Formula atlas

Add by writing both fractions over a common denominator, then add the numerators.

Percent means per hundred. A rate of p percent is the fraction p/100.

Replace the larger number by the remainder until the remainder is zero. The last nonzero remainder is the greatest common divisor.

Write a number as a significand between 1 and 10 times a power of ten. Adding the exponents multiplies the numbers.

One step: move the constant, then divide by the coefficient of x.

Rise over run. Positive slope climbs as x grows; zero slope is horizontal.

The two roots of ax² + bx + c = 0. The expression under the radical is the discriminant.

A logarithm converts multiplication into addition. It is the inverse of bˣ.

The middle term is easy to forget and is the whole reason completing the square works.

For a right triangle only. c is the side opposite the right angle.

Any side may be the base. The height is the perpendicular distance from the opposite vertex to the line of that base.

Circumference grows like the radius; area grows like the radius squared.

Volume and surface area. Notice S = dV/dr, a preview of calculus.

Corresponding sides stand in one common ratio k. Corresponding angles are equal.

Definitions in a right triangle. The angle θ is one of the acute angles, not the right angle.

True for every angle. It is the unit circle written in function notation.

Sides are proportional to the sines of the opposite angles.

Pythagoras, corrected by the included angle. If C = 90°, cos C = 0 and the correction vanishes.

Sine of a sum is not the sum of sines. The expansion mixes both functions.

Apply f, then feed the result to g. The domain is those x where f(x) lands in the domain of g.

The inverse undoes f. It exists when f is one-to-one on the domain you care about.

Adding exponents multiplies; multiplying exponents raises a power to a power.

Compute any logarithm from the natural logarithm, or from log base 10.

Amount after time t when a principal P grows at continuous rate r. If r is negative, this is decay.

Instantaneous rate of change at x = a, when this limit exists.

Bring down the exponent and reduce it by one. It holds for any real n where the expression makes sense.

Differentiate one factor at a time and add the two results.

Differentiate the outer function, then multiply by the derivative of the inner function.

The natural exponential is its own derivative. The logarithm’s derivative is the reciprocal, for x > 0.

Evaluate any antiderivative at the ends and subtract. This is the working rule for definite integrals.

Raise the exponent by one and divide by the new exponent. The +C records the family of antiderivatives.

The missing case n = −1. The absolute value keeps the formula valid for negative x.

Set u = g(x), so du = g′(x) dx. The integral becomes one in the variable u.

When f stays above g, the integrand is the vertical gap and the integral is the enclosed area.

Zero when the vectors are perpendicular. Also equal to the sum of componentwise products.

Signed area of the parallelogram spanned by the columns. Zero means the columns are parallel.

Exists exactly when ad − bc ≠ 0.

v is a direction A only stretches. The second equation finds the possible stretches λ.

The shadow of v on the line through u. Subtract it from v to get the perpendicular piece.

Differentiate with respect to x, holding y fixed.

The vector of first partials. It points uphill most steeply.

Rate of change in the direction of a unit vector u.

Signed volume under the surface z = f over the region D, or accumulated density if f is a density.

Divergence is a scalar measuring net outflow. Curl is a vector measuring local rotation.

The probability of A given that B occurred. Requires P(B) > 0.

Swap cause and evidence. The denominator is the total probability of the evidence.

Probability of k successes in n independent trials, each with success probability p.

Mean, and expected squared deviation from the mean.

The bell curve with mean 0 and standard deviation 1. General normals are shifts and scales of this one.

Adding sizes double-counts the overlap. Subtract it once.

Ways to choose k items from n when order does not matter.

Ways to choose k items from n when order matters.

The sum of all degrees is twice the number of edges.

Each term picks y from k of the n factors and x from the rest.

An implication and its contrapositive are the same statement. The converse is a different statement.

The negation of a universal claim is an existence claim about a failure.

The negation of an and is the or of the negations. There is a twin with and and or swapped.

Divide by h(y), multiply by dx, and integrate both sides. Check constant solutions h(y) = 0 separately.

For y′ + p(t) y = q(t), multiply through by μ. Then (μ y)′ = μ q.

Acceleration is proportional to displacement, toward the origin. Solutions oscillate at angular frequency ω.

The general solution. A and B are fixed by two initial conditions, position and velocity.

Growth rate r when scarce, shutting off as N approaches the carrying capacity K.

a and b leave the same remainder when divided by m.

When p is prime and p does not divide a. Powers repeat quickly modulo a prime.

When gcd(a, n) = 1. Fermat is the special case n = p, where φ(p) = p − 1.

Has a solution x exactly when gcd(a, m) = 1. Then x is the inverse of a modulo m.

The definition of aₙ → L. Every tolerance ε is eventually met and stays met.

Inputs close enough to a force outputs close to f(a).

For f continuous on [a, b] and differentiable on (a, b), some c strictly between a and b has derivative equal to the average slope.

Every nonempty set S of reals that is bounded above has a least upper bound in the reals.

The exponential of a pure imaginary number lies on the unit circle.

Distance from the origin in the complex plane.

A holomorphic function inside and on a simple closed curve is determined by its boundary values.

The integral is 2πi times the sum of residues at singularities inside the contour.

Associativity, identity, inverses. Closure is part of saying the operation maps the set to itself.

For a subgroup H of a finite group G. The index |G|/|H| counts the distinct cosets.

Elements collapsed to the identity. Trivial kernel means the homomorphism is injective.

Modular arithmetic gives a field exactly for prime moduli.

The definition of a continuous map between topological spaces.

Every open cover has a finite subcover. In Rⁿ, equivalent to closed and bounded.

The space cannot be split into two nonempty disjoint open pieces.

A reversible continuous deformation. The two spaces are topologically the same.

Length: positive, scales, and satisfies the triangle inequality. Zero only at the zero vector.

A generalization of the dot product. It defines a norm by ||x|| = √⟨x, x⟩.

The shape cut out by one polynomial equation. Several polynomials give an intersection of such shapes.

A functor preserves identities and composition. It is a homomorphism of categories.

A contraction on a nonempty complete metric space has a unique fixed point, and iteration converges to it from anywhere.

Velocity after time t when acceleration is constant.

Position after time t. The quadratic term is the signature of constant acceleration.

Net force and acceleration are parallel. Mass is the resistance to acceleration.

Mass times velocity. Conserved, as a vector, in an isolated system.

Magnitude of the acceleration of uniform circular motion, directed toward the center.

Attractive force between two point masses, along the line joining them.

Only the component of force along the displacement does work.

Kinetic energy, gravitational potential near Earth, and spring potential.

Equation of state of an ideal gas. Temperature must be in kelvin.

Heat required to change the temperature of a mass m by ΔT, with no phase change.

Frequency times wavelength. Holds for any periodic wave.

Wave speed set by tension and mass per length.

Magnitude of the force between two point charges. Direction is along the line joining them.

Voltage across an ohmic resistor, and the power it dissipates.

Series adds resistance. Parallel adds conductance.

Force on a moving charge. Magnitude qvB sin θ, direction perpendicular to v and B.

Induced emf equals the negative rate of change of magnetic flux.

Refraction at an interface. Angles are measured from the normal, not from the surface.

The factor by which a moving clock runs slow, and by which total energy exceeds rest energy.

A clock measures Δt₀ between two events at the same place in its frame. Everyone else measures a longer interval.

The relativistic relation. At rest it is E = mc². For light, m = 0 and E = pc.

Energy of one quantum of light.

Maximum kinetic energy of an ejected electron. Nothing is ejected if hf is below the work function φ.

Wavelength associated with a particle of momentum p.

Divide the mass by the molar mass. Keep the units: grams over grams per mole.

Moles of solute per liter of solution.

Amount of solute is unchanged when you add solvent.

The working equation for an ideal gas. T in kelvin.

Total pressure is the sum of partial pressures. Each partial pressure is the mole fraction times the total.

For aA + bB ⇌ cC + dD. Concentrations at equilibrium, each to the power of its coefficient.

The test for spontaneity at constant T and P. Negative means spontaneous in the forward direction.

The standard free-energy change fixes the equilibrium constant, and conversely.

Concentration falls exponentially. The half-life does not depend on the starting amount.

Rate constants grow with temperature. Eₐ is the activation energy.

At 25 °C, K_w = 1.0×10⁻¹⁴ and a neutral solution has pH 7.

Equilibrium constant for HA ⇌ H⁺ + A⁻. Larger Kₐ means a stronger weak acid.

pH of a buffer. The ratio is conjugate base over weak acid.

Both potentials are reduction potentials. The anode is the electrode where oxidation occurs.

Voltage and free energy describe the same reaction. F is the charge of one mole of electrons.

Cell voltage at nonstandard concentrations. At 25 °C the prefactor is about (0.059 V)/n when written with log base 10.

Constant per-capita growth. Doubling time is ln 2 / r.

Per-capita growth shrinks linearly as N approaches K.

Allele frequencies, and the genotype frequencies AA, Aa, aa under the null assumptions.

Frequency of heterozygotes. For a rare recessive, about 2√(disease frequency).

Initial rate versus substrate concentration for a simple enzyme.

The double-reciprocal form. Slope Km/Vmax, intercept 1/Vmax. Useful to see, noisy in practice.

Equilibrium potential of one ion. At 37 °C, about (61.5 / z) log₁₀([out]/[in]) in millivolts.

Characteristic time to diffuse a distance x. Quadratic in distance.

Fraction bound, for a cooperative binder such as hemoglobin. n > 1 is cooperativity.

How a biological quantity scales with body mass. Take logs: log Y = log a + b log M, a straight line.