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Elementary Math

Arithmetic

125 + 375 = 500

Number line

Fractions 

1/6 + 5/12 + 3/4

Pie chart

percent

convert 1/6 to percent

16.67%

Place values

place values of 6135

Number type arithmetic

1 + (even number * odd number) = odd integer

negative integer/positive integer = negative rational number

rational number + positive even = rational number

Word problems

Rachel has 17 apples. She gives 9 to Sarah. How many apples does Rachel have now?

Rachel has 17 apples.

Rachel gives 9 apples to Sarah.

How many apples does Rachel have?

Algebra 

Equation solving

solve x^2 + 4x + 6 = 0

x^3 - 4x^2 + 6x - 24 = 0  

(x - 4) (x^2 + 6) = 0

(x - 4/3)^3 + 2/3 (x - 4/3) - 560/27 = 0

Polynomials

x^3 + x^2 y + x y^2 + y^3

Rational functions

Properties as a real function

surjective onto R

Domain R (all real numbers)

Range R (all real numbers)

Parity all are even

Definite integrals

integral_(-1)^1 (-1 + x^2)/(1 + x^2) dx = 2 - π≈-1.14159

Simplifications

1/(1+sqrt(2))

simplify | x^5 - 20 x^4 + 163 x^3 - 676 x^2 + 1424 x - 1209

x (x (x ((x - 20) x + 163) - 676) + 1424) - 1209

Matrices

{{0,-1},{1,0}}.{{1,2},{3,4}}+{{2,-1},{-1,2}}

{{0, -1}, {1, 0}} . {{1, 2}, {3, 4}} + {{2, -1}, {-1, 2}}  = {{-1, -5}, {0, 4}}

Trace = 3

Determinant = -4

Inverse is  1/4(-4 | -5

0 | 1)

Characteristic polynomial

λ^2 - 3 λ - 4

Elgen values

λ_1 = 4

λ_2 = -1

Diagonalization

M = S.J.S^(-1)

where

M = (-1 | -5

0 | 4)

S = (1 | -1

0 | 1)

J = (-1 | 0

0 | 4)

S^(-1) = (1 | 1

0 | 1)

Calculus

Integrals  -  integrate sin x dx from x=0 to pi

integral_0^π sin(x) dx = 2

 

integrate x^2 sin^3 x dx

integral x^2 sin^3(x) dx = 1/108 (-81 (x^2 - 2) cos(x) + (9 x^2 - 2) cos(3 x) - 6 x (sin(3 x) - 27 sin(x))) + constant

Derivatives   - a derivative of x^4 sin x 

d/dx(x^4 sin(x)) = x^3 (4 sin(x) + x cos(x))

Sequences - sequence of Fibonacci numbers, 

the sequence in which each term is the sum of the two previous terms with F_0 = 0, F_1 = 1, F_n = F_(n - 1) + F_(n - 2)

0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, 2584, ...

Limits - lim_(x->0) (sin(x) - x)/x^3 = -1/6 

Sums - sum j^2, j=1 to 100

Products - product (k+2)/k, k=1..25

product_(k=1)^25 (k + 2)/k = 351

Series expansions - Taylor series sin x  

x - x^3/6 + x^5/120 + O(x^7)

(Taylor series)

vector Analysis - grad sin(x^2 y) = (2 x y cos(x^2 y), x^2 cos(x^2 y))

(x: first Cartesian coordinate | y: second Cartesian coordinate)

Application calculus - compute the area between y=|x| and y=x^2-6 

area between | y = abs(x)

y = x^2 - 6area between | y = abs(x)

integral_(-3)^3 (6 - x^2 + abs(x)) dx = 27

Integral transforms - 

Domain and Range - domain of f(x) = x/(x^2-1)

{x element R : x!=-1 and x!=1}

(assuming a function from reals to reals) 

del sin(x^2 y)

Continuity - is y = sin(x - 1.1)/(x - 1.1) + θ(x) continuous?

y = sin(x - 1.1)/(x - 1.1) + θ(x) is not continuous on its domain

(assuming a function from reals to reals)

{x element R : x!=11/10}

Geometry and Topology 

Plane geometry

annulus, inner radius=2, outer radius=5

5, 12, 13 triangle

Solid geometry

Coordinate geometry

line through (1,2) and (2,1)

Distance; from (1, 2) to (2, 1): sqrt(2)≈1.41421d

Midpoint; 

Geometric transformations

rotate 30 degrees

Rotation matrix

Moire patterns

angle between grids | 4° (degrees)

shift between grids | 0

CSC(X) is a cosecant function.

Polyforms

a plane geometric figure made by joining equal squares along common edges

order | count

1 | 1

2 | 1

3 | 2

4 | 5

5 | 12

a plane geometric figure made by joining 6 equal equilateral triangles along common edges

type | count

ignoring orientation (1-sided) | 19

including orientation (2-sided) | 12

  (equilateral triangle)

Topology

packing and covering problems

Curves and surfaces

Tilings