Undoing it is solving and requires new numbers
Addition | Subtraction | x+2=6: x=6-2=4,
x+6=2: x=2-6=-4 |
Integers
Negative numbers -4=2-6=3-7=... |
Functions: f(x)=x+c shifts | Expand addition to Integers | Geometry: Shifts (Horizontal/Vertical) |
Doing it again Undoing it is solving and requires new numbers
Multiplication | Division | x*2=6: x=6/2=3,
x*6=2: x=2/6=1/3=3^{-1} |
Rational numbers 1/3=2/6=3/9=... |
Multiplication and Addition
a(x+y)=ax+ay (Distributive Law) |
Functions: f(x)=cx scalings, linear, polynomial
|
Expand addition, multiplication to
Rational numbers |
Geometry: Scalings (Line, Plane) |
Doing it again Undoing it is solving and requires new numbers
Powers (Squaring, Cubing, etc.) x^2,x^3 | Roots (Square root, Cube root, etc.)
x^{1/2},x^{1/3} |
x^2=4: x=4^{1/2}=2,
x^2=2: x=2^{1/2}={\sqrt 2} |
Certain irrational and complex numbers |
Powers and Addition
(x+y)^2=x^2+2xy+y^2, Binomial Powers and Multiplication
|
Functions: | Expand addition, multiplication, powers to Irrational and Complex numbers | Geometry (Scalings, Rotations) |
Doing it again Undoing it is solving and requires new numbers
Exponentials
2^x, 3^x |
Logarithms
log_2 (y), log_3(y) |
2^x=8: x=3=log_2 (8)
2^x=3: x=log_2 (3) |
Certain irrational numbers |
Exponentials and Addition
a^{x+y}=a^x a^y Exponentials and Multiplication
|
Functions: | Expand addition, multiplication, exponentials to Irrational and Complex numbers | Geometry |
The pattern of Precalculus
Iteration (do it again) Inversion (Undo it/Solve) Expansion Interaction Kinds of numbers, Kinds of functions (linear,polynomial,rational,transcendental) Geometry
Addition (Natural Numbers)
Subtraction (Integers)
Solving x+2=6 and x+6=2
Multiplication (Natural Numbers/Integers)
Division (Rational Numbers)
Multiplication with Addition (Distributive Law) a(x+y)=ax+ay
Powers/Exponentials (Natural Numbers, Integers, Rational Number) Roots/Logarithms (Irrational numbers, Imaginary/Complex Numbers)
Power/Addition: Pascal/Binomial (x+y)^n= \sum_{j=0}^n ...
Power/Multiplication (xy)^n=x^n y^n
Exponential/Addition a^{x+y}=a^x a^y
Exponential/Multiplication a^{xy}={{a^x}^y}
And Calculus
Limit/Addition-Subtraction
Limit/Multiplication-Division
Limit/Iteration-Composition-Inversion
Derivative-Integral
Higher Derivative Multiple Integral
Derivative/Integral / Addition/Subtraction, Multiplication/Division, Iteration/Composition/Inversion, Power,Exponential,Circular functions
Connection to geometry, Connection to infinity