Equation library
Hear the formulas you study, read the way a lecturer would.
63 classic formulas, each with what it says in plain words, every symbol explained and a recording of how it is read aloud. They are read by the same engine that narrates your books.
Algebra and geometry
Triangles, polynomials, series and the complex exponential.
Pythagorean theorema² + b² = c²In a right triangle, the square of the longest side equals the sum of the squares of the other two.Quadratic formulax = (−b ± √(b² − 4ac)) / 2aThe two solutions of any quadratic equation ax² + bx + c = 0, from its three coefficients.Binomial theorem(x + y)ⁿ = ∑ₖ₌₀ⁿ C(n, k) xⁿ⁻ᵏ yᵏHow to expand a sum raised to a whole-number power, term by term.Law of cosinesc² = a² + b² − 2ab cos γThe third side of any triangle from two sides and the angle between them.Geometric series∑ₖ₌₀^∞ rᵏ = 1 / (1 − r)The sum of the endless series 1 + r + r² + r³ + …, when r is between −1 and 1.Euler's identitye^(iπ) + 1 = 0One equation that links the five most important constants in mathematics: e, i, π, 1 and 0.Euler's formulae^(iθ) = cos θ + i sin θThe complex exponential written as a point on the unit circle, joining exponentials and trigonometry.
Calculus and analysis
Derivatives, integrals, series and transforms.
Definition of the derivativef′(x) = lim_{h→0} (f(x + h) − f(x)) / hThe instantaneous rate of change of a function, as the limit of the slope over a shrinking interval.Fundamental theorem of calculus∫ₐᵇ f′(x) dx = f(b) − f(a)Integrating a rate of change over an interval gives the total change: integration undoes differentiation.Chain ruled/dx f(g(x)) = f′(g(x)) g′(x)How to differentiate a function of a function: the outer derivative times the inner one.Integration by parts∫ u dv = uv − ∫ v duThe product rule turned around: trade one integral for another that is easier to compute.Taylor seriesf(x) = ∑ₙ₌₀^∞ f⁽ⁿ⁾(a) (x − a)ⁿ / n!A function rebuilt from its derivatives at one point, as an infinite sum of powers.Fourier transformF(ω) = ∫₋∞^∞ f(t) e^(−iωt) dtA signal in time rewritten as the frequencies it contains.Gaussian integral∫₋∞^∞ e^(−x²) dx = √πThe area under the bell curve e^(−x²) over the whole real line is exactly the square root of pi.Stirling's approximationn! ≈ √(2πn) (n/e)ⁿA close estimate of n factorial for large n, without multiplying all the numbers out.
Probability and statistics
Conditional probability, distributions, estimators and bounds.
Bayes' theoremP(A|B) = P(B|A) P(A) / P(B)How to update the probability of a hypothesis A after seeing evidence B.Law of total probabilityP(A) = ∑ᵢ P(A | Bᵢ) P(Bᵢ)The probability of an event, built up from the probabilities of the separate cases that lead to it.Expected valueE[X] = ∑ᵢ xᵢ P(X = xᵢ)The long-run average of a random quantity: each possible value weighted by its probability.Varianceσ² = (1/n) ∑ᵢ (xᵢ − μ)²How spread out values are: the average squared distance from the mean.Standard error of the meanSE = σ / √nHow much a sample mean is expected to vary from sample to sample.z-scorez = (x − μ) / σHow many standard deviations a value lies above or below the mean.Normal distributionf(x) = (1 / (σ√(2π))) e^(−(x − μ)² / (2σ²))The density of the bell curve, centred on the mean μ with spread σ.Binomial distributionP(X = k) = C(n, k) pᵏ (1 − p)ⁿ⁻ᵏThe probability of exactly k successes in n independent tries, each with success probability p.Pearson correlation coefficientρ = Cov(X, Y) / (σ_X σ_Y)How closely two quantities move together in a straight line, on a scale from −1 to 1.Chebyshev's inequalityP(|X − μ| ≥ kσ) ≤ 1/k²A guarantee for any distribution: few values can lie far from the mean.Ordinary least squares estimatorβ̂ = (XᵀX)⁻¹XᵀyThe coefficients of the linear regression line that minimise the sum of squared errors.
Machine learning
Activations, losses, attention and reinforcement learning.
Scaled dot-product attentionAttention(Q, K, V) = softmax(QKᵀ / √dₖ)VThe core of the Transformer: each position mixes the values, weighted by how well its query matches each key.Softmax functionsoftmax(zᵢ) = e^(zᵢ) / ∑ⱼ e^(zⱼ)Turns a list of scores into probabilities that are positive and add up to one.Sigmoid (logistic) functionσ(x) = 1 / (1 + e^(−x))An S-shaped curve that squeezes any number into the range between 0 and 1.Cross-entropy lossL = −∑ᵢ yᵢ log(ŷᵢ)The standard loss for classification: how surprised the model is by the right answer.Mean squared errorMSE = (1/n) ∑ᵢ (yᵢ − ŷᵢ)²The average of the squared differences between predictions and true values.Gradient descent updateθ ← θ − η ∇L(θ)One step of the method that trains most machine-learning models: move the parameters downhill on the loss.Kullback–Leibler divergenceD_KL(P ‖ Q) = ∑ₓ P(x) log(P(x) / Q(x))How different one probability distribution is from another, measured in lost information.Bellman optimality equationV(s) = maxₐ (R(s, a) + γ ∑ₛ′ P(s′ | s, a) V(s′))The value of a state is the best immediate reward plus the discounted value of where you land next.ReLU activationReLU(x) = max(0, x)The rectified linear unit: pass positive inputs through unchanged and set negative ones to zero.F1 scoreF₁ = 2PR / (P + R)One number that balances precision and recall: their harmonic mean.Evidence lower bound (ELBO)log p(x) ≥ E_q[log p(x | z)] − D_KL(q(z | x) ‖ p(z))The objective of variational autoencoders: a lower bound on how well a model explains the data.Logistic regressionp = 1 / (1 + e^(−(β₀ + β₁x)))The probability of a yes-or-no outcome as an S-shaped function of a predictor.
Physics
Mechanics, electromagnetism, quantum mechanics and fluids.
Mass–energy equivalenceE = mc²Einstein's statement that mass is a form of energy: the rest energy of a body is its mass times the speed of light squared.Newton's second lawF = maThe net force on a body equals its mass times its acceleration.Newton's law of universal gravitationF = G m₁m₂ / r²Every two masses attract with a force proportional to both masses and falling with the square of the distance.Coulomb's lawF = (1 / (4πε₀)) q₁q₂ / r²The electric force between two point charges, falling with the square of the distance.Kinetic energyEₖ = ½mv²The energy a body has because it moves: half its mass times its speed squared.Schrödinger equationiħ ∂ψ/∂t = ĤψHow the quantum state of a system changes in time.Heisenberg uncertainty principleΔx Δp ≥ ħ/2Position and momentum cannot both be sharply defined at once.Gauss's law∇ · E = ρ / ε₀Electric field lines start and end on charges: the divergence of the field equals the charge density over ε₀.Faraday's law of induction∇ × E = −∂B/∂tA changing magnetic field creates a circulating electric field.Lorentz forceF = q(E + v × B)The force on a moving charge in electric and magnetic fields.Wave equation∂²u/∂t² = c² ∇²uHow a disturbance travels through a medium at speed c.Boltzmann's entropy formulaS = k_B ln ΩEntropy counts the microscopic arrangements that look the same from outside.Navier–Stokes equationρ(∂u/∂t + u · ∇u) = −∇p + μ∇²u + fNewton's second law for a flowing fluid: how velocity changes under pressure, viscosity and outside forces.Ideal gas lawPV = nRTPressure times volume equals the amount of gas times the gas constant times the temperature.Ohm's lawV = IRThe voltage across a resistor equals the current through it times its resistance.
Chemistry and biology
Buffers, reaction rates, enzymes and population genetics.
Henderson–Hasselbalch equationpH = pKa + log₁₀([A⁻] / [HA])The pH of a buffer from the acid's pKa and the ratio of base to acid.Arrhenius equationk = A e^(−Eₐ / (RT))How a reaction's rate constant rises with temperature.Michaelis–Menten equationv = V_max[S] / (K_m + [S])The rate of an enzyme reaction as the substrate concentration rises.Hardy–Weinberg equilibriump² + 2pq + q² = 1Genotype frequencies in a population that is not evolving, from the frequencies of two alleles.
Finance and economics
Interest, discounting, option pricing and risk.
Compound interestA = P(1 + r/n)^(nt)What a sum grows to when interest is added n times a year and itself earns interest.Net present valueNPV = ∑ₜ₌₁ᵀ Cₜ / (1 + r)ᵗFuture cash flows converted into today's money and added up.Present value of an annuityPV = C (1 − (1 + r)⁻ⁿ) / rWhat a stream of equal payments is worth today.Black–Scholes formulaC = S₀N(d₁) − Ke^(−rT)N(d₂)The price of a European call option on a stock that pays no dividends.Sharpe ratioS = (R_p − R_f) / σ_pReturn above the risk-free rate, per unit of volatility.Capital asset pricing model (CAPM)E(Rᵢ) = R_f + βᵢ(E(R_m) − R_f)The return investors should expect from an asset, given how it moves with the market.
Your own textbook, read like this.
Aethel reads every equation, figure and table in the books you upload. Free to start: five books and an hour of narration, no card needed.