the sequence 0, 1, 2, 3, 4.

Eyebrow, Eyebrow, Nose, Mouth. A variant of FOIL. Probably need a graphic.

Fencepost Rule. If one erects a fence 200' long and places posts at intervals of 10' apart, how many fenceposts are required? One needs 21, of course. The fencepost rule says that you should add 1 in similar circumstances. This pertains in particular to partitions of intervals into subintervals and sigma notation for sums. If one divides an interval into n subintervals, then one needs to designate n+1 points in that interval, including the endpoints. Similarly, if the index of a sum begins at i=5 and ends at i=20, then one is summing 20-5+1=16 values. The fencepost rule also helps to explain the coefficients used in the trapezoidal and Simpson's rules.

FOIL. What does the two-sided distributive law have to do with very thin sheets of metal?!

Hairy People. For every person in the world, count the number of hairs on his or her head. After you determine all these numbers, multiply all of them together. Email me for the answer.

Hexagon of Trigonometry.

Homotopy. If I am out walking my dog, then she is on a leash. If we go near a pole, and she goes on one side and I go on the other side, the leash is blocked by the pole. She can't figure out how to get back to me because she does not understand homotopy. My dog does not understand homotopy, but neither do many people.

Ideals Suck. In particular, they suck products.

Imaginary numbers. All numbers are imaginary. I use this idea when I need to talk about elements of the field of complex numbers, especially eigenvalues. Formally, the real axis is no more "real" than the imaginary axis. All numbers are all abstract constructs in our minds, be they real, complex, "imaginary", etc.

Indeterminate Forms of Limits. The hardest thing about learning L'Hospital's rule is learning that you don't usually need it. Thus, for instance, L'Hospital rule only applies when the limit has the form 0/0 or ∞/∞. In this vein, as far as I have been able to tell, there are only seven types of indeterminate forms for which you need either L'Hospital's rule or some algebraic trick which leads to L'Hospital's rule. Commit them to memory:

0/0, ∞/∞, 0⋅∞, 00, 1, ∞0, and ∞-∞

If a limit has one of these forms, then you need to do some work to evaluate it.

Induction. Induction is like falling dominoes.

Old enough to know. At some point you once learned that subtraction is the same as addition. In a similar vein, you also learned that division is the same as multiplication. You are now old enough to learn that multiplication is the same as addition.

The Pythagorean Theorem. The canonical proof-by-picture:

Reciprocals and Magnitude. The basic rule states:

1/Big = small and 1/small = Big.

In words, the reciprocal of a very large number is a very small number and the reciprocal of a very small number is a very large number. In this case, "small" means near zero and "Big" means far from zero. This is a nice rule because the equation stating the rule is concise and coherent.

The sun rises in the East. It is customary in mathematics to use the postive part of the x-axis as the polar axis. This means all angles are measured against this particular ray. But in what direction? Clock-wise or counter-clock-wise? It is customary in cartography to have North at the top of the map and East on the right-hand part of the map. The sun rises in the East, so we measure angles in the counter-clock-wise direction.

Short Exact Sequences. I need a graphic for this one. The Greek letter ι has a little hook and the Greek letter π has two stems. Similarly, an injection arrow has a little hook, and the projection arrow has two little arrowheads.

64/16=4. (Cancel the 6's!) This illustrates that one can get to the right answer through desperately faulty reasoning. The ends do not justify the means.

Socks and Shoes. Most groups are not commutative. One has to remember this when one wants a formula for the inverse of a product of two elements. In any group, whether or not it is commutative, one is always allowed to write (ab)-1=b-1a-1. In other words, if you want to perform the inverse of putting on your socks and shoes, first you take off your shoes and then you take off your socks.

SOHCAHTOA.
Sine is Opposite over Hypotenuse.
Cosine is Adjacent over Hypotenuse.
Tangent is Opposite over Adjacent.

Spheres. The Hopf fibration of the hypersphere amounts to the fact that the quotient space S3/S1 is homeomorphic to the ordinary sphere S2. In symbols, this is

S3/S1≈S2.

It looks like one is merely "cancelling" a factor of S.

Please Excuse My Dear Aunt Sally. This tells which operations have priority in a mathematical expression.

Vector Addition. The rule for adding vectors is to place the tip of one at the tail of the other. This is similar to what dogs do when they meet.

Zeroes of the Exponential Function. The only way ez could be zero was if z was in Hell.