Useful Notes / Curriculum

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 cheap, 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.
    • 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 x 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 "x" for multiplication? A variable is either shoved right up against a number like "3x=12", or parenthesis are used, like "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.
  • 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 simply write " 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 X^1/2, Cube (third power) roots are to the 1/3 power X^1/3 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.
    • 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 0-9 except for 7. The Mesopotamians figured this out.
• 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 then a slash, like this: Numerator/Denominator. It's lined up just like when 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.

Intermission 1

"Obvious" is the most dangerous word in mathematics.

—E.T.Bell

Basic math really is simple. The basis is in solid, relatable concepts like giving and receiving items, breaking items up and allocating 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. A scientific calculator is a good idea here, as it will handle more complex operations for you freeing you to work out thinking with algebra.

• 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. 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.
• 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 (Parenthesis^note you "clear" them from the inside out 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 operations^note multiplication comes before addition.
  • Associative: You can group addition and multiplication chains with parenthesis 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 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 order^note See the commutative property above for why 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 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. 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 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 of subtracted from them or have a parabola which doesn't cross the x-axis, your final answer will have no imaginary numbers in it.

Intermission 2

"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

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. 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.

• 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.
• 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.
• 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 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.
• 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 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", and sometimes called "Precalculus II" this is sometimes packaged with precalculus, sometimes taught on its own. It deals with functions in a circle and repeating functions in a wave, and then a couple non-trig concepts which have applications that refer to trig systems.

• 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, 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.
  • 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", 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.
• 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.

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.

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 in chairs.

• 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 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.
• 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 this point we're off the well-understood path of the young student and close to the edge of known reasoning. There's overlap but plenty to learn still.