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Useful Notes / Mathematics

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Welcome to the intersection of TV Tropes and mathematics! (FYI: The intersection of sets notation is correct, but everything on the left is just E = MC Hammer).
If in other sciences we should arrive at certainty without doubt and truth without error, it behooves us to place the foundations of knowledge in mathematics.
Roger Bacon

*takes a deep breath*

Mathematics is one of the oldest and most well-developed disciplines. It has some of the same features as science: there are multiple branches that all build on each other, old theories are not discarded, just absorbed into "more correct" or "more general" theories, and it has very rigorous notions of what is true and what can be claimed to be correct. Mathematics also avoids some of the major problems of science: you don't need huge amounts of background material to get into a topic, and you don't need lots of expensive equipment to do new mathematics.

It is also hated by roughly 99.9% of all people. This is the major reason that there aren't more scientists, doctors, engineers, or Wall Street workers. This is probably due to the way it is taught: 6 to 8 years of number crunching, most of which a calculator can do faster and with less chance of error, followed by massive, abstract generalizations (algebra and calculus). To make things worse, students are rarely told why they need to learn the current topic: science teachers tend to wait until after you've learned the math to show you the uses for it, and a lot of math teachers focus on the techniques rather than the applications. (Or they try to demonstrate the applications by assigning word problems — without realizing that "translate this word problem into math" is also a skill that needs to be taught, and that assigning said problems to students who don't have that skill won't help.) Worst of all, many elementary school teachers are poorly trained in mathematics, so they don't know good problem-solving techniques, they don't know the particular real-world applications of any given topic, and they don't like it enough to teach math for its own sake.

So what IS mathematics?

Mathematics is a tool, just like how a trope is a tool used to create a story.

Mathematics is a tool that serves as a model of reality.

To use the old and tired "Alice has 3 apples and Bob has 2. Bob gives all his apples to Alice. How many apples does Alice have then?" It's 3+2=5, so Alice has 5 apples in the end.

The point is we can figure out how many apples Alice will have if that happens without ever actually moving any apples in reality. All of math is doing simply that.

So... when you budget your money before going to the grocery store, or design an airplane or a building, or figure out how to launch a spacecraft to Mars, you're simply using math to model a possibility to see if everything works without actually doing it yet.

A multipurpose tool

Mind you there are thousands of different kinds of real-world tools for thousands of different purposes — math is no different. Math is one big, messy toolbox of both simple and complex tools.
  • Some tools are first-use friendly with obvious applications, like a hammer.
    • Addition and subtraction's uses are obvious. You do it before even learning what you're doing.
  • Some tools aren't helpful without extensive training, like an aircraft, and lead to disastrous results in the hands of the untrained.
    • Calculus is an incredibly helpful area of math, but it requires classes and classes to hone the abilities needed in it. You don't want someone who has only read Calculus for Dummies doing engineering or running a supercollider.
  • Some tools have no immediately obvious use at first glance, like a home computer to The Everyman pre-Internet.
    • Negative numbers and even the now-first-day-of-class concept of "zero" were all once considered pointless math diversions at one time. How could you have a negative number of apples? How can "nothing" be a number? But we use these concepts every day.
  • Some tools are outdated and or simply not often used anymore, like a town crier or a telephone switchboard operator, but would still work if needed.
    • In Pre-Calculus you learn a lot of stuff about functions so you can understand what Calculus simplifies, but doesn't replace.

What makes a math class so dull and "when am I ever going to use this???" is that they are teaching every tool and every use, not just the ones specific or obvious or even helpful to you — because no one knows what you'll need in life. For that, you have to first figure out where you want to go in life and then figure out how you will use math to achieve that purpose. Everyone does use some level of math down every path, from housewives to scientists.

Math is patterns

A pretty pattern of floor tiles is the same all the way across. If they're all square and laid neatly, they'll still line up on the other side of the floor.

The tools of math are patterns, just like that. The patterns math looks for help us make sense of the mess of life.

  • If you add two positive numbers the answer will ALWAYS be higher than either of the two numbers.
    • That's a pattern.
  • If you multiply a number by another you're lining up a tile pattern on a number of rows up and down by a number of rows side to side and then adding each individual tile up.
    • That's a pattern.
  • If you perform the same operation on both sides of an equation, the equation stays equal.
    • That's a pattern.

The patterns build on other patterns, making a very simple or a very complicated tapestry — and it works.

The "Math Path"

One concept/tool/pattern in math really does lead to another, though it's hard to see when you're actually on it.

On this page, there's a step-by-step list of all the things you learn in school which lead to complex math, starting from the 1-2-3's and up to calculus.

If you're looking for a specific "useful notes" about a part (tool) of math Like "What's multiplication?", that page is where to go.

You will cover everything on the math path, even though when you eventually get a job, you'll find there will be huge areas that you'll never actually use in that job. So why did you learn it? Math classes are for general consumption. You may never need long division or trigonometry in that job... BUT... suppose you're the one who figures out how to apply those areas to the job opening up new horizons to everyone and you end up well-known and awarded for your "Eureka!" Moment? You may not have figured it out if you hadn't first dipped your toes in the "unrelated" math!

Concepts versus calculation

Mathematics has concepts, just like every other subject, but it also has a Difficulty Spike in calculation. The problem with math classes is you can grasp all the concepts but still stumble on the calculations — and your math teacher will mostly be looking for calculations on exams, because it shows you understood and applied the concepts.

On a non-math, interpersonal level, the concept of kindness towards others is easy to grasp. Actually doing it when a person you completely hate needs an act of kindness? That's the hard part.

Math is exactly the same way. It's one thing to explain that "algebra is just a method to find missing values". That's easy, and everyone can identify with trying to find something they are missing. Now having a problem like "Find the value of x in 3x+7=25" slapped down in front of you is a completely different matter. That's the hard part.

Math and Science both have the same problem. The "cool stuff" used to generate interest in the subjects has little to do with the nuts and bolts of actually working the subject, and that "nuts and bolts" is boring old calculation. Math has playing with mobius strips and fun logical-thinking puzzles, and science has Mentos geysers and other flashy demonstrations. The problem is when you actually get into the math or science class, what you will be doing in class is crunching numbers and learning how different equations were nailed together and simplified to create new equations. Nowhere near as fun, but a thousand times more helpful to math and science for opening up new horizons and thus leading to new flashy demonstrations.

Why can't you just use a calculator?

This is a question your math teacher will roll their eyes at and often say something about how if you find yourself without a calculator, you'll know how to find the answer. To be fair to you, we do live in a world where computers and calculators are everywhere. It is unlikely that in everyday life you'd find yourself without one and need to do higher math in an emergency.

However, a calculator is just Dumb Muscle doing the heavy-lifting in math. There's more to math than calculating, which is all a "calculator" does. For example:

  • A calculator does not know what order the problem you are working on needs to be solved in.
  • A calculator cannot look at the answer and ask "does this make sense according to the problem I have?"
  • A calculator won't tell you that you forgot to enter a value.
  • A calculator can't suggest a better, simpler way of approaching the problem.

Those are examples of the "math thinking" your teacher is trying to inspire, and your calculator cannot do.

The correct answer to "Why can't you just use a calculator?" is that just having the right answer isn't the point. It's you understanding how you get there, which is what your teacher wants to see.

If you get to higher math, like calculus, you'll even find yourself butting up against the intellectual capacity of your calculator. They only handle so many digits at a time, use only so many decimal places before saying "that's enough".

  • Get any decent calculator and type "1.000000001" (8 zeros) and pressing enter and you'll get exactly that number.
  • Now type that with an additional zero "1.0000000001" (9 zeros) and you'll get just "1".

Why? Because as far as the calculator is concerned, the second is insignificant. It's the difference between 1 and a billionth, and one and ten billionth. This is not a problem in most math you do... but in higher math calculations, these issues can add up, ending in the wrong answer — and the Real Life problems calculus handles can be expensive, destructive and even deadly when the math is done incorrectly.

There's no hurry

It does not matter how slowly you go so long as you do not stop.
Only in class or at your job is there a time limit. Many people find math much more enjoyable when they are doing it on their own, at their own speed, for whatever reason they choose.

Believe it or not, for many people, math is a hobby. You don't even need to buy anything or go anywhere.

Is math the language of the universe?

There has never been much controversy over whether 1 + 2 = 3. The question of how and whether we can truly prove 1 + 2 = 3 , which wobbles uneasily between mathematics and philosophy, is another story.
— Jordan Ellenberg

You'll see this idea here and there - that math is the foundation of reality. Is it true? Kinda yes and kinda no.

Yes: math — as a pattern — helps us understand our universe through science. Math helps us rationally understand and prove science.

No: Math is a pattern that closely follows reality, but we don't know for sure if reality follows math or math follows reality. Oddly enough, math has models (asymptotes) that show that some knowledge can be increased for eternity yet never get to full understanding. That's deep into philosophy, that discussion.

Again, math is best viewed as a tool, not an end.

How to be successful in math classes

  • Do your homework. Show every step.
    • Sadly, the best way to learn math is to roll up your sleeves and do it with paper and pencil, even when you don't want to. Many have tried to find a better way, but all have failed at that.
    • If you get the wrong answer but have shown your work on how you got there, the teacher may see that you got concepts correct even if the input was wrong, and give you partial credit.
  • Learn how to use your calculator.
    • We mean take some time to look at the instructions, and read about operations you don't understand yet.
    • Before the new millennium, teachers did not like calculators in classes — these teachers learned without them and felt that practice made a good math mind. Now, they're accepted in higher math as it's unlikely to find yourself without a computer of some sort. Before scientific calculators made trig easy, students had to have tables for the sine, cosine and tangent! But don't worry (or do!) — there are lots of things a teacher can ask you to calculate that a calculator can't do - like solve for x, or giving an exact fractional answer, and the old standby: "show your work". A calculator will help tremendously if you can't do the more complex basic operations in your head, though!
  • When you make a mistake on your homework, don't crumple up the paper and start anew — find the problem and note it on the paper to the teacher where and what you did wrong. Then do it correctly and show your work. That's the kind of thinking math is trying to foster, not simply "getting the right answer".
    • You will find there's a mistake you keep making over and over again. Now that you know what it is, you can work on that!
  • On a test, look at a word problem and circle or highlight the important things. Underline the actual question that needs to be answered — it may be buried in the question!
    • If you list the total cabbages that Alice and Bob bought individually at the store and not notice that the question actually wants to know how many 300-cabbage loads it will take to transport them in Bob's car, you've actually stopped a step short of the true answer and you'll be wrong.
  • Accept that you will make mistakes.
    • If you go to any Ivy-league math class with the very tip-top minds, you'll see even the professors making simple mistakes — missing a negative sign, skipping a step, forgetting to carry a one, using the wrong equation. If they can do that AND get a doctorate, you can surely pass a low-level class such as algebra.
  • Don't compare yourself to people who seem to have "math brains". Yes, there are some who simply get it. But that doesn't make them any better or worse than you. Classes aren't weeding people like you out — people weed themselves out. All of math is logical, work at it and you'll get it. Just because you can't rattle off the factors of 256 without thinking about it first doesn't mean you won't make a fine mathematician.

Alternative Title(s): Math, Maths