Math Reference & Interactive Demos

A collection of math lessons, formulas, and trivia compiled by Jackson Otto. Scroll down for interactive physics-based demonstrations.

Algebra Fundamentals

The Quadratic Formula

For any equation of the form ax² + bx + c = 0, the solutions are:

x = (-b ± √(b² - 4ac)) / 2a

The expression under the radical, b² - 4ac, is called the discriminant. If it's positive, there are two real solutions. If zero, one repeated solution. If negative, two complex solutions.

Laws of Exponents

Rule	Formula	Example
Product	a^m · aⁿ = a^m+n	2³ · 2⁴ = 2⁷ = 128
Quotient	a^m / aⁿ = a^m-n	5⁶ / 5² = 5⁴ = 625
Power	(a^m)ⁿ = a^mn	(3²)³ = 3⁶ = 729
Zero	a⁰ = 1	99⁰ = 1
Negative	a^-n = 1/aⁿ	2^-3 = 1/8

Factoring Patterns

a² - b² = (a + b)(a - b) — Difference of Squares

a³ + b³ = (a + b)(a² - ab + b²) — Sum of Cubes

a³ - b³ = (a - b)(a² + ab + b²) — Difference of Cubes

Math Trivia: The word "algebra" comes from the Arabic al-jabr, meaning "reunion of broken parts," from a 9th-century book by mathematician al-Khwarizmi.

Geometry

Area Formulas

Shape	Formula
Circle	A = πr²
Triangle	A = ½bh
Trapezoid	A = ½(b₁ + b₂)h
Parallelogram	A = bh
Ellipse	A = πab

Volume Formulas

Solid	Formula
Sphere	V = (4/3)πr³
Cylinder	V = πr²h
Cone	V = (1/3)πr²h
Rectangular Prism	V = lwh

The Pythagorean Theorem

a² + b² = c²

In a right triangle, the square of the hypotenuse equals the sum of the squares of the other two sides. Common Pythagorean triples: (3, 4, 5), (5, 12, 13), (8, 15, 17), (7, 24, 25).

Math Trivia: There are over 370 known proofs of the Pythagorean theorem, including ones by Leonardo da Vinci and U.S. President James Garfield.

Trigonometry

SOH-CAH-TOA

Function	Definition	Mnemonic
sin(θ)	Opposite / Hypotenuse	SOH
cos(θ)	Adjacent / Hypotenuse	CAH
tan(θ)	Opposite / Adjacent	TOA

Key Identities

sin²(θ) + cos²(θ) = 1

sin(2θ) = 2 sin(θ) cos(θ)

cos(2θ) = cos²(θ) - sin²(θ)

Unit Circle Reference

Angle	sin	cos	tan
0°	0	1	0
30°	1/2	√3/2	√3/3
45°	√2/2	√2/2	1
60°	√3/2	1/2	√3
90°	1	0	undefined

Math Trivia: The equals sign (=) was invented in 1557 by Welsh mathematician Robert Recorde, who was tired of writing "is equal to" over and over.

Pre-Calculus & Calculus Concepts

Limits

The limit describes the value a function approaches as the input approaches some value. A classic example:

lim (x→0) sin(x)/x = 1

Derivative Rules

Rule	Formula
Power Rule	d/dx [xⁿ] = nx^n-1
Product Rule	d/dx [fg] = f'g + fg'
Quotient Rule	d/dx [f/g] = (f'g - fg') / g²
Chain Rule	d/dx [f(g(x))] = f'(g(x)) · g'(x)

Common Integrals

∫ xⁿ dx = xⁿ⁺¹/(n+1) + C (n ≠ -1)

∫ sin(x) dx = -cos(x) + C

∫ cos(x) dx = sin(x) + C

∫ 1/x dx = ln|x| + C

∫ e^x dx = e^x + C

Math Trivia: Newton and Leibniz independently invented calculus in the late 1600s, leading to one of the most famous disputes in the history of science.

Number Theory & Fun Facts

Types of Numbers

Prime numbers under 100: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97.

Perfect numbers equal the sum of their proper divisors: 6 (1+2+3), 28 (1+2+4+7+14), 496, 8128.

Fibonacci sequence: 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610...

Math Trivia: A pizza that has radius "z" and height "a" has volume Pi · z · z · a.

Math Trivia: 111,111,111 × 111,111,111 = 12,345,678,987,654,321 — a perfect palindrome.

Math Trivia: If you shuffle a deck of 52 cards, the number of possible orderings (52!) is larger than the number of atoms on Earth.

Math Trivia: Zero is the only number that cannot be represented in Roman numerals.

Math Trivia: The number 1729 is known as the Hardy-Ramanujan number — it is the smallest number expressible as the sum of two cubes in two different ways: 1³ + 12³ and 9³ + 10³.

Quick Reference: Math Symbols

Symbol	Meaning	Example
∑	Summation	∑(i=1 to 5) i = 15
∏	Product	∏(i=1 to 4) i = 24
∞	Infinity	lim (x→∞)
Δ	Change in	Δy / Δx
∇	Gradient	∇f(x,y)
∈	Element of	x ∈ R
⊂	Subset	A ⊂ B
∴	Therefore	x=2 ∴ 2x=4

Math reference compiled for educational purposes.