Useful Notes / Math Curriculum

"Obvious" is the most dangerous word in mathematics.

— E.T. Bell

The above quote is important because people who already know the math view things as obvious that we, the rank and file, don't. It's easy for experts to forget that it took them a long time to understand a concept, that they struggled with the very concepts they are trying to explain, and to forget the very long and difficult hike up the mountain of maths.

But it's not really that unpleasant of a journey, and the view is really spectacular from up there.

The following is a trip down the general mathematic curriculum (classroom plans) through calculus. 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. Statistics and geometry are only lightly covered where they come up on the path to calculus.

Feel free to read past the point where you understand to. There are no tests or comprehension questions here, but you may learn something new, and may realize you know more math than you thought!

Main concepts are in bold. Feel free to skim anything that doesn't have a bold term in it, it's just trivia and tips.

Basic math (Arithmetic)

"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 cheap, basic calculator will get you through this stuff.

Basic math (Arithmetic)

• 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. The single digits 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.
    • Bertrand Russell and Alfred Whitehead's Principia Mathematica spends several hundred pages proving the validity of the proposition that 1 + 1 = 2, so yeah — that's as complex an explanation as you're getting here.
  • 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 × 4 = 12.
    • In algebra you'll see expressions like "3x + 7 = 25". The x here is a letter x and an unknown variable multiplied by 3. See why algebra doesn't use "×" for multiplication? A variable is either shoved right up against a number like "3x = 12", or parentheses are used, like "3(4) = 12" or "(3)4 = 12" or "(3)(4) = 12". You'll also see a dot being used "3 ⋅ 4 = 12", but if drawn too low it can be confused with a decimal.^note Historically in the United Kingdom and Ireland, this "middle dot" would usually be used as a decimal point to prevent it from "disappearing" in the ruled line, whereas the period (UK full stop) would be used for multiplication. The period had been ruled to be used as a decimal point by the UK Ministry of Technology in 1968, though.
  • Division is the reverse of multiplication. You're breaking a number into equal groups and then taking one of those groups. Since 12 is 3 + 3 + 3 + 3, 12 divided by four is 3. That 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 (÷).^note Note that this sign is not widespread outside the Anglo-American world — some countries use it to denote subtraction (in taxes); some use it to denote a range of values.
    • 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.
    • Dividing a number by any number less than one (but more than zero) results in a larger number than you started with, which seems to violate common sense if you think of division as "making things smaller". It does make sense, though — you just need to understand division with fractions to see why.
    • 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 3 × 3 × 3 × 3 = 81. Much easier to simply write "3⁴".
    • Roots or "radicals" are the reverse of exponents. What's the "fourth root" of 81? 81 = 3 × 3 × 3 × 3. 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 X^1/2, Cube (third power) roots are to the 1/3 power X^1/3 and so on.
• 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.
• Factors and multiples revolve around divisibility. For example, in the equation "12 ÷ 4 = 3", 12 is a (common) multiple of 3 and 4, 3 and 4 are factors of 12, and 12 is divisible by 3 and 4. Also the divisibility rules and methods to find the greatest/highest common factor (GCF/HCF) as well as the least/lowest common multiple (LCM).
• Prime numbers are any whole counting number (or "natural numbers" to mathematicians) which are larger than 1 and only divisible by 1 and itself. Each number larger than 1 can be expressed as a product of prime numbers. Composite numbers are also numbers larger than 1, but are divisible by more than two numbers.
• You'll learn about fractions, decimals, ratios and percentages, which are typically interchangeable.
  • 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 where you can't put one over the other (like here on TV Tropes); the top goes first and then a slash, like this: Numerator/Denominator. It's lined up just like when using a division symbol.
    • You can simplify a fraction by dividing both the numerator and the denominator by one of their common factors. In fact, if you multiply/divide both terms by a common non-zero number, the fraction won't change its value.
    • Fractions with common denominators can be added/subtracted, 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.
    • Dividing fractions (yes, we're dividing division!) is done by multiplying against the second fraction's reciprocal (flip it upside down). The number 0 has no reciprocal.
  • A ratio is a quantitative relation between two or more terms. For example, if there are 7 girls and 9 boys, then the ratio of girls to boys is 7/9, or 7:9.
  • A percentage is basically a ratio with 100 as the denominator. Thus one percent (1%) means 1/100 and 100% means one. A "million percent", as many people say, surely means 10,000 (times).
  • 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.
  • While on decimals, here's something you won't get until you take college science class. How many decimals should you use when your calculation spits out a big bunch? YOU can use as many as you like, but the true answer is learning about "significant figures" (or significant digits, or "sig figs"). It's a little complicated but that topic is "the rules for where to round off decimals".
• 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 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.
• You also learn your shapes, laying down the basis for geometry, and later trigonometry.
  • An important part of geometry is learning how to find the perimeter (length of the outline; called the "circumference" for a circle) and area of simple shapes.
  • You'll learn how to measure angles in degrees.
    • There are 360 degrees in a circle. That number is used because 360 has a HUGE number of factors, which makes it easy to divide by. 360 is also divisible by every number from 1-9 except for 7. The Mesopotamians figured this out.
  • Geometry and basic math are pretty tightly connected, so geometry isn't really a separate area, when it comes down to it.

"Dividing one number by another is mere computation; knowing what to divide by what is mathematics."

— Jordan Ellenberg

Basic math really is simple. The basis is in solid, relatable concepts like giving and receiving items, breaking items up and grouping them. You can move on into adulthood with a grasp of basic math and get along fine in life, fall in love, raise a family, pay your bills, etc. Students are expected to at least dip their toes into algebra, though — and some people are surprised when they actually understand it.

From here on out, each step in math is kind of like a spinning merry-go-round of concepts. All the concepts are interconnected and step-by-step just like the teacher says... but you have to climb onto the already-spinning merry-go-round first, and it doesn't slow down. Then, once you get on, you'll find yourself pulled in different directions, and feel a little queasy from time to time. Don't worry, though — these are all normal side effects of abstract thought, and you'll eventually acclimate.

Algebra

"Dear algebra, stop asking us to find your X, they're not coming back."

— Unknown

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. (This part is sometimes its own class - Algebra 1)

After that you'll learn about how to plot all the possible answers for an equation on a graph, leading you to the equations of lines and curves — which are the first steps to understanding equations called functions. (Algebra 2)

A scientific calculator is a good idea here, as it will handle more complex operations thus freeing your mind to work on thinking with algebra.

Algebra

• Algebra introduces variables, which are denoted by letters (Usually "x", but any letter or symbol will do). Also called "unknowns".
  • Important: When using letters as variables, always make sure all instances of that letter only ever represent the same variable. If you've found X equals 7, then every X in that problem must equal 7. Use other letters if you need to represent other numbers.
  • 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. When you have multiple X variables (as you'll have in parabolas) you can name them Alice and Bob (X_Alice and X_Bob) to differentiate the two but still reminding everyone that they are both among the x-values, too.
  • While we're on variables, it's important to understand that there is a lot of overlap in variables. There are only 26 letters in the English alphabet and 24 in the Greek alphabet. Even with using all of those, we don't have a variable that always means the same thing. C in physics is the speed of light. In geometry C is used to denote a distance to a foci in an ellipse. So you can never be certain what a variable represents in an equation without context.
• 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.^note Some places, such as Vietnam, demonstrate this rule in a different way: when a term switches sides in an equation, it must also switch its sign (to the opposite). Thus "+" turns into "-" (or "×" turns into "÷") and vice versa. In the equation above, you would switch the 3 to the right side and its sign to "-", so we have "x = 8 - 3" or "x = 5". 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.
• "Check your work". When you were a kid, all you had to do was say that "3 + 4 = 7", and you were right — you could look it up on the addition tables to prove it. Now you must prove you are correct by plugging the value you found into x and then doing the math again. This is annoying but it gives you more algebra practice and (annoyingly) sometimes you'll find your answer was indeed wrong.
  • This is an introduction to proofs. You'll finally be able to say "My answer is absolutely correct and here's why.", which is what mathematicians have to do to justify a job where they can often sit around in their pajamas all day playing with numbers.
• In the first few days they'll gloss over some properties of algebra. Don't shrug 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 (Parentheses^note you "clear" them from the inside out, and/or usually in the order of ( ) — [ ] — { } then Exponents then Multiplication and Division then Addition and Subtraction).^note Variants include the British BODMAS/BIDMAS, in which B means Brackets, O means Orders and I means Indices but the higher in math you go, the more complicated that order of operations will become with each new math concept.
    • If you've ever heard someone mumble or say "Please Excuse My Dear Aunt Sally", it's a mnemonic to remember PEMDAS.
    • You've probably come across the meme on the Internet that tosses an equation at you and then the comments section falls into arguments about order of operations, such as "8 ÷ 2(2 + 2) = ?". These are thrown out there for people who like to watch the arguing. Fact is, there is no single correct answer without any context. Without any explanation on what you're doing (you're obviously not using this equation for engineering, after all), you technically CAN do the equation however you want. But don't bother commenting unless you want a fight — just keep scrolling past it, or use set notation (below) to list every possible answer and annoy everyone.
  • Commutative: As long as you follow the order of operations^note multiplication comes before addition, you can add numbers in any order or multiply them in any order you want. 2 + 7 will equal 7 + 2, and 2 x 7 is the same as 7 x 2.
  • Associative: You can group addition and multiplication chains with parentheses in any order you want, but still following the order of operations^note multiplication comes before addition.
    • The two above properties don't apply to subtraction, but if you add negative numbers instead of subtracting positive ones ("3 + (-5) = x" instead of "3 - 5 = x") you can use them!
  • Distributive: You can multiply using groups in parentheses 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 × 2 = 4)
    • Two negative numbers multiplied together equals a positive number. (You read that right. -2 × -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 order^note See the commutative property above for why will give you a negative number. (-2 × 2 = -4)
    • Remember this opposite sign rule when doing fractions. If the signs match, the whole fraction is positive. If the top and bottom signs don't match, the whole fraction is negative.
• Math and English will collide as you spend some time learning how to translate word problems into an equation you can solve.
  • While we are on word problems no one is expecting you to actually come across some of the problems you will see in class and on tests. "Bob is 16 and is four times as old as Alice. How old will Bob be when he is twice Alice's age?"^note "Why would this matter?" or "Who cares?" Are perfectly legitimate "real life" responses to this question in most circumstances - but NOT in your math class. You see these problems because they requires multiple steps of algebra to solve. (Assign values, find a formula, solve the formula, check your work, etc)
    • THEN the big lesson to be learned is this: the unknown you will be looking for will not actually be the final answer to the question.^note The Bob and Alice question is a real word problem you can solve, and the answer is Bob will be 24, and Alice will be 12. The unknown you'll need to solve it for is "how many years before or after right now will they reach that age?''
• You'll learn about polynomials which are mixes of variables and numbers. You'll never stop using these, and when you're introduced to new maths up ahead on the road to calculus you'll see how nice they are and you'll find yourself wishing everything in math was as easy and cooperative as polynomials.
  • 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 multiply or divide an inequality by a negative number, you must flip the inequality sign around the other way.
  • You'll also have to show what set of numbers the inequality refers to, using a "set", speaking of which..
• You'll learn sets. This topic isn't really doing math, but cleaning up and showing your answer(s) or information clearly — but you'll have to do math to get them numbers to put into sets!
  • Example: What temperature can humans live at? You already know there's more than a single temperature. There's the set of numbers near "room temperature" where we're comfortable, the set of numbers where we need a jacket or air conditioning, and we have the set of many, many temperatures (to -273.15°C downward and infinity upward) where humans burn to death or freeze to death.
• 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 horizontal number 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... and you're back to using fractions anyway, only worse — you put a fraction in your fraction with "3/4 over 1". Just use the fraction in the first place, okay?
    • Protip: The capital N for "Negative" has a diagonal crossbar which also has a negative slope.
    • LOOKING AHEAD: The derivative equation in calculus - where you'll actually start to "do calculus stuff" in calculus - is a slope equation. See, curves have a constantly changing slope. You'll learn how to find those at any particular point on the particular curve 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.
  • This is where "X₁ and X₂" come in handy as both answers will be on the X-axis.
  • Something you can understand now that will come up more in precalculus: The exponent on a variable is equal to the number of answers it will have.^note More precisely: Every polynomial equation of degree n with complex number coefficients has n (complex) solutions. Solutions may repeat. This rule is also known as the Fundamental Theorem of Algebra. X¹ (Or simply "X") has only one answer. X² has two answers. X³ has three answers. X⁴⁷⁸ has 478 answers!
• 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^note In basic mathematics, it's impossible to squareroot a negative number since multiplying any number by itself gives a positive answer. One day, a mathematician basically went "Okay, but what if there was such a thing?", hence "Imaginary". But as it turns out, this impossible number has many mathematical uses and is key to understanding things like electrical currents.. Think of them as the "guts" of numbers — you'll come across them as you pull numbers apart but they'll sort out before the answer when you put the numbers back together. In regular-old algebra unless you're studying complex numbers^note numbers with imaginary numbers added or subtracted from them; (3+i) would be one example. or have a parabola which doesn't cross the x-axis, your final answer will have no imaginary numbers in it.

"M.A.T.H.: Mental Abuse To Humans"

— Unknown

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 STEM^note science, technology, engineering, and mathematics fields and the more badass of the statistics and business math folks will move onward though precalculus.

Precalculus

Functions and concepts

"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. There's Difficulty Spike to calculus, and this class tries to soften that. That would be great if it weren't for the introduction of many concepts that require algebra to understand, and while they stretched out algebra over several years before college (algebra 1 and 2) now you're getting concepts that could each use an entire course to understand properly all crammed into one course.

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. A graphing calculator is a good idea from here, but you don't HAVE to have one (they're pricey!), you can draw things out on paper — indeed your teacher will require it.

Precalculus: Functions and concepts

• Functions are equations likened to machines where you put something in^note an x value "domain" on the aforementioned cartesian plane and when you do the algebra it spits out an answer^note as an y value "range" on the cartesian plane.
  • A simple function is y = x + 1 . You plug different numbers into x (like "4") and get an answer (here it's "5"). As functions get more complicated, this is still the basic point. You can make a chart of every single possible answer to y = x + 1. It's silly with such a simple equation (it's just a diagonal line) but when you get to complex equations dealing with science and finance you can look and see the trends.
  • Functions are the basis of every line graph you see.
    • The data points you see were usually found by math using an already existing function (equation).
    • The more complicated side is when they were found by doing research and then using math to see if there's a pattern connecting them. This is a computation-heavy subject in statistics called regression analysis where you average out the data points which you found into the closest possible function (equation).
  • All the work with lines and parabolas in algebra were functions. From here on out you stop using a "y" and instead use function notation: a fancy letter f with an x in parenthesis " f(x) "
    • The parenthesis in f(x) isn't multiplication here - you draw the "f" bigger or use a smaller "(x)"
    • f(x) = x + 1 means "You can plug number x into function f". You're about to do operations on functions themselves, so you're allowed to use other letters for the functions now, too. g(x), h(x), x(x). So... here we go...
• 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.
  • You need to know your algebra rules here because one mistake will topple your whole answer.
  • g(x), h(x), x(x) simply clarify which function is which. You're still going to have to write them all out longhand and do all the work on paper. Sorry.
  • LOOKING AHEAD: The derivative equation in calculus - where you'll actually start to "do calculus stuff" in calculus - requires operations on functions.
• 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 = 4^x.
  • Logs are actually super simple to use (You'll get a chart of the uses), 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.
  • There is a log function for every number, but your calculator only has buttons for the two that come up a lot.
    • "log" is Log₁₀.
    • "ln" is Log_e and is called the "natural log". It uses "Euler's number"^note "Euler" pronounced as "oil-er", named after the mathematician. e is special like pi, but instead equals "2.71828...".
  • HISTORY: before calculators, people used slide rules (which were themselves invented in the 1600's), basically sliding rulers with logarithmic scales.
  • HISTORY: If you add two logs together, you get the log of the number they give when multiplied.
    • Using a calculators' log₁₀ button you'll see that, log₁₀ 2 = 0.3010 and log₁₀ 3 = 0.4771. We know that 2 × 3 = 6. Add those two together (log₁₀ 2 + log₁₀ 3)and you get 0.7781... which equals log₁₀ 6.
    • Before calculators people would sometimes use big books with tables of logs to double-check their multiplication - to multiply 2 by 3 you just had to look up those logs, add them, and then look up the matching answer in the table.
• 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.
• You'll learn about asymptotes, which are parts of functions (curved lines) that approach but will never-ever-ever get to a value. They'll curve closer and closer for infinity, never reaching the next point.
  • Before you think of this as a weird math concept with no real-world equivalent, remember "there's always room for improvement", and "perfection is unattainable no matter how hard to push towards it". Asymptotes also show diminishing returns.
• Matrices are rectangular arrays of numbers in columns and rows. Taken as a set, they can have operations performed 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

Trigonometry is often shortened to "trig", and sometimes called "Precalculus II". This is sometimes packaged with precalculus, sometimes taught on its own.

It starts with a good, deep look at triangles and circles — the two simplest shapes. From there it moves into deeper uses of angles and repeating functions in a wave, and then a couple non-trig concepts which have applications that refer to trig systems.

Precalculus: Trigonometry

• The Greek letter theta, Θ, is the generic "x" used in textbooks for an unknown 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, but curved around the circle. You can always fit 6.28 "radius lengths", or 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 best done with fractions involving pi.
  • "What the heck use is that? Degrees work great!" Well... if you know a wheel on a car has a radius of 5 units, and the wheel turns around exactly five times, then you know the wheel has travelled a distance of (5 units × 6.28 radians × 5 rotations) units, or 157 units. This is handy in mechanical engineering — wheels, gears, conveyor belts, etc. Degrees don't tell you distance, only angles. But you can convert from one to the other.
  • Even though your calculator will give you decimals for radians, your teacher will want the fraction. (We told you that 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.
  • If you ever hear someone say or mumble to themselves "sohcahtoa"^note "so-ka-toe-ah", it's a mnemonic to help remember which trig functions go with which sides of a right triangle. "soh" means "Sine (equals) Opposite side (over) Hypotenuse" and so on.)
  • HISTORY: These days you use a calculator to find these functions, but before the calculator people had to look them up on printed tables.
• Reversing sine, cosine and tangent is to take the arc function, (arcsine, arccosine, arctangent), which you'll see is shortened to sin^-1, cos^-1, tan^-1.
  • NOTE: The -1 here does not represent the exponent "-1".
  • Use the regular function (sine, cosine and tangent) if you have the Θ (degrees or radians) already.
  • Use the arc function (sin^-1, cos^-1, tan^-1) when you're dividing a side of a right triangle by another side to find the Θ (degrees or radians).
• 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 make a fraction with the number one over 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.
  • An identity is an example of something glossed over in lower math classes but is treated as if you've had it drilled into your head later. An identity is different from an equation in that you can replace the variables with any number and both sides will still be equivalent. For example:
    • A^B(A^C) = A^B+C is true no matter what three numbers you assign to mean A, B and C.
    • A + B = C is not an identity as you can easily assign numbers to make it not true, like 1 + 7 = 2.
• Vectors look like what were called "rays" back in geometry. A vector isn't merely an arrow, it has a magnitude ("strength", the length of the vactor - and it's 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.
• Parametric Equations are taking two equations and making them into functions and graphs. They do this by relating those two coordinates to a third called "t" as it usually means "time".
  • A circle is not a function, as a function can only have one range (y-axis) value. Parametric functions can take two half-circle functions (the bottom half and the top half) and make them into a circle.

"If you're going through Hell, keep going."

— Winston Churchill

Generally people don't take precalculus without the intention of moving on to calculus, so without further ado...

Calculus

"Calculus isn't scary, but the amount of algebra you have to do for it is scary."

— Unknown

Simply put, calculus is the study of change the same way arithmetic is the study of numbers and geometry is the study of shapes. It turns out that various functions (the ones you learned over algebra and precalculus) are related to each other. You're going to learn how the graph of one thing is related to the graph of another.

Calculus 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. It just requires all of the above to get here, and even with that it can be rough. All the concepts from basic math to precalculus? You're about to revisit them all and see how they apply to calculus. Calculus does not replace, it only refines. And boy, what a refining job!

It is suggested that you know your order of operations from algebra before you get started here. You'll not only be using it a lot, but to save space (and thus paper and ink) in the textbooks the algebra steps tend to be omitted as "obvious". This is one of the reasons calculus looks so confusing, but for someone in higher math to point out that they divided both sides of the equation by 3, it's like pointing out that People Sit on Chairs.

Calculus

• At the start is the idea of the limit. Curves don't have a slope^note The "rise over run" from back in algebra like a line — you need two points for a line! But... you can calculate two points to be to just short of an 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.
  • To "take the limit" is to reduce a number to so infinitely small that it can simply be thrown out as irrelevant. It's not zero, just so very small it can be treated like a zero, or just ignored. It sounds sloppy, but this sloppiness has proven the test of time and leads into some amazing math gymnastics.
  • You're going to learn rules for manipulating limits that won't really come up much in most lines of work. Humor your calculus teacher or professor anyway and do them.

Many don't make it this far, so here's some stuff to help you understand the little we've put here.

• Differential calculus runs whole functions through a formula called the derivative equation, which is the "rise over run" slope fraction from algebra on steroids — see the fraction in the picture?
  • The equation uses the limit notation, hence why you need to know it.
  • A derivative is an equation that has been run through the derivative equation, like a meat grinder.
  • This is where you need to know your order of operations from algebra and the operations on functions from precalculus.
  • 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 acceleration is the derivative of velocity, which is the derivative of distance.
  • You'll learn several rules for working with derivatives, sum, product, quotient, and chain. Each one is a teeny bit more complicated than the last. The chain rule is very annoying, and sadly it's every bit as important as it is annoying.
  • The power rule is a shortcut to taking derivatives with polynomials. 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.
  • Warning: there's no official derivative notation. There are quite a few different ways people use. Just use what your teacher tells you to, then go your own way. In the picture, the derivative is shown by dx/dy and an apostrophe by the f in the derivative equation: f '(x)
• Integral calculus or "The one with that funny curly symbol"^note It's actually a stylized "s" for "summation" 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.
  • To integrate is to do the antiderivative — the opposite of a derivative.
  • There's a power rule for integrals, too, saving you a ton of algebra.
  • When you take enough derivatives, polynomial terms with X⁰ eventually disappear. When you integrate they pop out of nowhere. This is called adding a constant and it usually denoted as "+C". Just like accidentally having your calculator in the wrong mode in trigonometry is a common mistake to make even for experts, forgetting to add a constant is a common mistake, too.
• Using the integral and the derivative on a function is very handy in pretty much everything. When you apply the derivative or the integral to a function, the new function crosses the x axis exactly where the precise top or bottom of the "hump" was in the old equation.
  • Simple example — if you take the derivative of a parabola, the line that results crosses the x-axis at the same x value where the parabola was highest or lowest.
  • When building a tall building you want just enough structure to support the tower, but not so much that you're crowding out the rooms on the lower floors with thick supports that aren't really necessary. With calculus you can chart out a graph of the different sizes of supports and find that sweet spot where you're spending exactly the right price to have safe structure without overbuilding and spending extra money on materials you don't actually need.
    • Calculus lets us build skyscrapers because of this. Old buildings were stone because it was safer to overbuild than to underbuild. Calculus helped engineers find just the right amount we needed.

"...in mathematics, you don't understand things. You just get used to them."

— John Von Neumann

There's more math beyond that, but the class sizes get small and the topics get specialized. You deal with all sorts of things, but they all branch off of the above somewhere.

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