Khan Academy Algebra

Dividing by zero

0 / 1 = 0 1 / 0.1 = 10 1 /0.001 1 / -0.1 1/0 =

Simplify

Simplify the expression so it cannot be broken down any further while still being a single expression

Many numerators over a denominator

a+b a b ----- = --- + --- c c c

Dividing By Zero

"Undefined" b/c literally has no good definition. No one knows what happens. 5 / 0 •You have 5 cookies. You MUST give them away (/). No one exists to receive you cookies. You're screwed.

(a+b)(a-b)

(a+b)(a-b) = a^2 - b^2

(a+b)^2

(a+b)^2 = (a^2 + 2ab + b^2)

(x+a)(x-a)

(x+a)(x-a) = (x^2-a^2) Remember you can go backwards as well!

(x+a)^2 "x" = x

(x+a)^2 = (x^2+2ax+a^2)

Independent/Dependent Variables

Independent variables exist. The value of the dependent variable depends on what the independent value variable is •So for functions, the input would be independent and the output dependent because the output depends on the input

Taking Square Root

Linear: x Quadratic: x^2 Cubic: x^3 ... etc. Remember both +- square roots

Formula

Relationship between variables e.g., formula for volume of sphere, F = ma

Evaluate

When a variable is a specific value, what is the solution it the equation

Coordinate Plane

X/y coord. plane called "Cartesian Plane" b/c Rene Deacartes •Coord planes important because it allows algebra to be expressed in a geometry way. And because algebra is manipulation of any number, it is now possible to manipulate shapes •Descartes made a chart that generates shapes based on how points relate to each other

Why no multiplication sign?

•X is a commonly used variable because not many words start with X so it doesn't get confused for something, and the multiplication sign is "x"

Algebra Origins

~820AD, Baghdad Founder: al-Kwarizmi -Made general rules •Diophantus -Specific problems "Restoration/completion" •Used to balance equations

Why Are Letters Used In Algebra

•Easier to keep track of than anything else •Letters can represent a specific thing, so if that things value changes we know it still represents the same thing but a different magnitude

"Abstract"

•Focusing on ideas and concepts rather than physical reality In real life, people call cube-looking things "cubes". But the idea of a cube means six congruent sides •Think of essence in philosophy. A big, smooth rock can be called a "chair" because it has the essence of a chair •Renaissance Art = Very Realistic Abstract Art = Capturing non-physical ideas like emotion and texture

Beauty/Importance of Abstraction in Mathematics

•Generalising a problem allows the solution to be applied to any similar problem •Interesting how the way quantities interact with each other is the same across contexts e.g. Multiplying decimals. Works for finding the discount price (final price = discount times initial price) and F = m•a. This means that because of the abstraction these two equations have similar properties and can be manipulated similarly!

Point-Slope Form

•Useful way of writing equation of a line (y-b)= m(x-a) m = slope a = X coord. of another point b = Y coord. of another point •Useful when you know the slope and a point to fill (a, b)

Standard Form

•Useful way of writing equation of a line ax + by = c a = b = c = •Useful for easily knowing x&y intercepts

Slope-Intercept Form

•Useful way of writing equation of a line y=mx+b m = slope b = y-intercept

Evaluating Unknown Variables Using Structure

•When a group of variables represent a single value (X+Y=37), use that group of variables as a variable for the number itself