SkyLineWay.com - Differentiation - Emergent Properties

Math Quick Reference

Notable: Graphs, Number Tables, Equations

2^Exponent

Result	Exponent
2	1
4	2
8	3
16	4
32	5
64	6
128	7
256	8
512	9
1024	10
2048	11
4096	12
8192	13

10^Exponent

Result	Exponent
10	1
100	2
1000	3
10000	4
100000	5
1000000	6
10000000	7
100000000	8
1000000000	9
10000000000	10
100000000000	11
1000000000000	12
10000000000000	13

e^Exponent

Result	Exponent
2.7182818284	1
7.3890560986096	2
20.085536921879	3
54.5981500284	4
148.41315908646	5
403.42879344016	6
1096.6331582617	7
2980.9579865237	8
8103.0839259913	9
22026.465790022	10
59874.141700892	11
162754.79137658	12
442413.39188399	13

Addition Progression

Quadratic Equation: Circle

Properties of Equations

Equality

a=a

a=b <=> b=a

a=b <=> a+c=b+c

a=b <=> a-c=b-c

a=b <=> a*c=b*c

a=b <=> a/c=b/c

a=b <=> f(a)=b <=> f(b)=a

Algebraic

a+b=b+a <=> ab=ba

(a+b)+c <=> a+(b+c)

a(b+c) <=> ab+ac

a+0=a <=> a*1=a

a+(-a)=0 <=> a*(1/4)=1

a=b <=> a/c=b/c

a=b <=> f(a)=b <=> f(b)=a

Logarithm/Exponent

Core Inversion

b^log_b(x)=x

log_b(b^x)=x

b^0=1 <=> log_b(1)=0

b^1=b <=> log_b(b)=1

Logarithm

log_b(xy)=log_b(x)+log_b(y) <=> (b^x)*(b^y)=b^(x+y)

log_b(x/y)=log_b(x)-log_b(y) <=> (b^x)/(b^y)=b^(x-y)

log_b(x^n)=n*log_b(x)

Exponent

(b^x)*(b^y)=b^(x+y)

(b^x)/(b^y)=b^(x-y)

(b^x)^y=b^(xy)

Equality

b^x=b^y <=> x=y

log_b(x)=log_b(y) <=> x=y

Math Subjects(Grouped by Scope)

Situational Application

Applying these equations to different scenarios and different domains. Materials, shapes, and other simple properties. Scale, rotation, etc...

Dimensional Application

Translation of equations into 2 dimensions and 3 dimensions or more. Dimensions, Matrix, and Arrays.

						String Array Example: Char_X_XX_XX_XX
						Where X can be numerical string data, or even parser required string data.
						This example can have up to 4 dimensions, aka properties.
						A parser could use the underscore to denote the groups of X data to be parsed as group data.

Substrate Paradigm Application

Equations applied to properties of emergent properties of space/time/fields. Overlapping concepts accrossed domains. Even intertwined/coupled with correlation or inverse correlation. Frames, relative reference, coherence.

Top Level Math Concepts:

Hand Written
Long Hand
Short Hand
Encapsulation
Pattern Recognition
Records/Logs/Logarithms
Relationships and Equations are closely related
Word Problems
Math may or may not be inherent
Nearly Everything is Emergent

Bonus

If there was no big bang, then how could there be baryonic acoustic oscillations?