The Joy of X

math involves both invention and discovery: we invent the concepts but discover their consequences
a x b = b x a - Commutative Law
(a x b) x c = a x (b x c) - Associative Law
a x (b + c) = ab + ac - Distributive Law
p x q =/= q x p - in nature, where p and q represent the momentum and position of a quantum particle
different ways to describe a part of a whole: fraction, percent, decimal
fractions always yield decimals that terminate of eventually repeat periodically
almost all decimals are irrational and their digits look statistically random
in the base 10 number system, there is no single symbol reserved for number 10
10 is marked by a position - the tens place - instead of a symbol
dots and dashes in the Morse code were technological forerunners of today's zeros and ones

variables are what distinguishes algebra from arithmetic
Identity - a kind of a formula
(a + b)^2 = a^2 + 2ab + b^2
(a + b)(c + d) = ac + ad + bc + bd
complex numbers - real and imaginary numbers bonded together
i^2 = -1
The Fundamental Theorem of Algebra - the roots of any polynomial are always complex numbers
fractal - an intricate shape whose inner structure repeats at finer and finer scales
Complex Dynamics - a blend of chaos theory, complex analysis, and fractal geometry
mathematical modeling - making simplifying assumptions when solving word problems
x = (-b ± √(b^2 - 4ac)) / 2a - quadratic formula is the solution to any
ax^2 + bx + c = 0 - quadratic equation
exponential functions and logarithms are inverses of each other

a^2 + b^2 = c^2 - Pythagorean theorem - implies that space is flat, as opposed to curved
parabolas and ellipses are both cross-sections of the surface of a cone
conic sections - curves obtained by cutting the surface of a cone with a plane: circles, ellipses, parabolas, hyperbolas
since wave - horizontal or vertical excursions of something moving in a circle
nature's most basic mechanism of pattern formation - the emergence of sinusoidal structure from a background of bland uniformity
Pi = C / D
A = Pir^2

calculus is the mathematics of change
derivative - how fast something is changing - local rate of change
integral - how much it's accumulating - cumulative rate of change
things always change slowest at the top or the bottom
Snell's Law - describes how light rays bend when they pass from air into water
when something changes steadily, at a constant rate, algebra works beautifully - distance equals rate times time
differential equations - how interlinked variables change from moment to moment, depending on their current values
the laws of physics are always expressed in the language of differential equations
three-body problem - contains the seeds of chaos and makes its behavior unpredictable in the long run, hence it's intractable
vector - a step that carries you from one place to another
Maxwell's electromagnetic wave unified three previously unrelated phenomena: electricity, magnetism, and light

distribution - things that seem random and unpredictable when viewed in isolation often turn out to be lawful and predictable when viewed in aggregate
normal distribution - an idealized version of the bell curve
power-law distribution - have heavy tails
conditional probability - the probability that some event A happens, given the occurrence of some other event B

the percentage of prime numbers decreases as numbers increase
encryption algorithms rely on the difficulty of decomposing an enormous number into its prime factors
group theory - bridges the arts and sciences; it addresses something the two cultures share - an abiding fascination with symmetry
topology - offshoot of geometry where two shapes are regarded as the same if you can deform one into the other continuously
geodesics - like circles, are the natural generalization of straight lines
Georg Cantor proved that some infinities are bigger than others

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