Term Glossary

a cheat sheet for the casino

Here’s a quick reference for terms and constants you’ll run into throughout this book. Not exhaustive, just enough to get you oriented. Think of it as a cheat sheet for the casino.

Terms

Absolute Value – How far a number is from zero, regardless of direction. Always positive. Think of it as distance, not position.

Algorithm – A step-by-step procedure for solving a problem. Recipes are algorithms. So is long division.

Associativity – The rule that says grouping doesn’t matter: (a + b) + c = a + (b + c). Not all operations have this.

Axiom – A statement accepted as true without proof. The starting rules of any mathematical system. You gotta start somewhere.

Base (number systems) – The number of unique digits in a counting system. We use base 10 (decimal). Computers use base 2 (binary).

Bijection – A perfect one-to-one pairing between two sets. Every element matches exactly one partner. No leftovers.

Binary – Base-2 number system using only 0 and 1. The language of computers. On or off, true or false.

Boolean – A value that’s either true or false. Named after George Boole. The foundation of logic and computing.

Cardinality – The size of a set. How many elements it contains. Infinity has different cardinalities, which is wild.

Cartesian Plane – The x-y coordinate grid. Named after Descartes, who connected algebra and geometry by giving shapes equations.

Coefficient – The number multiplied by a variable. In 3x, the 3 is the coefficient. It scales things.

Commutativity – The rule that says order doesn’t matter: a + b = b + a. Multiplication has it, division doesn’t.

Complement – Everything NOT in a given set. If your set is even numbers, the complement is odd numbers.

Constant – A fixed value that doesn’t change. Opposite of a variable. Pi and e are famous constants.

Convergence – When a sequence or series approaches a specific value as it goes on forever. It settles down.

Cosine (cos) – A trig function. The x-coordinate on the unit circle. The horizontal component of rotation.

Degree – In angles, a unit of measurement (360 in a full circle). In polynomials, the highest exponent.

Derivative – The rate of change of a function at any point. How fast something is moving right now. Calculus stuff.

Determinant – A single number computed from a matrix that tells you about its behavior—whether it flips, scales, or collapses space.

Dimension – The number of independent directions in a space. A line is 1D, a plane is 2D, our world is 3D. Math goes higher.

Distributive Property – a(b + c) = ab + ac. Multiplication distributes over addition. One of the most used rules in algebra.

Domain – The set of all valid inputs for a function. What you’re allowed to plug in.

Element – A thing inside a set. If the set is {1, 2, 3}, then 2 is an element.

Empty Set (∅) – The set with nothing in it. The foundation of all of set theory. Zero in set form.

Equation – A statement that two expressions are equal. The backbone of algebra.

Euler’s Number (e) – Approximately 2.71828. The base of natural logarithms. Shows up everywhere in growth and decay. One of the most important constants in math.

Exponent – How many times a number is multiplied by itself. In 2³, the 3 is the exponent.

Factorial (n!) – n multiplied by every positive integer below it. 5! = 120. Counts the number of ways to arrange things.

Field – A set with addition and multiplication that both behave nicely—commutative, associative, with identities and inverses.

Function – A rule that takes an input and gives exactly one output. A mathematical machine.

Golden Ratio (φ) – Approximately 1.618. Shows up in art, architecture, nature, spirals. It’s the ratio where the whole is to the large part as the large part is to the small.

Graph – A visual representation of a function or data set. Also, in discrete math, a collection of nodes and edges.

Group – A set with one operation that’s closed, associative, has an identity, and every element has an inverse. The simplest algebraic structure.

Homeomorphism – In topology, a continuous deformation between shapes. If you can stretch one into the other without cutting or gluing, they’re homeomorphic.

Hypotenuse – The longest side of a right triangle. The one across from the 90-degree angle.

Identity – An element that does nothing under an operation. 0 for addition, 1 for multiplication.

Imaginary Number (i) – The square root of -1. Doesn’t exist on the number line but is essential for complex numbers, physics, and engineering.

Infinity (∞) – Not a number. A concept representing something without bound. There are different sizes of infinity.

Integer – Whole numbers including negatives: ...–2, –1, 0, 1, 2... No fractions, no decimals.

Integral – The opposite of a derivative. Accumulation. The area under a curve. Calculus’s other big tool.

Intersection (∩) – The elements that two sets have in common. The overlap.

Inverse – The element that undoes another. The inverse of 3 under addition is –3. Under multiplication, it’s 1/3.

Irrational Number – A number that can’t be written as a fraction. Its decimal goes forever without repeating. Pi, e, and √2 are irrational.

Limit – The value a function approaches as the input gets closer to some point. The foundation of calculus.

Linear – Relating to a straight line. A linear equation graphs as a line. A linear system plays by proportional rules.

Logarithm – The inverse of exponentiation. log₂(8) = 3 because 2³ = 8. Asks: what power gets me there?

Matrix – A rectangular grid of numbers. Used in linear algebra to represent transformations and systems of equations.

Mean (Average) – Add everything up, divide by how many there are. The center of gravity of a data set.

Median – The middle value when data is sorted. Less affected by outliers than the mean.

Nash Equilibrium – In game theory, a state where no player benefits from changing their strategy alone. A stable standoff.

Natural Number – The counting numbers: 0, 1, 2, 3... (whether 0 counts depends on who you ask).

Null Set – Same as the empty set. Contains nothing. Often written as ∅ or {}.

Operator – A symbol representing a mathematical operation. +, –, ×, ÷ are operators. So is d/dx in calculus.

Ordered Pair – Two values in a specific order, like (x, y). Position matters. (3, 5) is not (5, 3).

Parabola – The U-shaped curve you get from a quadratic equation. Every thrown ball follows one.

Payoff Matrix – In game theory, a table showing what each player gets for each combination of strategies.

Pi (π) – Approximately 3.14159. The ratio of a circle’s circumference to its diameter. Irrational, transcendental, everywhere.

Polynomial – An expression with variables and exponents added together: 3x² + 2x – 5. The workhorses of algebra.

Prime Number – A number greater than 1 that’s only divisible by 1 and itself. The atoms of arithmetic.

Probability – A number between 0 and 1 measuring how likely an outcome is. 0 = impossible, 1 = certain.

Proof – A logical argument that demonstrates a statement is true beyond doubt. The gold standard in math.

Pythagorean Theorem – a² + b² = c². The relationship between the sides of a right triangle. One of the oldest and most famous results in math.

Radian – An angle measurement based on the radius of a circle. 2π radians = 360 degrees. Math prefers radians.

Range – The set of all possible outputs of a function. What comes out the other side.

Rational Number – Any number that can be written as a fraction of two integers. Includes decimals that terminate or repeat.

Real Number – Any number on the number line. Rationals and irrationals combined. Everything except imaginary numbers.

Ring – An algebraic structure with two operations (like addition and multiplication) where addition forms a group and multiplication is associative.

Sample Space – In probability, the set of all possible outcomes of an experiment. Every roll of the dice, every flip of the coin.

Scalar – A single number, as opposed to a vector or matrix. It scales things.

Set – A collection of distinct objects. The most fundamental structure in mathematics.

Sine (sin) – A trig function. The y-coordinate on the unit circle. The vertical component of rotation.

Slope – Rise over run. How steep a line is. The rate of change between two points.

Standard Deviation – How spread out data is from the mean. Small = clustered. Large = scattered.

Subset – A set entirely contained within another set. Every set is a subset of itself. The empty set is a subset of everything.

Tangent (tan) – A trig function. Sine divided by cosine. Also, a line that touches a curve at exactly one point.

Theorem – A statement that has been proven true using axioms and logic. Graduated from conjecture.

Topology – The study of properties that survive stretching and bending. Geometry without measurement. Shape without size.

Transcendental Number – A number that isn’t the solution to any polynomial equation with integer coefficients. Pi and e are transcendental.

Transformation – A function that moves, rotates, scales, or otherwise changes geometric objects. Matrices describe them.

Union (∪) – Everything in either set combined. The merger.

Variable – A symbol representing an unknown or changeable quantity. Usually x, y, or z. A placeholder for possibility.

Vector – A quantity with both magnitude and direction. An arrow in space. The building block of linear algebra.

Vertex – A corner point. Where edges meet in a shape, or the peak/valley of a parabola.

Zero-Sum Game – A game theory situation where one player’s gain equals the other’s loss. Poker is zero-sum. Trade isn’t.

Key Constants

0 (Zero) – The additive identity. The empty set’s cardinality. The boundary between positive and negative. More important than any other number.

1 (One) – The multiplicative identity. The successor of zero. The building block.

π (Pi) ≈ 3.14159 – Circumference divided by diameter. Shows up in circles, waves, probability, and places you’d never expect.

e (Euler’s Number) ≈ 2.71828 – The base of natural growth. Compound interest, radioactive decay, probability—e is the universe’s favorite growth rate.

φ (Phi / Golden Ratio) ≈ 1.61803 – The ratio where a/b = (a+b)/a. Fibonacci spirals, sunflower seeds, ancient architecture. Nature’s aesthetic.

i (Imaginary Unit) – √(–1). Opens up the complex plane. Essential for quantum mechanics, signal processing, and electrical engineering.

∞ (Infinity) – Not a number, a concept. There’s countable infinity (integers) and uncountable infinity (real numbers). Yes, some infinities are bigger.

√2 ≈ 1.41421 – The first known irrational number. The diagonal of a unit square. Blew the Pythagoreans’ minds.