# Useful Notes / Mathematics

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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

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.

...That's it, full stop. 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 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 disatrous 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.

What makes math classes so dull and "when am I ever going to use this???" is that they are teaching every tool and every use, not 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.

The most common depiction of mathematics in media is of dry lectures in high school, where the teacher is 100% unaware of the class and assigns tons of homework. If the setting is college, then the professor is insane, the lectures are only a little more interesting, the homework is a lot less but very hard, and it still is an unenjoyable experience. Occasionally, a mathematician character will be shown working, but the math is rarely explained.

These depictions tend to include a lot of random equations (or meaningless pieces of equations) scrawled across blackboards. A lot of these equations actually come from physics, as advanced mathematics is more prone to forming Walls of Text than walls of symbols. (See E = MC Hammer for examples.)

Here's a trip down the general math curriculum through college. You're currently somewhere on this path, or can point to a spot you gave up on it. Mind you there are lots of side paths not shown below - here we're trying to show how one concept moves on to the next level, how simple addition leads up to high-level calculations.

Basic math
He say; one and one and one is three
The Beatles "Come Together"
This is kid stuff. Here you worked your way through the operations and got some background in simple math. What you learned here will help you in general life situations. A basic calculator will get you through this stuff.
• First, the numbers: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 each representing a larger number of objects from nothing to nine.
• Note that "10" isn't one of those, though it's usually counted. 0-9 are the only numbers allowed in the ones place. "10" is the "zero" in the tens place.
• We could rattle on about numbers and bases and radixes, include pi and imaginary numbers, inclusive and exclusive counting... but we're not going to. Take it as an example of how complicated even the simplest math is to explain fully using the proper math terms but also in a way that young children can understand.
• The six operations. There's a lot of boring rote memorization committing these to memory in school, but you'll be using these forever.
• Addition is putting two groups of numbers together and "adding" up the total. 3 things and 4 things is a group of seven things, so 3+4=7 and 4+3=7.
• Subtraction is the opposite, taking away a group from another group. Seven things, take away four things leaves three things, so 7-4=3 and also 7-3=4.
• Multiplication is adding the same number to itself multiple times. 3+3+3+3=12 is saying "three four times is twelve", so 3 x 4 = 12.
• In algebra you'll see expressions like "3x+7=25". The x here is a letter x and an unknown variable. See why algebra doesn't use "x" for multiplication? Parenthesis are used, like 3(4)=12. You'll also see a dot being used, but if drawn too low it can be confused with a decimal.
• Division is the reverse of multiplication. You're breaking a number into equal groups and then taking one of those groups. 12 divided by four is 3+3+3+3. Three is our answer.
• Division is frustratingly shown in multiple ways and in different arrangements. There's fractions, ratios, long division, and using the backslash / or the division symbol: ÷.
• The division symbol is not a "funny plus sign". Look at it closely and it's a dot over a dot. It represents a fraction.
• The Internet makes a joke out of division by zero, that it's a Reality-Breaking Paradox. It's actually not, it's simply undefined - calculators will keep running the division forever trying to finish the infinite operation, so they're told to just throw up an "error" message on the screen instead of even trying, that's all.
• "Undefined" here in realty means something like "smells like purple" or "Page 398 of a 200-page book". It simply doesn't exist or doesn't make any sense, since no number multiplied by zero can give another number besides zero.
• Older primary school students and teenagers will learn:
• Exponents or "powers" are for multiplication what multiplication was for addition. Three to the fourth power is 3x3x3x3=81. Much easier to do " 34 ".
• Roots or "radicals" are the reverse of exponents. What's the "fourth root" of 81? 81=3x3x3x3. There are four threes, therefore three is the answer.
• Roots use the radical sign which looks like a long division symbol, but can also be shown by a fractional power. Square roots are anything to the half power X1/2, Cube (third power) roots are to the 1/3 power X1/3 and so on.
• You learn that little operations like 3+5=8 are called equations.
• Sadly, it won't be until algebra that it's explained why it's called that: both sides are equal. If you add the 3+5 side to each other you get 8=8. It's equal. Even pre-algebra "catch-up" classes for adults looking to go to college later in life have them using a colorful abstract system of "balancing shapes", but often doesn't explain why or the goal at the end.
• Now and then you'll get a "fill in the blank" problem like "3+_=8. Put the missing number in the blank." This is more algebra foreshadowing, where the "blank" will become an "x".
• You'll get started in inequalities, but they won't be called that. Your teacher will show you the little "mouth" or < and > signifying "less than" and "greater than". Just open the mouth at the larger number, and you'll be right! It's easy now, but you'll have plenty of time to hate inequalities come algebra.
• Word problems pop up, offering a bridge between simply adding three to five and actually using the addition in a Real Life setting. Converting a problem from spoken language or a thought processes is something people do without thinking about it, and important part of math - otherwise math would simply be a silly theoretical concept and we'd still be hiding from predators in caves.
• You also learn how to use number lines, which seem pointless at the time aside from teaching how to count and introducing negative numbers. They pop up again in algebra, but vertically, too!
• You also learn your shapes, laying down the basis for geometry, and later trigonometry.
• Geometry and basic math are pretty tightly connected, so geometry isn't really a separate area, when it comes down to it.
• You'll learn how to measure angles in degrees.
• You'll learn about fractions and decimals.
• The top of a fraction is the numerator. The bottom is the denominator. Pretend the bottom is a shredder the top falls into.
• When typing a fraction in an environment you can't put one over the other (like here on TV Tropes), the top goes first and the a slash, like this: Numerator/Denominator. It's lined up just like using a division symbol.
• Fractions with common denominators can be added, different denominators cannot. Finding common denominators is a pain, but remember when doing math on your own only your teacher insists on you finding the lowest common one. Go ahead and use a big fat one, it'll still work.
• Multiplication with fractions is far easier, you just multiply the two numerators and the two denominators to each other and leave them in their fraction spots. Done.
• Sadly, no - before you even ask, you can never simply give up fractions for decimals. Some things in higher math are actually made easier using fractions, such as slopes and rational expressions. Sorry.
• Decimals are rarely exact. 1/7 is 0.142857142...(and on and on) You can't write it out exactly as a decimal - you'll never stop! This isn't a problem in everyday life, but it's a major issue in higher math and sciences. Writing "1/7" - one operation backward from writing it out - is exact. Hence it's a rational number, or a ratio.

Algebra
Generally given to teenagers, algebra helps us look at math problems in a new light, showing that some back door work in unexpected directions can give an answer which seemed originally impossible to know. A scientific calculator is a good idea here.
• Algebra introduces variables, which are denoted by letters (Usually "x", but any letter or symbol will do).
• As you go forward in math, you'll start seeing variables with a subscript: "X1, X2, X3...". These aren't math operations. They're simply nametags when you want to say that one "X" is different from the other. You can even put whole words and sentences down there if you like - physics has the coefficient of maximum static friction: µmaximum static friction but as you can see its overuse jumbles up already-confusing math equations.
• You spent your young years learning 3+5=8, but what if you have 3 things and then one day you recount and find you have 8 things? How many things did you gain? Now you have 3+x=8, which is weird, and the teacher won't accept "Well, the answer is 5 because when we did 3+_=8 back in primary school the answer was 5 in the blank and that's what's missing." algebra teaches you how to solve it and far more complicated problems.
• Answer: Subtract a 3 from both sides of the equal sign and the equation becomes "x=5", thus x is indeed 5.
• So to do algebra, put simply, is to do the same operation to both sides of an equation. In doing this you can manipulate an equation to leave the unknown variable on one side of the equal symbol and the amount that it is equal to on the other.
• Another use is you can take any complex equation involving lots of variables and juggle it around to put whatever one of those variables you don't know on one side to solve for it using the variables you do know.
• In the first few days they'll gloss over some properties of algebra. Don't shrug them them off. They're glossed over because the math teacher is already good with math, bored with talking about this, and is more concerned with getting attendance right.
• The order of operations for solving simple algebra equations (this is important to avoid notational ambiguity).
• One easy reminder is PEMDAS (Parenthesisnote  then Exponents then Multiplication then Division then Addition then Subtraction.) but the higher in math you go, the more complicated that order of operations will become with each new math concept.
• Commutative: You can add or multiply in any order you want, but still following the order of operationsnote .
• Associative: You can group addition and multiplication chains with parenthesis in any order you want, but still following the order of operationsnote .
• Distributive: You can multiply using groups in parenthesis by "distributing" the numbers.
• Under this is where you learn the FOIL method, factoring polynomials, etc.
• The sign rules. Nail these down in your head, you're gonna use them a lot.
• Two positive numbers multiplied together equals a positive number. (duh: 2 times 2 = 4)
• Two negative numbers multiplied together equals a positive number. (You read that right. -2 times -2 = 4, just like two positives, that's because when you say no to a no, it means yes, another way to illustrate this is because when you lose debt, you're actually gaining credit, so it makes sense, you can verify this more rigorously using the distributive rule.)
• Two opposite sign numbers, that's a negative and a positive multiplied together in any ordernote  will give you a negative number. (-2 times 2 = -4)
• You'll learn about polynomials which are mixes of variables and numbers. You'll never stop using these, so get used to them.
• The "normal number" part is called the coefficient. There's more of a vocabulary connected to polynomials, but most of the words don't come up a lot even in class and "coefficient" is the one you really ought to know.
• Inequalities come back, in that you'll learn you can use algebra on them just like equations - the main difference you need to know is this: whenever you divide by both sides, you turn the inequality symbol around the other way.
• You'll also have to show what set of numbers the inequality refers to.
• You'll learn about rational expressions, which is a fancy way of saying "fractions on crack".
• Protip: Don't think of it as "rational" like "reasonable". Think RATIO-nal, as they are ratios and not very reasonable.
• The coordinate plane ("Cartesian plane") comes into play here as the already-familiar line has a vertical number line tacked on.
• The horizontal is the x-axis, the vertical the y-axis.
• This is an introduction to functions, which will come up in precalculus. The functions you will learn (but won't be called functions yet) are lines and curves (you can define a line as a curve with a very big radius, so everything is a curve in this definition).
• Lines have a slope, meaning they tilt up to the right (positive slope) or down to the right (negative slope).
• A slope is best expressed (shown) as a fraction. The "rise" (up and down) over the "run" (left and right). A line with a slope of 3/4 is just "go up three, go right four" on a coordinate plane. This is a little harder to figure out with just ".75", but you CAN use .75 over 1. It's just harder without whole numbers.
• Protip: The capital N for "Negative" has a diagonal crossbar which also has a negative slope.
• Curves have a constantly changing slope - you'll learn how to find those in calculus.
• You'll learn that some equations have more than one answer, for example:
• Parabolas can have two answers - like when you toss a ball up in the air and catch it. When is it in your hand? At the beginning AND end of the toss.
• You'll learn how to solve with two unknown variables.
• For each unknown variable, you need another equation with both variables that can't be simplified to the other equation.
• You'll learn the Pythagorean Theorem of A2+B2=C2, and dealing with the sides of any triangle with one 90 degree angle. This is incredibly useful now with coordinates and later in trigonometry.
• You'll learn a complicated-looking equation called the quadratic formula that you'll swear you'll never commit to memory but will find yourself doing it anyway. It's that handy.
• The more complex polynomials you'll see in precalculus can be factored down to ones you can use this equation on.
• Imaginary numbers begin to appear here, as there's no reason to take the square root of -1 in a basic math class. They're denoted by the letter "i", mean "the square root of -1", and are actual numbers, not imaginary. Think of them as the "guts" of numbers - you'll come across them as you pull numbers apart but before the answer they'll sort out when you put the the numbers back together. In regular-old algebra unless you're studying complex numbersnote  or have a parabola which doesn't cross the x-axis, your final answer will have no imaginary numbers in it.

Intermission
At this point, many students will stop dead and vow never to find a variable again if they don't have to. Some will veer off into light statistics classes (probability and data) and light business math classes (computing interest and accounting). Serious STEMnote  fields and the more badass of the statistics and business math folks will move onward though.

Precalculus
A generating function is a clothesline on which we hang up a sequence of numbers for display.
—Herbert Wilf
This is college-level (or advanced-placement high school) stuff here. Precalculus deals with functions and trigonometry. It's a preparation for calculus. A graphing calculator is a good idea from here. There's Difficulty Spike to Calculus, and this class tries to soften that. Instead of simply finding "x", we're now talking about ranges of answers, and "minimum and maximum" will hound you the rest of your math days.
• Functions are equations likened to machines where you put something in note  and when you do the algebra it spits out an answer note .
• You'll learn to do all the basic math operations on functions - yes, equations plus/minus/times/divided by equations, and a new system, composing functions - running one through the other.
• Under functions come logarithms or simply "logs", which will help you solve the only simple "x variable" problem algebra can't solve: where the variable is an exponent, like 16=4x.
• Logs are actually super simple to use but they aren't taught until now because they are based in exponent functions, and math teachers don't want to teach you something you don't have the background for no matter how helpful it may be.
• Long division, by now a distant memory from elementary school which you hoped was just a bad dream, comes back to rear its ugly head again in the division of polynomials - helpful in factoring them. The good news is you'll learn a new division type, synthetic division, which will save you from that in most cases.
• Vectors look like what were called "rays" back in geometry. A vector isn't merely an arrow, it has a magnitude ("strength", and is always positive) and a direction (usually in degrees).
• These are important in physics classes. With a little trig, you can turn that vector into straight x and y vectors, and then back again.
• REAL WORLD CONCEPT: When you use a video game joystick or gamepad, the device is just paying attention to how much x and y vectors you are giving the pad... and then converting it to a diagonal vector.
• Matrices are rectangular arrays of numbers in columns and rows. Taken as a set, they can have operations preformed on them.

Trigonometry

Secant, tangent, cosine, sine
Three point one four one five nine
Genius Bonus sports chant from a few of the nerdier colleges such as MIT and RPI.
Often shortened to "trig", this is sometimes packaged with precalculus, sometimes taught on its own. It deals with functions in a circle and repeating functions in a wave.
• The Greek letter theta, Θ, is the generic "x" used in textbooks for an angle in trig. Anytime you're just dealing with one angle it's called that, just like how when you only have one variable in algebra it's called "x". And while we're on angles...
• You have a new measure of angles come into play, just when you're comfy with degrees. The radian is equal to the radius of a circle, and there are about 6.28 radians around the circumference of a circle, no matter what the radius of the circle is. 6.28 is twice pi (3.14), and measurements in radians are done with fractions involving pi.
• Even though your calculator will give you decimals for radians, your teacher will want the fraction. (We told you you weren't escaping fractions!)
• You WILL screw up a lot of calculations because your calculator was in radian mode when you needed degree mode, and vice versa. Protip: If you use ANY trigonometric functions in your calculation, check what mode your calculator is in before pressing enter/equal!
• The easier part of trig is how any shape can be broken up into triangles with a square corner (right triangles), and with those you can use functions called tangent, cosine and sine to find missing sides and angles.
• Compared to algebra coordinates, the cosine is the angle to the X axis, the sine the angle to the y axis. You can keep it straight because just like (x,y), (cosine,sine) is/are in alphabetical order.
• The harder part is learning the details behind repeating functions above and their inverse pals: respectively the cotangent, secant and cosecant.
• The reason your calculator doesn't have buttons for those is because it doesn't need them - they're more important for functions and understanding identities. If you really need to do one on your calculator, just divide the number one by its inverse function: hence cotangent Θ is 1/(tangent Θ).
• You'll learn sinusoidal functions which repeat forever. These are extremely handy when dealing with waves and wheels/gears.
• Then you move on to trigonometric identities which work like puzzles, using them to simplify trig equations down to something more manageable.

Calculus
Mathematics as we know it and as it has come to shape modern science could never have come into being without some disregard for the dangers of the infinite.
—David Bressoud
This is considered the Final Boss of math by those who are going to college for their bachelor's degree in a math-related field. For scientists, engineers, mathematicians, and those headed for graduate school in those fields, it's simply the Disc-One Final Boss. It's viewed almost as a mystical art, but it's really not.
• At the start is the idea of the limit. You can't find the slope of an infinitely small point - you need two points for a line! But... you can calculate two points to be to just short of that infinitely small point. And there's the start.
• Remember how curves have constantly changing slopes? Calculus will help you find the slope at any one singular point on the curve using exactly that.
• Differential calculus runs whole functions through a formula called the derivative formula.
• This may sound weird, but a simple example is the function for the area of a circle (πr2), when put through the derivative equation, gives you the formula for the circumference of a circle (2πr). Thus the two are related - the circumference of a circle is the derivative of the area of a circle. And the connections keep coming: In physics you'll learn that distance is the derivative of velocity, which is the derivative of acceleration.
• There's a "cheat" for some derivatives called the the power rule. If you take the exponent and put it at the front of the polynomial, then subtract one from that exponent and leave that as the exponent, it's the derivative. Using our circle example: πr2 becomes 2πr1 or simply 2πr!
• Going in the opposite direction is to "integrate" and is called the antiderivative.
• Integral calculus or "The one with that funny curly symbol"note  uses what is called the Fundamental theorem of calculus to calculate the area under a curve, like the ones you learned about in algebra and precalculus.
• Calculus deals with minimums and maximums. It may not be able to give you an exact answer, but it can give you a range of correct answers you can work with.
• A common calculus problem gives you a sheet of square cardboard that some must be cut away from the corners to make a box and then asks you "what size box would give the largest capacity but cut away the least amount cardboard?"
Onwards
Mathematics is, in many ways, the most precious response that the human spirit has made to the call of the infinite
—Cassius J. Keyser
It moves on from there. It's hard to nail down specific areas because by the point we're off the beaten path of the young student and close to the edge of known reasoning. There's overlap but plenty to learn still.

Finally....

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, but no one has found a better way.
• 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 the important things. Underline the actual question that needs to be answered - it may be buried in the question!
• 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 doesn't mean you won't make a fine mathematician.

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