Use Good Notation (Visual)

Let me start with a story. This one is very famous, I promise.

You probably know the story of how calculus was developed, the back-stabbing story of Newton & England vs. Leibniz & the rest of Europe. Way back in the fall of 1666, Isaac Newton (he hadn't yet been knighted, so he wasn't a sir) thought about particles traveling along paths. To accurately calculate such things, he invented the concept of fluxions, what we would now relate to derivatives. These he represented with $\dot{x}$, a dot on top, and we still use this notation to represent differentiation in physics, usually with respect to time. To represent antidifferentiation, he just put the dot on the other variable. Then, Leibniz came along with similar ideas. He did not know about Newton's ideas, because Newton basically never published anything. The main difference was he represented derivatives with the $\frac{d}{dx}$ operator and antiderivatives with $\int$, a deformed "s" for "sum".

The story continues, but can you guess what notation most of Europe chose?

Almost every country chose Leibniz's, the lone holdout being England because Newton was English. The reason for this was simple. Leibniz's notation had two properties: it was visually intuitive and visuallyadaptable.

First, the intuition. At it's heart a derivative is a slope, a ratio of two quantities. The $\frac{d}{dx}$ shows this visually, even if it is technically not a fraction. The "d" in the derivative could also clearly be seen, and it reminded one of "difference" or "derivative" or "differentiation". Similarly, the "s" in the integral reminds us the integral is fundamentally a "s"-um.

But Leibniz's notation really shows its value in it's adaptability. The integral notation can be easily changed to represent different concepts. Stringing two integrals together say "Take two consecutive integrals!" Adding numbers or expressions to the top or bottom of the integral says "Let this be the boundary for your area calculation!" The derivative is even better. Not only can the visual representation of a derivative as a fraction allow us to think of isolating the "dx" or just the "d" (which in fact leads us to think of the "d" as a small geometric patch, or perhaps a minuscule difference, or etc.), but also, we change the fraction or add more letter to the fraction. If I take the derivative of y with respect to x, I just write $\frac{dy}{dx}$, and I can treat dy as its own thing. What's more, the higher derivative compositions just involve putting an exponent, so $\frac{d^\pi}{dx^\pi}$ is the $\pi$th derivative (Irrational derivative? Try taking e to the power of the derivative operator.). I can even change the d to something else, like $\partial$ or $\Delta$, to symbolize something related, like a partial derivative. Thus, Leibniz's notation is highly adaptable and promotes creative thought.

The best notation clearly and concisely shows your intent, yet allows room for change.

For example, if you wanted to represent time, you would probably choose $t$. Then, the initial time could be $t_{initial}$ or $t_i$. If you are choosing a variable, the variable should either be something like $x, y, z$, or should be the first letter of the relevant word, as long as it that is not i or e, which are numbers. There are some conventions you should follow. There is an implied "usually" in the following statements.

The first few letters (a, b, c, d...) are (usually) constants. The last few are variables. The middle few are parameters. i-n are integers. i, j, and k are counters, i.e. they keep track of the numbers 1 to whatever, like in summation notation. You can use them for number that are counted from something to something else. They are also quaternions. p, q, r, s are roots. a, b, c, d, p, q, r, s are used in fractions. a and b are used in sequences and series and sums

Here are some specific uses. e and i are numbers. Very important ones, I hasten to add. Probably the most important, I desperately try to correct. t means time. s mean position (I'm not sure why. It shouldn't mean speed, which is usually v). It is also the standard variable for studies on the Riemann hypothesis and the zeta function ($\zeta()$). v means velocity. This has been stolen for use as speed, but they are not the same. a means acceleration. And Ackermann. And Abel. And an associative A-list A-name assembly. $\mathbb{N}$, $\mathbb{Z}$, $\mathbb{Q}$, $\mathbb{R}$, and $\mathbb{C}$ mean natural numbers, integers (this is from german zahl, says math folklore, which suprisingly exists), rationals ("q"-uotient), reals, and complex numbers.

There are a host of other functions you can memorize with their key letter. Most aren't even English.

Now that you know the key to good symbolic notation, lets examine some very famous notation.

A relational operator is just something that compares two other things. For example, there is the greater than operation $>$ which actually goes from large to small. The famous equality sign, $=$ was originally conceived because it looks like two equal and parallel lines. Combining the two logically gives $>=$, the "greater than or equal to sign." In fact, the equals sign was popularized by Robert Recorde. He also popularized the plus sign, which is actually related to the ampersand ("&") symbol. Both are abbreviations of the Latin "et" meaning "and".

Euler was a very prolific symbol inventor. He invented the usage of a letter and parentheses to denote functions like $f(x)$. This actually is also very intuitive because $f$ is actually an operator, so $f(x)$ means the application of $f$ to $x$. He used $\Sigma$ ("s"-igma) for "s"-ummation, $i$ for the "i"-maginary unit, and $e$ for and "e"-xponential constant.