Every Math concept in the high school Science Bowl study set (grades 9-12), with a short definition for each. Read through them here, then drill them as flashcards or practise real Math questions.
Form ax² + bx + c = 0. Quadratic formula: x = (−b ± √(b²−4ac)) / 2a. Discriminant Δ = b²−4ac: Δ > 0 → two real roots; Δ = 0 → one real root; Δ < 0 → two complex roots. Vertex at x = −b/2a. Sum of roots = −b/a; product of roots = c/a.
log_b(x) = y ↔ bʸ = x. ln(x) = log_e(x). Rules: log(ab) = log a + log b; log(a/b) = log a − log b; log(aⁿ) = n·log a; log_b(1) = 0; log_b(b) = 1. Change of base: log_b(x) = ln(x)/ln(b). Inverse of exponential: b^(log_b(x)) = x.
SOH-CAH-TOA: sin = opp/hyp, cos = adj/hyp, tan = opp/adj. Pythagorean identity: sin²θ + cos²θ = 1. Other identities: 1 + tan²θ = sec²θ; 1 + cot²θ = csc²θ. Law of Sines: a/sinA = b/sinB. Law of Cosines: c² = a² + b² − 2ab·cosC. Radians: 180° = π rad.
Instantaneous rate of change. Power rule: d/dx[xⁿ] = nxⁿ⁻¹. Product: (uv)' = u'v + uv'. Quotient: (u/v)' = (u'v − uv')/v². Chain rule: d/dx[f(g(x))] = f'(g(x))·g'(x). d/dx[sin x] = cos x; d/dx[eˣ] = eˣ; d/dx[ln x] = 1/x. Critical points where f'(x) = 0 or undefined.
Antiderivative / area under curve. Power rule: ∫xⁿdx = xⁿ⁺¹/(n+1) + C (n ≠ −1). Fundamental Theorem: ∫[a,b] f(x)dx = F(b) − F(a). Key: ∫eˣdx = eˣ + C; ∫(1/x)dx = ln|x| + C; ∫cos x dx = sin x + C; ∫sin x dx = −cos x + C. Integration by substitution and by parts.
P(A) = favorable / total. P(A and B) = P(A)·P(B) for independent events. P(A or B) = P(A) + P(B) − P(A∩B). Combinations C(n,r) = n!/[r!(n−r)!] — order doesn't matter. Permutations P(n,r) = n!/(n−r)! — order matters. Conditional: P(A|B) = P(A∩B)/P(B).
Arithmetic: aₙ = a₁ + (n−1)d; sum Sₙ = n(a₁+aₙ)/2. Geometric: aₙ = a₁·rⁿ⁻¹; sum Sₙ = a₁(1−rⁿ)/(1−r). Infinite geometric series: S = a₁/(1−r) if |r| < 1. Fibonacci: aₙ = aₙ₋₁ + aₙ₋₂ (1,1,2,3,5,8,13,…). Telescoping series: most terms cancel.
i = √(−1); i² = −1; i³ = −i; i⁴ = 1. Form z = a + bi. Addition: add real and imaginary parts. Multiplication: FOIL (use i² = −1). Modulus |z| = √(a²+b²). Conjugate: z̄ = a − bi; z·z̄ = |z|². Euler: e^(iθ) = cosθ + i sinθ. Division: multiply by conjugate over conjugate.
Mean μ = Σx/n. Median: middle value. Mode: most frequent. Variance σ² = Σ(x−μ)²/n. Standard deviation σ = √(variance). Normal distribution: 68–95–99.7 rule (1σ, 2σ, 3σ). Correlation r ∈ [−1, +1]. Linear regression: line of best fit. Outliers can skew mean but not median.
Rectangular arrays of numbers. Matrix multiplication: rows × columns (NOT commutative). det of 2×2 [a b; c d] = ad − bc. Inverse A⁻¹ exists iff det ≠ 0. Identity matrix I: A·I = A. Solve systems via Gaussian elimination or Cramer's Rule. Eigenvalue λ: Av = λv (eigenvector v).
Quantities with magnitude and direction. Component form: v = ⟨a, b, c⟩. Magnitude |v| = √(a²+b²+c²). Dot product: u·v = |u||v|cosθ = Σuᵢvᵢ (scalar; zero if perpendicular). Cross product: u×v is perpendicular to both; |u×v| = |u||v|sinθ. Unit vector: v/|v|.
lim_{x→a} f(x) = L if f(x) approaches L as x → a (from both sides). L'Hôpital's Rule: for 0/0 or ∞/∞ indeterminate forms, lim f/g = lim f'/g'. Continuity at a: f(a) defined, limit exists, and lim_{x→a} f(x) = f(a). Squeeze theorem: if g(x) ≤ f(x) ≤ h(x) and g,h → L, then f → L.
Prime numbers have exactly two factors: 1 and themselves. Fundamental Theorem of Arithmetic: every integer > 1 has a unique prime factorization. GCD: greatest common divisor; LCM: least common multiple; GCD × LCM = product of two numbers. Modular arithmetic: a ≡ b (mod n) means n divides (a−b). Euler's totient function φ(n).
Circle: (x−h)² + (y−k)² = r². Ellipse: x²/a² + y²/b² = 1 (a > b; foci at ±c where c² = a²−b²). Parabola: y = ax² + bx + c; focus/directrix definition. Hyperbola: x²/a² − y²/b² = 1; asymptotes y = ±(b/a)x. All are cross-sections of a double cone.
Long division of polynomials mirrors long division of integers. Synthetic division: shortcut when dividing by (x − c). Remainder Theorem: dividing f(x) by (x − c) gives remainder f(c). Factor Theorem: (x − c) is a factor of f(x) iff f(c) = 0. Rational Root Theorem: rational roots of aₙxⁿ + … + a₀ are ±(factors of a₀)/(factors of aₙ).
Domain: set of valid inputs. Range: set of outputs. Composition: (f∘g)(x) = f(g(x)). Inverse: f⁻¹ exists if f is one-to-one (passes horizontal line test); f(f⁻¹(x)) = x. Transformations: f(x) + k (vertical shift), f(x + h) (horizontal shift), af(x) (vertical stretch), f(bx) (horizontal compression).
(a + b)ⁿ = Σ C(n,k) aⁿ⁻ᵏ bᵏ for k = 0 to n. The kth term (starting from 1) = C(n,k−1)·aⁿ⁻⁽ᵏ⁻¹⁾·bᵏ⁻¹. Pascal's triangle rows give the coefficients. C(n,k) = n!/[k!(n−k)!]. Used in probability distributions (binomial distribution) and approximations.
Set: unordered collection of distinct elements. Union A ∪ B: elements in A or B. Intersection A ∩ B: elements in both. Complement Aᶜ: elements not in A. Empty set ∅. Subset: A ⊆ B if every element of A is in B. |A ∪ B| = |A| + |B| − |A ∩ B| (inclusion-exclusion). Venn diagrams visualize set relationships.
Optimize a linear objective function subject to linear inequality constraints. Feasible region: all points satisfying all constraints (convex polygon). Optimal solution always occurs at a vertex (corner point) of the feasible region. Used in operations research, resource allocation, and supply chain optimization.
Equations involving a function and its derivatives. Separable ODEs: dy/dx = f(x)g(y) → separate and integrate. Exponential growth/decay: dy/dt = ky → y = y₀eᵏᵗ. Second-order linear ODEs model springs (y'' + ky = 0 → y = A cos√k·t + B sin√k·t). Initial conditions determine particular solutions.
Points described by (r, θ): r = distance from origin, θ = angle from positive x-axis. Conversion: x = r cosθ, y = r sinθ; r = √(x²+y²), θ = arctan(y/x). Circle r = a. Rose curves: r = a cos(nθ). Cardioid: r = a(1 + cosθ). Area in polar: A = ½∫r²dθ.
f(x) = p(x)/q(x) where p, q are polynomials. Vertical asymptote where q(x) = 0 (and p(x) ≠ 0). Horizontal asymptote: if deg(p) < deg(q) → y = 0; deg(p) = deg(q) → y = ratio of leading coefficients; deg(p) > deg(q) → oblique asymptote. Holes where both p and q have a common factor.
|x| = x if x ≥ 0; −x if x < 0. |x| < a means −a < x < a. |x| > a means x > a or x < −a. Triangle inequality: |a + b| ≤ |a| + |b|. Solving quadratic inequalities: factor, find roots, test intervals. Sign chart method: track sign changes at roots and asymptotes.
Curve defined by x = f(t) and y = g(t) where t is a parameter. Eliminate the parameter to get Cartesian form. Derivative: dy/dx = (dy/dt)/(dx/dt). Arc length: L = ∫√[(dx/dt)²+(dy/dt)²] dt. Circle: x = r cosθ, y = r sinθ. Projectile: x = v₀cosθ·t, y = v₀sinθ·t − ½gt².
Graphs consist of vertices (nodes) and edges. Degree of vertex: number of edges connected. Euler path: visits every edge exactly once (exists if exactly 0 or 2 odd-degree vertices). Euler circuit: Euler path starting and ending at same vertex (0 odd-degree vertices). Trees: connected graphs with no cycles; n vertices, n−1 edges.
Taylor series: f(x) = Σ f⁽ⁿ⁾(a)/n! · (x−a)ⁿ; approximates a function around point a. Maclaurin series: Taylor series centered at 0. Common Maclaurin series: eˣ = 1 + x + x²/2! + x³/3! + …; sin x = x − x³/3! + x⁵/5! − …; cos x = 1 − x²/2! + x⁴/4! − …; ln(1+x) = x − x²/2 + x³/3 − … (|x| ≤ 1). Radius of convergence determined by ratio test. Truncating gives polynomial approximations.
U-substitution: reverse of chain rule; let u = g(x), du = g'(x)dx. Integration by parts: ∫u dv = uv − ∫v du (choose u by LIATE: Logarithm, Inverse trig, Algebraic, Trig, Exponential). Trigonometric substitution: for √(a²−x²) use x = a sinθ; for √(a²+x²) use x = a tanθ; for √(x²−a²) use x = a secθ. Partial fractions: decompose rational functions into simpler fractions before integrating. Tabular integration: efficient for repeated integration by parts.
Functions of multiple variables f(x, y, z). Partial derivatives: ∂f/∂x holds other variables constant. Gradient ∇f = (∂f/∂x, ∂f/∂y, ∂f/∂z): points in direction of steepest ascent. Directional derivative: rate of change in direction of unit vector û. Double integrals: ∫∫f(x,y) dA; can switch to polar coordinates. Triple integrals: volume and mass calculations. Line integrals: ∫F·dr along a curve. Green's, Stokes', and divergence theorems connect line/surface/volume integrals.
Direct proof: assume hypothesis, show conclusion follows logically. Proof by contrapositive: prove 'not Q → not P' instead of 'P → Q'. Proof by contradiction: assume negation of conclusion, derive contradiction. Mathematical induction: base case (n=1 true) + inductive step (if n=k true, then n=k+1 true) → true for all positive integers. Strong induction: assume all cases up to k. Proof by cases: exhaust all possible cases. Counterexample: single example that disproves a universal claim.
Discrete distributions: Binomial P(X=k) = C(n,k)pᵏ(1−p)ⁿ⁻ᵏ; μ = np, σ² = np(1−p). Poisson P(X=k) = e^(−λ)λᵏ/k!; μ = σ² = λ (rare events in interval). Continuous distributions: Normal (bell curve; 68-95-99.7 rule). Exponential: time between Poisson events; memoryless property. Uniform: equal probability on interval. Central Limit Theorem: sample means approach normal distribution regardless of population distribution as n → ∞.
Base-2 (binary): digits 0,1; used in computers. Base-8 (octal): digits 0–7. Base-16 (hexadecimal): digits 0–9 and A–F. Conversion: repeatedly divide by base, collect remainders. Modular arithmetic: a ≡ b (mod n) means n | (a−b). Arithmetic mod n: (a+b) mod n, (ab) mod n. Fermat's Little Theorem: if p prime, aᵖ ≡ a (mod p). Chinese Remainder Theorem: system of congruences with coprime moduli has unique solution mod product. Applications: cryptography, hashing, calendar calculations.
Inclusion-exclusion principle: |A∪B∪C| = |A|+|B|+|C| − |A∩B| − |A∩C| − |B∩C| + |A∩B∩C|. Stars and bars: distributing n identical objects into k distinct bins = C(n+k−1, k−1). Derangements: permutations with no fixed point; D(n) = n!(1 − 1/1! + 1/2! − … + (−1)ⁿ/n!). Catalan numbers: Cₙ = C(2n,n)/(n+1); count valid parenthesizations and paths. Pigeonhole principle: if n+1 items in n containers, at least one container has ≥2 items.
Euler's formula: e^(iθ) = cosθ + i sinθ. Euler's identity: e^(iπ) + 1 = 0. De Moivre's theorem: (cosθ + i sinθ)ⁿ = cos(nθ) + i sin(nθ). Polar form of complex number: z = r e^(iθ) = r(cosθ + i sinθ). Multiplication in polar form: multiply moduli, add arguments. Roots of unity: solutions to zⁿ = 1 are equally spaced on unit circle. nth roots: n solutions, spaced 2π/n apart. Fourier series uses complex exponentials as basis functions.
Eigenvalue equation: Av = λv (A is matrix, λ is eigenvalue, v is eigenvector). Find eigenvalues: solve det(A − λI) = 0 (characteristic polynomial). Eigenvectors: solve (A − λI)v = 0. Diagonalization: A = PDP⁻¹ where D is diagonal matrix of eigenvalues, P has eigenvectors as columns. Symmetric matrices have real eigenvalues and orthogonal eigenvectors. Applications: principal component analysis (PCA), differential equations, Google PageRank, quantum mechanics.
Equations defining sequences in terms of previous terms. Linear recurrences with constant coefficients: aₙ = c₁aₙ₋₁ + c₂aₙ₋₂ + … Characteristic equation method: substitute aₙ = rⁿ to get characteristic polynomial; roots r give solution. Fibonacci: aₙ = aₙ₋₁ + aₙ₋₂; closed form (Binet's formula): Fₙ = (φⁿ − ψⁿ)/√5 where φ = (1+√5)/2 (golden ratio). Generating functions: represent sequence as power series coefficients to solve recurrences.
Defined using exponentials: sinh x = (eˣ − e⁻ˣ)/2; cosh x = (eˣ + e⁻ˣ)/2; tanh x = sinh x / cosh x. Identity: cosh²x − sinh²x = 1. Derivatives: d/dx[sinh x] = cosh x; d/dx[cosh x] = sinh x. Inverse: arcsinh x = ln(x + √(x²+1)); arccosh x = ln(x + √(x²−1)) (x ≥ 1). Applications: catenary curve (hanging chain); special relativity (rapidity); solutions to Laplace's equation.
Mathematical study of strategic decision-making. Nash equilibrium: no player can improve their payoff by unilaterally changing strategy; can have multiple or no pure-strategy equilibria. Prisoner's dilemma: individually rational choices lead to collectively suboptimal outcomes. Dominant strategy: best regardless of opponent's choice. Zero-sum games: one player's gain = other's loss (solved by minimax theorem). Mixed strategies: probabilistic choices. Applications: economics, evolutionary biology (evolutionarily stable strategies), political science.
Improper integral: infinite limits or discontinuous integrand. Type 1 (infinite limits): ∫[a,∞) f(x)dx = lim_{t→∞} ∫[a,t] f(x)dx. Type 2 (discontinuity): ∫[a,b] f(x)dx where f is discontinuous at endpoint. Comparison test: if 0 ≤ f(x) ≤ g(x) and ∫g converges, so does ∫f. p-integral: ∫[1,∞) x^(−p)dx converges iff p > 1. Gamma function: Γ(n) = ∫[0,∞) x^(n−1)e^(−x)dx = (n−1)! for positive integers.
Boolean algebra uses values 0 (false) and 1 (true). Operations: AND (A·B), OR (A+B), NOT (A'). De Morgan's Laws: (A·B)' = A'+B'; (A+B)' = A'·B'. Logic gates: AND, OR, NOT, NAND, NOR, XOR (A⊕B = A'B + AB'). Truth tables enumerate all input combinations. Sum of products (SOP) and product of sums (POS) forms. Karnaugh maps simplify Boolean expressions. Applications: digital circuit design, computer architecture, cryptography.
Algorithms for approximating solutions to mathematical problems. Newton-Raphson method: xₙ₊₁ = xₙ − f(xₙ)/f'(xₙ); fast convergence (quadratic) for root-finding. Bisection method: bracket root, halve interval; slower but guaranteed. Euler's method: numerical ODE solving; yₙ₊₁ = yₙ + h·f(xₙ, yₙ). Runge-Kutta (RK4): more accurate ODE solver. Trapezoidal rule and Simpson's rule: numerical integration. Error analysis: truncation error (method) vs roundoff error (finite precision).