Algebra

Numbers

Natural numbers: 1, 2, 3, etc...
Integers: Positive & Negative natural numbers, plus 0.
Rational: any number <math>a/b \,</math>, where a and b are integers.
Real: All rationals plus everything else, like <math>\sqrt(2)\,</math> and <math>\pi\,</math>

Add, subtract any two numbers
- Know how to add and subtract fractions (not trivial)
- Know how to work with negatives
Multiply, divide
- Multiplying negatives
- division == fraction!
powers, roots
- Multiplying powers adds the exponents if the base is the same.
- Dividing subtracts
logarithms, e, natural log
- These multiply special.

Use letters to represent unknown numbers.
If you do the same operation to an equation, the equation remains equal.
- Be careful if you don't see '='. The thing may have to flip to keep it all true.
Define equations to limit the values of these numbers.
- N equations and N unknowns will give precisely 1 answer (if the equations are not dependent on each other.)
- Less than N equations means you have an infinite number of answers, but the combinations are limited.
Know how to multiple polynomials. Ex, <math>(x + a)(y -b)</math>
Know how to factor polynomials.
- <math>ax^2 + bx + c = 0 \text{ solving for x gives } x = \frac{-b \pm \sqrt{b^2-4ac}}{2a} \,</math>
- Completing the square: Note <math>(a + b)^2 = a^2 + 2ab + b^2</math>, so try to make the RHS appear, and replace with the LHS. Ex: <math>a^2 + 3ab - b^2 = a^2 + 2ab + b^2 + ab - 2b^2 = (a + b)^2 + ab - 2b^2</math>
What happens if you get <math>\sqrt{-1}</math>? That's Complex Number. (Don't write it off: just ignore it for now.)

Using graph paper, choose an origin, and draw vertical and horizontal lines through it. This is your axes.
You can graph any equation of 2 variables by changing one variable and plotting it.
You should know how to graph parabolas, hyperbolas, etc...
Adding two graphs to each other moves points up or down accordingly.
Multiplying graphs tends to squeeze or zoom in areas by the factor accordingly.
Draw hollow circles where you can't plot the point because it doesn't exist.
Draw hollow and filled circles at discontinuous points.