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729 results for “ac”

Formulae (716)

Percentage

\% = \frac{\text{part}}{\text{whole}} \times 100

What fraction of a whole a part represents, out of 100.

Arithmetic & BasicsBasic

Percent of a Number

\text{value} = \frac{p}{100} \times N

Find p percent of a number N.

Arithmetic & BasicsBasic

Percentage Change

\% \text{change} = \frac{\text{new} - \text{old}}{\text{old}} \times 100

Increase or decrease of a value as a percentage.

Arithmetic & BasicsBasic

Fraction to Decimal

\frac{a}{b} = a \div b

Convert a fraction into a decimal by dividing.

Arithmetic & BasicsBasic

Adding Fractions

\frac{a}{b} + \frac{c}{d} = \frac{ad + bc}{bd}

Add two fractions using a common denominator.

Arithmetic & BasicsBasic

Multiplying Fractions

\frac{a}{b} \times \frac{c}{d} = \frac{ac}{bd}

Multiply numerators and denominators straight across.

Arithmetic & BasicsBasic

Dividing Fractions

\frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \times \frac{d}{c}

Divide by multiplying by the reciprocal (flip and multiply).

Arithmetic & BasicsBasic

Average (Mean)

\text{average} = \frac{\text{sum of values}}{\text{number of values}}

Add all values and divide by how many there are.

Arithmetic & BasicsBasic

Ratio

a : b = \frac{a}{b}

Compares two quantities by division.

Arithmetic & BasicsBasic

Proportion (Cross Multiply)

\frac{a}{b} = \frac{c}{d} \Rightarrow ad = bc

Two equal ratios; solve using cross multiplication.

Arithmetic & BasicsBasic

Unit Rate

\text{unit rate} = \frac{\text{total}}{\text{quantity}}

Cost or amount for a single unit.

Arithmetic & BasicsBasic

Speed = Distance ÷ Time

s = \frac{d}{t}

How fast something moves.

Arithmetic & BasicsBasic

Order of Operations (PEMDAS)

P \to E \to (M \div D) \to (A \div S)

Parentheses, Exponents, Multiply/Divide, Add/Subtract.

Arithmetic & BasicsBasic

Place Value

N = \sum d_i \times 10^{i}

Each digit’s value depends on its position.

Arithmetic & BasicsBasic

Discount Price

\text{price} = N \left(1 - \frac{p}{100}\right)

Final price after a percentage discount.

Arithmetic & BasicsBasic

Sales Tax Total

\text{total} = N \left(1 + \frac{t}{100}\right)

Price after adding a percentage tax.

Arithmetic & BasicsBasic

Tip Amount

\text{tip} = \frac{p}{100} \times \text{bill}

Gratuity as a percentage of the bill.

Arithmetic & BasicsBasic

Mixed Number to Fraction

a\frac{b}{c} = \frac{ac + b}{c}

Convert a mixed number to an improper fraction.

Arithmetic & BasicsBasic

Reciprocal

\frac{1}{x}

The multiplicative inverse of a number.

Arithmetic & BasicsBasic

Percent to Fraction

p\% = \frac{p}{100}

Convert a percentage to a fraction.

AlgebraIntermediate

Quadratic Formula

x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}

Finds the roots of any quadratic equation ax² + bx + c = 0.

AlgebraIntermediate

Discriminant

\Delta = b^2 - 4ac

Determines the nature of the roots of a quadratic equation.

AlgebraIntermediate

Logarithm Quotient Rule

\log_b\left(\frac{x}{y}\right) = \log_b(x) - \log_b(y)

The logarithm of a quotient equals the difference of logarithms.

AlgebraIntermediate

Change of Base Formula

\log_b(x) = \frac{\log_k(x)}{\log_k(b)}

Converts a logarithm from one base to another.

AlgebraBasic

Difference of Squares

a^2 - b^2 = (a+b)(a-b)

Factors the difference of two perfect squares.

AlgebraBasic

Perfect Square Trinomial

(a \pm b)^2 = a^2 \pm 2ab + b^2

Expands the square of a binomial.

AlgebraIntermediate

Sum of Cubes

a^3 + b^3 = (a+b)(a^2 - ab + b^2)

Factors the sum of two cubes.

AlgebraIntermediate

Difference of Cubes

a^3 - b^3 = (a-b)(a^2 + ab + b^2)

Factors the difference of two cubes.

AlgebraBasic

Arithmetic Series Sum

S_n = \frac{n}{2}(a_1 + a_n)

Sum of the first n terms of an arithmetic sequence.

AlgebraIntermediate

Geometric Series Sum

S_n = a_1 \frac{1 - r^n}{1 - r}

Sum of the first n terms of a geometric sequence (r ≠ 1).

AlgebraIntermediate

Infinite Geometric Series

S = \frac{a_1}{1 - r}, \quad |r| < 1

Sum of an infinite geometric series that converges.

AlgebraIntermediate

Completing the Square

ax^2+bx+c = a\left(x+\frac{b}{2a}\right)^2 + c - \frac{b^2}{4a}

Rewrites a quadratic in vertex form.

AlgebraIntermediate

Vieta's Formulas

x_1 + x_2 = -\frac{b}{a}, \quad x_1 x_2 = \frac{c}{a}

Relates the roots of a quadratic to its coefficients.

AlgebraBasic

FOIL Method

(a+b)(c+d) = ac + ad + bc + bd

Multiply two binomials: First, Outer, Inner, Last.

AlgebraBasic

Quotient of Powers

\frac{a^m}{a^n} = a^{m-n}

Dividing powers with the same base subtracts exponents.

AlgebraBasic

Negative Exponent

a^{-n} = \frac{1}{a^n}

A negative exponent means reciprocal.

AlgebraIntermediate

Fractional Exponent

a^{m/n} = \sqrt[n]{a^m}

A fractional exponent is a root.

AlgebraIntermediate

Inverse Variation

y = \frac{k}{x}

y changes inversely with x.

AlgebraAdvanced

Sum of First n Cubes

\sum_{k=1}^{n} k^3 = \left(\frac{n(n+1)}{2}\right)^2

Closed form for the sum of the first n cubes.

AlgebraIntermediate

Sum of Cubes

a^3 + b^3 = (a+b)(a^2 - ab + b^2)

Factoring formula for a sum of two cubes.

AlgebraIntermediate

Difference of Cubes

a^3 - b^3 = (a-b)(a^2 + ab + b^2)

Factoring formula for a difference of two cubes.

AlgebraIntermediate

Logarithm Quotient Rule

\log_b\frac{x}{y} = \log_b x - \log_b y

The log of a quotient is the difference of the logs.

AlgebraAdvanced

Change of Base

\log_b x = \frac{\log_k x}{\log_k b}

Convert a logarithm to a different base.

AlgebraAdvanced

Harmonic Mean

HM = \frac{n}{\sum 1/x_i}

Reciprocal of the average of reciprocals.

AlgebraIntermediate

Completing the Square

x^2 + bx = \left(x+\frac{b}{2}\right)^2 - \frac{b^2}{4}

Rewrite a quadratic as a perfect square plus a constant.

AlgebraIntermediate

Parabola Vertex

x = -\frac{b}{2a}

x-coordinate of the vertex of a quadratic.

AlgebraAdvanced

Infinite Geometric Series

S = \frac{a}{1-r}, \; |r|<1

Sum of an infinite geometric series that converges.

AlgebraIntermediate

Finite Geometric Series

S_n = a\frac{1-r^n}{1-r}

Sum of the first n terms of a geometric series.

AlgebraIntermediate

Rational Exponent

a^{m/n} = \sqrt[n]{a^m}

A fractional exponent is a root and a power combined.

AlgebraBasic

Negative Exponent

a^{-n} = \frac{1}{a^n}

A negative exponent gives the reciprocal.

AlgebraBasic

Quotient of Powers

\frac{a^m}{a^n} = a^{m-n}

Divide powers with the same base by subtracting exponents.

AlgebraAdvanced

System of Two Linear Equations

x = \frac{c_1 b_2 - c_2 b_1}{a_1 b_2 - a_2 b_1}

Cramer-rule solution for two simultaneous equations.

AlgebraAdvanced

Partial Fraction Setup

\frac{1}{(x-a)(x-b)} = \frac{A}{x-a} + \frac{B}{x-b}

Decompose a rational expression into simpler fractions.

Geometry & TrigonometryBasic

Area of a Triangle

A = \frac{1}{2} b h

Area of a triangle from base and height.

Geometry & TrigonometryBasic

Area of a Trapezoid

A = \frac{1}{2}(a+b)h

Area of a trapezoid with parallel sides a and b.

Geometry & TrigonometryBasic

Volume of a Sphere

V = \frac{4}{3}\pi r^3

The volume enclosed by a sphere.

Geometry & TrigonometryBasic

Volume of a Cone

V = \frac{1}{3}\pi r^2 h

The volume of a cone.

Geometry & TrigonometryBasic

Volume of a Pyramid

V = \frac{1}{3} B h

Volume of any pyramid.

Geometry & TrigonometryBasic

Surface Area of a Sphere

A = 4\pi r^2

The total surface area of a sphere.

Geometry & TrigonometryBasic

Surface Area of a Cylinder

A = 2\pi r(r + h)

Total surface area of a closed cylinder.

Geometry & TrigonometryIntermediate

Law of Sines

\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}

Relates sides to opposite angles in any triangle.

Geometry & TrigonometryIntermediate

Tangent Identity

\tan\theta = \frac{\sin\theta}{\cos\theta}

Defines tangent in terms of sine and cosine.

Geometry & TrigonometryBasic

Area of a Sector

A = \frac{1}{2} r^2 \theta

Area of a circular sector (θ in radians).

Geometry & TrigonometryBasic

Midpoint Formula

M = \left(\frac{x_1+x_2}{2}, \frac{y_1+y_2}{2}\right)

Midpoint of a line segment.

Geometry & TrigonometryBasic

Slope of a Line

m = \frac{y_2 - y_1}{x_2 - x_1}

Steepness of a straight line.

Geometry & TrigonometryBasic

Area of a Square

A = a^2

Space enclosed by a square.

Geometry & TrigonometryBasic

Area of a Rhombus

A = \frac{1}{2} d_1 d_2

Area from the two diagonals of a rhombus.

Geometry & TrigonometryBasic

Area of a Kite

A = \frac{1}{2} d_1 d_2

Area from the two diagonals of a kite.

Geometry & TrigonometryIntermediate

Area of a Regular Polygon

A = \frac{1}{2} a P

Area from apothem and perimeter.

Geometry & TrigonometryIntermediate

Space Diagonal of a Cuboid

d = \sqrt{l^2 + w^2 + h^2}

Longest diagonal through a rectangular box.

Geometry & TrigonometryBasic

Surface Area of a Cube

A = 6a^2

Total surface area of a cube.

Geometry & TrigonometryBasic

Surface Area of a Cuboid

A = 2(lw + lh + wh)

Total surface area of a rectangular box.

Geometry & TrigonometryIntermediate

Surface Area of a Cone

A = \pi r(r + l)

Total surface area of a cone with slant height l.

Geometry & TrigonometryIntermediate

Curved Surface of a Cylinder

A = 2\pi r h

Side (curved) surface area of a cylinder.

Geometry & TrigonometryIntermediate

Interior Angle (Regular Polygon)

\theta = \frac{(n-2)\times 180^\circ}{n}

Measure of each interior angle of a regular polygon.

Geometry & TrigonometryIntermediate

Cotangent Identity

\cot\theta = \frac{\cos\theta}{\sin\theta}

Cotangent as the ratio of cosine to sine.

Geometry & TrigonometryIntermediate

Reciprocal Trig Identities

\sec\theta = \tfrac{1}{\cos\theta}, \ \csc\theta = \tfrac{1}{\sin\theta}

Secant and cosecant as reciprocals.

Geometry & TrigonometryAdvanced

Tangent Double Angle

\tan(2\theta) = \frac{2\tan\theta}{1 - \tan^2\theta}

Double-angle identity for tangent.

Geometry & TrigonometryIntermediate

Area of Regular Polygon

A = \frac{1}{2} a p

Area from apothem and perimeter.

Geometry & TrigonometryIntermediate

Each Interior Angle (Regular)

\theta = \frac{(n-2)\times 180\degree}{n}

Measure of one interior angle of a regular polygon.

Geometry & TrigonometryBasic

Exterior Angle (Regular)

\theta = \frac{360\degree}{n}

Measure of one exterior angle of a regular polygon.

Geometry & TrigonometryIntermediate

Number of Diagonals

D = \frac{n(n-3)}{2}

Count of diagonals in an n-sided polygon.

Geometry & TrigonometryAdvanced

Frustum Volume

V = \frac{1}{3}\pi h(R^2 + Rr + r^2)

Volume of a truncated cone.

Geometry & TrigonometryAdvanced

Spherical Cap Volume

V = \frac{\pi h^2}{3}(3R - h)

Volume of a slice cut from a sphere.

Geometry & TrigonometryAdvanced

Circumradius

R = \frac{abc}{4K}

Radius of a triangle circumscribed circle.

Geometry & TrigonometryAdvanced

Inradius

r = \frac{K}{s}

Radius of a triangle inscribed circle.

Geometry & TrigonometryAdvanced

Tangent Half-Angle

\tan\frac{\theta}{2} = \frac{1-\cos\theta}{\sin\theta}

Expresses the tangent of a half angle.

Geometry & TrigonometryBasic

Secant Definition

\sec\theta = \frac{1}{\cos\theta}

Reciprocal of cosine.

Geometry & TrigonometryIntermediate

Sector Area (Radians)

A = \frac{1}{2}r^2\theta

Area of a circular sector with angle in radians.

TrigonometryBasic

Sine Ratio (SOH)

\sin\theta = \frac{\text{opposite}}{\text{hypotenuse}}

Ratio of the side opposite an angle to the hypotenuse.

TrigonometryBasic

Cosine Ratio (CAH)

\cos\theta = \frac{\text{adjacent}}{\text{hypotenuse}}

Ratio of the side adjacent to an angle to the hypotenuse.

TrigonometryBasic

Tangent Ratio (TOA)

\tan\theta = \frac{\text{opposite}}{\text{adjacent}}

Ratio of the opposite side to the adjacent side.

TrigonometryIntermediate

Cosecant

\csc\theta = \frac{1}{\sin\theta}

Reciprocal of sine.

TrigonometryIntermediate

Secant

\sec\theta = \frac{1}{\cos\theta}

Reciprocal of cosine.

TrigonometryIntermediate

Cotangent

\cot\theta = \frac{1}{\tan\theta} = \frac{\cos\theta}{\sin\theta}

Reciprocal of tangent.

TrigonometryBasic

Tangent Identity

\tan\theta = \frac{\sin\theta}{\cos\theta}

Tangent equals sine divided by cosine.

TrigonometryIntermediate

Sine of a Difference

\sin(A-B) = \sin A\cos B - \cos A\sin B

Angle subtraction formula for sine.

TrigonometryIntermediate

Cosine of a Difference

\cos(A-B) = \cos A\cos B + \sin A\sin B

Angle subtraction formula for cosine.

TrigonometryAdvanced

Tangent of a Sum

\tan(A+B) = \frac{\tan A + \tan B}{1 - \tan A\tan B}

Angle addition formula for tangent.

TrigonometryAdvanced

Tangent of a Difference

\tan(A-B) = \frac{\tan A - \tan B}{1 + \tan A\tan B}

Angle subtraction formula for tangent.

TrigonometryAdvanced

Double Angle: Tangent

\tan 2\theta = \frac{2\tan\theta}{1 - \tan^2\theta}

Tangent of twice an angle.

TrigonometryAdvanced

Half Angle: Sine

\sin\frac{\theta}{2} = \pm\sqrt{\frac{1-\cos\theta}{2}}

Sine of half an angle.

TrigonometryAdvanced

Half Angle: Cosine

\cos\frac{\theta}{2} = \pm\sqrt{\frac{1+\cos\theta}{2}}

Cosine of half an angle.

TrigonometryAdvanced

Half Angle: Tangent

\tan\frac{\theta}{2} = \frac{1-\cos\theta}{\sin\theta}

Tangent of half an angle.

TrigonometryAdvanced

Product: sin A cos B

\sin A\cos B = \tfrac{1}{2}[\sin(A+B) + \sin(A-B)]

Product-to-sum identity.

TrigonometryAdvanced

Product: cos A cos B

\cos A\cos B = \tfrac{1}{2}[\cos(A-B) + \cos(A+B)]

Product-to-sum identity.

TrigonometryAdvanced

Product: sin A sin B

\sin A\sin B = \tfrac{1}{2}[\cos(A-B) - \cos(A+B)]

Product-to-sum identity.

TrigonometryAdvanced

Sum to Product: sin A + sin B

\sin A + \sin B = 2\sin\tfrac{A+B}{2}\cos\tfrac{A-B}{2}

Sum-to-product identity.

TrigonometryAdvanced

Sum to Product: sin A − sin B

\sin A - \sin B = 2\cos\tfrac{A+B}{2}\sin\tfrac{A-B}{2}

Sum-to-product identity.

TrigonometryAdvanced

Sum to Product: cos A + cos B

\cos A + \cos B = 2\cos\tfrac{A+B}{2}\cos\tfrac{A-B}{2}

Sum-to-product identity.

TrigonometryAdvanced

Sum to Product: cos A − cos B

\cos A - \cos B = -2\sin\tfrac{A+B}{2}\sin\tfrac{A-B}{2}

Sum-to-product identity.

TrigonometryIntermediate

Law of Sines

\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}

Relates sides of any triangle to the sines of opposite angles.

TrigonometryAdvanced

Law of Tangents

\frac{a-b}{a+b} = \frac{\tan\tfrac{A-B}{2}}{\tan\tfrac{A+B}{2}}

Relates two sides and their opposite angles.

TrigonometryIntermediate

Triangle Area (SAS)

\text{Area} = \tfrac{1}{2}ab\sin C

Area from two sides and the included angle.

TrigonometryBasic

Degrees to Radians

\text{rad} = \text{deg} \times \frac{\pi}{180}

Convert an angle from degrees to radians.

TrigonometryBasic

Radians to Degrees

\text{deg} = \text{rad} \times \frac{180}{\pi}

Convert an angle from radians to degrees.

CalculusIntermediate

Power Rule (Derivative)

\frac{d}{dx}x^n = n x^{n-1}

Derivative of a power function.

CalculusIntermediate

Chain Rule

\frac{d}{dx}f(g(x)) = f'(g(x))\,g'(x)

Derivative of a composite function.

CalculusIntermediate

Quotient Rule

\left(\frac{f}{g}\right)' = \frac{f'g - fg'}{g^2}

Derivative of a quotient of two functions.

CalculusIntermediate

Derivative of Sine

\frac{d}{dx}\sin x = \cos x

Derivative of the sine function.

CalculusIntermediate

Derivative of Cosine

\frac{d}{dx}\cos x = -\sin x

Derivative of the cosine function.

CalculusIntermediate

Derivative of e^x

\frac{d}{dx}e^x = e^x

The exponential function is its own derivative.

CalculusIntermediate

Derivative of ln x

\frac{d}{dx}\ln x = \frac{1}{x}

Derivative of the natural logarithm.

CalculusIntermediate

Power Rule (Integral)

\int x^n\,dx = \frac{x^{n+1}}{n+1} + C, \; n \ne -1

Antiderivative of a power function.

CalculusAdvanced

Derivative (Limit Definition)

f'(x) = \lim_{h \to 0}\frac{f(x+h)-f(x)}{h}

Formal definition of the derivative.

CalculusAdvanced

Taylor Series

f(x) = \sum_{n=0}^{\infty}\frac{f^{(n)}(a)}{n!}(x-a)^n

Expands a function as an infinite polynomial around a.

CalculusAdvanced

Maclaurin Series for e^x

e^x = \sum_{n=0}^{\infty}\frac{x^n}{n!}

Taylor series of the exponential at 0.

CalculusAdvanced

L'Hôpital's Rule

\lim_{x\to c}\frac{f(x)}{g(x)} = \lim_{x\to c}\frac{f'(x)}{g'(x)}

Evaluates indeterminate limits 0/0 or ∞/∞.

CalculusAdvanced

Surface of Revolution

S = 2\pi\int_a^b f(x)\sqrt{1+(f'(x))^2}\,dx

Surface area generated by revolving a curve about the x-axis.

CalculusBasic

Constant Rule (Derivative)

\frac{d}{dx}(c) = 0

The derivative of any constant is zero.

CalculusBasic

Constant Multiple Rule

\frac{d}{dx}(c\,f) = c\,f'

Constants factor out of a derivative.

CalculusIntermediate

Second Derivative

f''(x) = \frac{d}{dx}\left(f'(x)\right)

The derivative of the first derivative; measures concavity.

CalculusIntermediate

Derivative of Tangent

\frac{d}{dx}\tan x = \sec^2 x

Derivative of the tangent function.

CalculusAdvanced

Average Value of a Function

f_{avg} = \frac{1}{b-a}\int_a^b f(x)\,dx

The mean value of a continuous function over an interval.

CalculusBasic

Power Rule

\frac{d}{dx}x^n = nx^{n-1}

Derivative of a power function.

CalculusIntermediate

Quotient Rule

\left(\frac{u}{v}\right)' = \frac{u'v - uv'}{v^2}

Derivative of a quotient of two functions.

CalculusIntermediate

Chain Rule

\frac{dy}{dx} = \frac{dy}{du}\frac{du}{dx}

Derivative of a composite function.

CalculusBasic

Derivative of Sine

\frac{d}{dx}\sin x = \cos x

The derivative of sine is cosine.

CalculusBasic

Derivative of Cosine

\frac{d}{dx}\cos x = -\sin x

The derivative of cosine is negative sine.

CalculusIntermediate

Derivative of Exponential

\frac{d}{dx}e^x = e^x

The exponential function is its own derivative.

CalculusIntermediate

Derivative of Natural Log

\frac{d}{dx}\ln x = \frac{1}{x}

The derivative of the natural logarithm.

CalculusBasic

Power Rule for Integration

\int x^n\,dx = \frac{x^{n+1}}{n+1} + C

Antiderivative of a power function (n≠-1).

CalculusIntermediate

Integral of 1/x

\int \frac{1}{x}\,dx = \ln|x| + C

Antiderivative of the reciprocal function.

CalculusAdvanced

Mean Value Theorem

f'(c) = \frac{f(b)-f(a)}{b-a}

Guarantees a point where the tangent equals the average slope.

CalculusAdvanced

Average Value of a Function

\bar{f} = \frac{1}{b-a}\int_a^b f(x)\,dx

Mean value of a continuous function over an interval.

CalculusAdvanced

Partial Derivative

\frac{\partial f}{\partial x}

Rate of change with respect to one variable, holding others fixed.

CalculusAdvanced

Gradient Vector

\nabla f = \left(\frac{\partial f}{\partial x}, \frac{\partial f}{\partial y}\right)

Vector of partial derivatives pointing in the steepest ascent.

Statistics & ProbabilityBasic

Arithmetic Mean

\bar{x} = \frac{1}{n}\sum_{i=1}^{n} x_i

The average of a data set.

Statistics & ProbabilityIntermediate

Variance

\sigma^2 = \frac{1}{n}\sum_{i=1}^{n}(x_i - \bar{x})^2

Average squared deviation from the mean.

Statistics & ProbabilityIntermediate

Standard Deviation

\sigma = \sqrt{\frac{1}{n}\sum_{i=1}^{n}(x_i-\bar{x})^2}

Square root of the variance; typical spread.

Statistics & ProbabilityIntermediate

Z-Score

z = \frac{x - \mu}{\sigma}

Number of standard deviations a value is from the mean.

Statistics & ProbabilityIntermediate

Bayes' Theorem

P(A|B) = \frac{P(B|A)P(A)}{P(B)}

Updates a probability given new evidence.

Statistics & ProbabilityIntermediate

Combinations

\binom{n}{r} = \frac{n!}{r!(n-r)!}

Number of ways to choose r items from n (order irrelevant).

Statistics & ProbabilityIntermediate

Permutations

P(n,r) = \frac{n!}{(n-r)!}

Number of ordered arrangements of r items from n.

Statistics & ProbabilityAdvanced

Normal Distribution

f(x) = \frac{1}{\sigma\sqrt{2\pi}}e^{-\frac{(x-\mu)^2}{2\sigma^2}}

Probability density of the bell curve.

Statistics & ProbabilityAdvanced

Poisson Distribution

P(X=k) = \frac{\lambda^k e^{-\lambda}}{k!}

Probability of k events in a fixed interval.

Statistics & ProbabilityIntermediate

Conditional Probability

P(A|B) = \frac{P(A \cap B)}{P(B)}

Probability of A given B has occurred.

Statistics & ProbabilityAdvanced

Pearson Correlation

r = \frac{\sum (x_i-\bar{x})(y_i-\bar{y})}{\sqrt{\sum(x_i-\bar{x})^2\sum(y_i-\bar{y})^2}}

Measures linear correlation between two variables.

Statistics & ProbabilityAdvanced

Standard Error of the Mean

SE = \frac{\sigma}{\sqrt{n}}

Standard deviation of the sample mean.

Statistics & ProbabilityBasic

Factorial

n! = n\times(n-1)\times\cdots\times 2\times 1

Product of all positive integers up to n.

Statistics & ProbabilityIntermediate

Coefficient of Variation

CV = \frac{\sigma}{\bar{x}}\times 100\%

Relative variability as a percentage of the mean.

Statistics & ProbabilityAdvanced

Covariance

\text{cov}(X,Y) = \frac{1}{n}\sum (x_i-\bar{x})(y_i-\bar{y})

Measures how two variables vary together.

Statistics & ProbabilityIntermediate

Coefficient of Variation

CV = \frac{\sigma}{\mu}\times 100

Relative variability as a percentage of the mean.

Statistics & ProbabilityBasic

Percentile Rank

PR = \frac{\text{below}}{n}\times 100

Percentage of values below a given score.

Statistics & ProbabilityAdvanced

Margin of Error

E = z\frac{\sigma}{\sqrt{n}}

Half-width of a confidence interval for the mean.

Statistics & ProbabilityAdvanced

Confidence Interval (Mean)

\bar{x} \pm z\frac{\sigma}{\sqrt{n}}

Range likely to contain the population mean.

Statistics & ProbabilityAdvanced

Z-Test Statistic

z = \frac{\bar{x} - \mu_0}{\sigma/\sqrt{n}}

Tests whether a sample mean differs from a hypothesized value.

Statistics & ProbabilityAdvanced

Chi-Square Statistic

\chi^2 = \sum \frac{(O-E)^2}{E}

Tests goodness of fit between observed and expected counts.

Statistics & ProbabilityIntermediate

Weighted Mean

\bar{x}_w = \frac{\sum w_i x_i}{\sum w_i}

Average where values carry different weights.

Statistics & ProbabilityAdvanced

Odds Form of Bayes

O(A|B) = O(A)\times \frac{P(B|A)}{P(B|\neg A)}

Updates odds using a likelihood ratio.

Statistics & ProbabilityIntermediate

Sample Variance

s^2 = \frac{1}{n-1}\sum(x_i-\bar{x})^2

Unbiased estimate of population variance.

Probability & CombinatoricsBasic

Permutations

P(n,r) = \frac{n!}{(n-r)!}

Number of ordered arrangements of r items from n.

Probability & CombinatoricsBasic

Combinations

C(n,r) = \binom{n}{r} = \frac{n!}{r!(n-r)!}

Number of unordered selections of r items from n.

Probability & CombinatoricsBasic

Factorial

n! = n \times (n-1) \times \cdots \times 2 \times 1

Product of all positive integers up to n.

Probability & CombinatoricsIntermediate

Permutations of a Multiset

\frac{n!}{n_1! n_2! \cdots n_k!}

Arrangements of n items where some are identical.

Probability & CombinatoricsBasic

Classical Probability

P(A) = \frac{\text{favorable outcomes}}{\text{total outcomes}}

Probability of an event with equally likely outcomes.

Probability & CombinatoricsIntermediate

Conditional Probability

P(A|B) = \frac{P(A \cap B)}{P(B)}

Probability of A given that B has occurred.

Probability & CombinatoricsAdvanced

Bayes Theorem

P(A|B) = \frac{P(B|A)\,P(A)}{P(B)}

Reverses conditional probabilities using prior knowledge.

Probability & CombinatoricsAdvanced

Poisson Probability

P(X=k) = \frac{\lambda^k e^{-\lambda}}{k!}

Probability of k events in a fixed interval given average rate λ.

Probability & CombinatoricsAdvanced

Normal Distribution PDF

f(x) = \frac{1}{\sigma\sqrt{2\pi}} e^{-\frac{(x-\mu)^2}{2\sigma^2}}

Bell-curve probability density function.

Probability & CombinatoricsBasic

Odds

Odds = \frac{P(A)}{1 - P(A)}

Ratio of the probability of an event to its non-occurrence.

Probability & CombinatoricsAdvanced

Catalan Number

C_n = \frac{1}{n+1}\binom{2n}{n}

Counts many recursive structures such as balanced parentheses.

Data Science & MLBasic

Mean Squared Error (MSE)

MSE = \frac{1}{n}\sum_{i=1}^{n}(y_i - \hat{y}_i)^2

Average of squared differences between predicted and actual values.

Data Science & MLBasic

Root Mean Squared Error (RMSE)

RMSE = \sqrt{\frac{1}{n}\sum_{i=1}^{n}(y_i - \hat{y}_i)^2}

Square root of MSE, in the same units as the target variable.

Data Science & MLBasic

Mean Absolute Error (MAE)

MAE = \frac{1}{n}\sum_{i=1}^{n}|y_i - \hat{y}_i|

Average of absolute differences between predictions and actual values.

Data Science & MLIntermediate

Coefficient of Determination (R²)

R^2 = 1 - \frac{\sum(y_i-\hat{y}_i)^2}{\sum(y_i-\bar{y})^2}

Proportion of variance in the target explained by the model.

Data Science & MLBasic

Classification Accuracy

Accuracy = \frac{TP + TN}{TP + TN + FP + FN}

Fraction of predictions that are correct.

Data Science & MLIntermediate

Precision

Precision = \frac{TP}{TP + FP}

Of all positive predictions, the fraction that were correct.

Data Science & MLIntermediate

Recall (Sensitivity)

Recall = \frac{TP}{TP + FN}

Of all actual positives, the fraction correctly identified.

Data Science & MLIntermediate

F1 Score

F_1 = 2\cdot\frac{Precision \cdot Recall}{Precision + Recall}

Harmonic mean of precision and recall.

Data Science & MLIntermediate

Specificity

Specificity = \frac{TN}{TN + FP}

Of all actual negatives, the fraction correctly identified.

Data Science & MLIntermediate

Sigmoid Function

\sigma(x) = \frac{1}{1 + e^{-x}}

Maps any real number to the range (0, 1); used in logistic regression.

Data Science & MLAdvanced

Softmax Function

\sigma(z_i) = \frac{e^{z_i}}{\sum_{j=1}^{K} e^{z_j}}

Converts a vector of scores into a probability distribution.

Data Science & MLBasic

ReLU Activation

f(x) = \max(0, x)

Rectified Linear Unit: outputs the input if positive, else zero.

Data Science & MLAdvanced

Binary Cross-Entropy Loss

L = -\frac{1}{n}\sum[y\log(\hat{y}) + (1-y)\log(1-\hat{y})]

Loss function for binary classification.

Data Science & MLIntermediate

Cosine Similarity

\cos\theta = \frac{A \cdot B}{\|A\|\|B\|}

Measures similarity between two vectors by the angle between them.

Data Science & MLBasic

Euclidean Distance

d = \sqrt{\sum_{i=1}^{n}(a_i - b_i)^2}

Straight-line distance between two points in n-dimensional space.

Data Science & MLBasic

Z-Score Normalization

z = \frac{x - \mu}{\sigma}

Standardizes features to zero mean and unit variance.

Data Science & MLBasic

Min-Max Normalization

x' = \frac{x - x_{min}}{x_{max} - x_{min}}

Rescales features to the range [0, 1].

Data Science & MLIntermediate

TF-IDF Weight

w_{t,d} = tf_{t,d} \times \log\frac{N}{df_t}

Term importance: frequency in a document scaled by rarity across documents.

Linear AlgebraAdvanced

Inverse of a 2×2 Matrix

A^{-1} = \frac{1}{ad-bc}\begin{pmatrix} d & -b \\ -c & a \end{pmatrix}

Inverse of a 2×2 matrix (ad−bc ≠ 0).

Linear AlgebraIntermediate

Trace of a Matrix

\text{tr}(A) = \sum_i a_{ii}

Sum of the diagonal entries of a square matrix.

Linear AlgebraAdvanced

Vector Projection

\text{proj}_{\vec{b}}\vec{a} = \frac{\vec{a}\cdot\vec{b}}{|\vec{b}|^2}\vec{b}

Projection of vector a onto vector b.

Linear AlgebraIntermediate

Matrix Transpose

(A^T)_{ij} = a_{ji}

Reflects a matrix across its main diagonal.

Linear AlgebraAdvanced

Determinant (3×3)

\det = a(ei-fh) - b(di-fg) + c(dh-eg)

Determinant of a 3×3 matrix by cofactor expansion.

Linear AlgebraIntermediate

Matrix Trace

\text{tr}(A) = \sum_i a_{ii}

Sum of the diagonal entries of a square matrix.

Linear AlgebraIntermediate

Angle Between Vectors

\cos\theta = \frac{\mathbf{a}\cdot\mathbf{b}}{\|a\|\|b\|}

Angle from the dot product and magnitudes.

Linear AlgebraAdvanced

Rank-Nullity Theorem

\text{rank}(A) + \text{nullity}(A) = n

Relates rank and null space dimension to columns.

Linear AlgebraBasic

Unit Vector

\hat{\mathbf{v}} = \frac{\mathbf{v}}{\|\mathbf{v}\|}

Vector of length one in the same direction.

PhysicsBasic

Newton's Second Law

F = ma

Force equals mass times acceleration.

PhysicsBasic

Kinetic Energy

KE = \frac{1}{2}mv^2

Energy of a moving object.

PhysicsBasic

Power

P = \frac{W}{t}

Rate of doing work.

PhysicsBasic

Kinematics: Velocity

v = u + at

Final velocity under constant acceleration.

PhysicsBasic

Kinematics: Displacement

s = ut + \frac{1}{2}at^2

Displacement under constant acceleration.

PhysicsBasic

Kinematics: v²

v^2 = u^2 + 2as

Velocity–displacement relation under constant acceleration.

PhysicsAdvanced

Coulomb's Law

F = k\frac{q_1 q_2}{r^2}

Electrostatic force between two charges.

PhysicsAdvanced

Snell's Law

n_1\sin\theta_1 = n_2\sin\theta_2

Refraction of light between two media.

PhysicsIntermediate

Newton's Gravitation

F = G\frac{m_1 m_2}{r^2}

Attractive force between two masses.

PhysicsAdvanced

Simple Pendulum Period

T = 2\pi\sqrt{\frac{L}{g}}

Period of a simple pendulum.

PhysicsAdvanced

Centripetal Force

F = \frac{mv^2}{r}

Force needed to keep an object in circular motion.

PhysicsBasic

Density

\rho = \frac{m}{V}

Mass per unit volume.

PhysicsBasic

Pressure

P = \frac{F}{A}

Force per unit area.

PhysicsAdvanced

Doppler Effect

f' = f\frac{v \pm v_o}{v \mp v_s}

Observed frequency shift due to relative motion.

PhysicsBasic

Speed

v = \frac{d}{t}

Speed equals distance divided by time.

PhysicsBasic

Acceleration

a = \frac{\Delta v}{t}

Rate of change of velocity.

PhysicsBasic

Frequency and Period

f = \frac{1}{T}

Frequency is the inverse of the period.

PhysicsBasic

Mechanical Advantage

MA = \frac{F_{out}}{F_{in}}

Force amplification of a simple machine.

PhysicsBasic

Hooke Law

F = -kx

Restoring force of a spring proportional to displacement.

PhysicsIntermediate

Rotational Kinetic Energy

KE = \frac{1}{2}I\omega^2

Kinetic energy of a rotating body.

PhysicsAdvanced

Gravitational Potential Energy (General)

U = -\frac{GMm}{r}

Potential energy of two masses at distance r.

PhysicsAdvanced

Orbital Velocity

v = \sqrt{\frac{GM}{r}}

Speed needed for a circular orbit.

PhysicsAdvanced

Escape Velocity

v_e = \sqrt{\frac{2GM}{r}}

Minimum speed to escape a gravitational field.

PhysicsIntermediate

Spring Oscillation Period

T = 2\pi\sqrt{\frac{m}{k}}

Period of a mass on a spring.

PhysicsBasic

Friction Force

f = \mu N

Resistive force between surfaces.

PhysicsAdvanced

Terminal Velocity

v_t = \sqrt{\frac{2mg}{\rho A C_d}}

Constant fall speed when drag balances gravity.

PhysicsIntermediate

Projectile Range

R = \frac{v^2\sin 2\theta}{g}

Horizontal distance of a projectile on level ground.

PhysicsIntermediate

Centripetal Force

F = \frac{mv^2}{r}

Inward force needed for circular motion.

PhysicsBasic

Power from Work

P = \frac{W}{t}

Rate of doing work.

PhysicsIntermediate

Elastic Potential Energy

U = \frac{1}{2}kx^2

Energy stored in a stretched or compressed spring.

Physics Laws & TheoremsBasic

Newton's First Law (Inertia)

\sum F = 0 \Rightarrow v = \text{const}

An object stays at rest or in uniform motion unless acted on by a net force.

Physics Laws & TheoremsBasic

Newton's Second Law

F = ma = \frac{dp}{dt}

Net force equals mass times acceleration (rate of change of momentum).

Physics Laws & TheoremsBasic

Newton's Third Law

F_{AB} = -F_{BA}

For every action there is an equal and opposite reaction.

Physics Laws & TheoremsIntermediate

Archimedes' Principle

F_b = \rho g V

Buoyant force equals the weight of fluid displaced.

Physics Laws & TheoremsIntermediate

Pascal's Law

P = \frac{F_1}{A_1} = \frac{F_2}{A_2}

Pressure applied to a confined fluid transmits equally in all directions.

Physics Laws & TheoremsAdvanced

Bernoulli's Principle

P + \tfrac{1}{2}\rho v^2 + \rho g h = \text{const}

Faster-moving fluid has lower pressure (energy conservation in flow).

Physics Laws & TheoremsIntermediate

Law of Universal Gravitation

F = G\frac{m_1 m_2}{r^2}

Every mass attracts every other mass.

Physics Laws & TheoremsAdvanced

Kepler's First Law

r = \frac{a(1-e^2)}{1 + e\cos\theta}

Planets orbit the Sun in ellipses with the Sun at one focus.

Physics Laws & TheoremsAdvanced

Kepler's Second Law

\frac{dA}{dt} = \text{const}

A line from a planet to the Sun sweeps equal areas in equal times.

Physics Laws & TheoremsIntermediate

Zeroth Law of Thermodynamics

A = B, \ B = C \Rightarrow A = C

Two systems in thermal equilibrium with a third are in equilibrium with each other.

Physics Laws & TheoremsBasic

Charles's Law

\frac{V_1}{T_1} = \frac{V_2}{T_2}

At constant pressure, volume of a gas is proportional to absolute temperature.

Physics Laws & TheoremsBasic

Gay-Lussac's Law

\frac{P_1}{T_1} = \frac{P_2}{T_2}

At constant volume, pressure of a gas is proportional to absolute temperature.

Physics Laws & TheoremsAdvanced

Stefan–Boltzmann Law

j = \sigma T^4

Total energy radiated by a black body scales with the fourth power of temperature.

Physics Laws & TheoremsBasic

Ohm's Law

V = IR

Voltage across a resistor equals current times resistance.

Physics Laws & TheoremsIntermediate

Coulomb's Law

F = k\frac{q_1 q_2}{r^2}

Electrostatic force between two point charges.

Physics Laws & TheoremsAdvanced

Gauss's Law

\oint \vec{E} \cdot d\vec{A} = \frac{Q}{\varepsilon_0}

Electric flux through a closed surface is proportional to the enclosed charge.

Physics Laws & TheoremsAdvanced

Faraday's Law of Induction

\varepsilon = -\frac{d\Phi_B}{dt}

A changing magnetic flux induces an electromotive force.

Physics Laws & TheoremsAdvanced

Lenz's Law

\varepsilon \propto -\frac{d\Phi_B}{dt}

Induced current opposes the change in flux that produced it.

Physics Laws & TheoremsIntermediate

Lens Maker's / Thin Lens

\frac{1}{f} = \frac{1}{v} - \frac{1}{u}

Relates focal length to object and image distances for a thin lens.

Physics Laws & TheoremsIntermediate

Snell's Law

n_1 \sin\theta_1 = n_2 \sin\theta_2

Light bends when passing between media of different refractive index.

Physics Laws & TheoremsAdvanced

Heisenberg Uncertainty Principle

\Delta x \, \Delta p \ge \frac{\hbar}{2}

Position and momentum cannot both be known with unlimited precision.

Physics Laws & TheoremsBasic

Newton First Law

\sum F = 0 \Rightarrow \text{constant } v

An object stays at rest or in uniform motion unless acted upon.

Physics Laws & TheoremsBasic

Newton Third Law

F_{AB} = -F_{BA}

Every action has an equal and opposite reaction.

Physics Laws & TheoremsBasic

Hooke Law

F = -kx

Elastic restoring force is proportional to displacement.

Physics Laws & TheoremsAdvanced

Faraday Law of Induction

\varepsilon = -\frac{d\Phi_B}{dt}

A changing magnetic flux induces an EMF.

Physics Laws & TheoremsIntermediate

Archimedes Principle

F_b = \rho V g

Buoyant force equals the weight of displaced fluid.

Thermodynamics & HeatBasic

Charles's Law

\frac{V_1}{T_1} = \frac{V_2}{T_2}

Volume is proportional to temperature at constant P.

Thermodynamics & HeatIntermediate

Gay-Lussac's Law

\frac{P_1}{T_1} = \frac{P_2}{T_2}

Pressure is proportional to temperature at constant V.

Thermodynamics & HeatIntermediate

Combined Gas Law

\frac{P_1 V_1}{T_1} = \frac{P_2 V_2}{T_2}

Combines Boyle, Charles and Gay-Lussac.

Thermodynamics & HeatAdvanced

Carnot Efficiency

\eta = 1 - \frac{T_C}{T_H}

Maximum efficiency of a heat engine.

Thermodynamics & HeatAdvanced

Entropy Change

\Delta S = \frac{Q}{T}

Reversible entropy change at temperature T.

Thermodynamics & HeatAdvanced

Heat Conduction (Fourier)

Q = \frac{kA\Delta T\,t}{d}

Heat conducted through a slab over time.

Thermodynamics & HeatBasic

Celsius to Fahrenheit

F = \tfrac{9}{5}C + 32

Convert Celsius to Fahrenheit.

Thermodynamics & HeatAdvanced

RMS Molecular Speed

v_{rms} = \sqrt{\frac{3RT}{M}}

Root-mean-square speed of gas molecules.

Thermodynamics & HeatAdvanced

Average KE of a Gas Molecule

\bar{E} = \tfrac{3}{2}k_B T

Mean translational kinetic energy per molecule.

Thermodynamics & HeatAdvanced

Coefficient of Performance

COP = \frac{Q_C}{W}

Efficiency measure of a refrigerator or heat pump.

Optics & WavesBasic

Wave Period

T = \frac{1}{f}

Time for one complete oscillation.

Optics & WavesIntermediate

Snell's Law

n_1\sin\theta_1 = n_2\sin\theta_2

Refraction of light between two media.

Optics & WavesIntermediate

Refractive Index

n = \frac{c}{v}

Ratio of light speed in vacuum to that in a medium.

Optics & WavesAdvanced

Critical Angle

\theta_c = \sin^{-1}\frac{n_2}{n_1}

Angle beyond which total internal reflection occurs.

Optics & WavesIntermediate

Thin Lens Equation

\frac{1}{f} = \frac{1}{d_o} + \frac{1}{d_i}

Relates focal length to object and image distances.

Optics & WavesIntermediate

Lens Magnification

M = -\frac{d_i}{d_o}

Image size relative to object size.

Optics & WavesIntermediate

Mirror Equation

\frac{1}{f} = \frac{1}{d_o} + \frac{1}{d_i}

Focal length relation for curved mirrors.

Optics & WavesBasic

Lens Power

P = \frac{1}{f}

Optical power in dioptres (f in metres).

Optics & WavesAdvanced

Diffraction Grating

d\sin\theta = m\lambda

Condition for constructive interference maxima.

Optics & WavesAdvanced

Double-Slit Fringe Spacing

\Delta y = \frac{\lambda L}{d}

Spacing of interference fringes on a screen.

Optics & WavesAdvanced

Doppler Effect (Sound)

f' = f\frac{v \pm v_o}{v \mp v_s}

Observed frequency shift from relative motion.

Optics & WavesIntermediate

Sound Intensity Level

\beta = 10\log_{10}\frac{I}{I_0}

Loudness in decibels relative to threshold.

Optics & WavesIntermediate

Photon Energy (Wavelength)

E = \frac{hc}{\lambda}

Photon energy from its wavelength.

Optics & WavesAdvanced

Standing Wave (String)

f_n = \frac{nv}{2L}

Harmonic frequencies of a fixed string.

Optics & WavesAdvanced

Wave Speed on a String

v = \sqrt{\frac{T}{\mu}}

Speed depends on tension and linear density.

Electricity & MagnetismIntermediate

Coulomb's Law

F = k\frac{q_1 q_2}{r^2}

Force between two point charges.

Electricity & MagnetismBasic

Electric Field

E = \frac{F}{q}

Force per unit charge at a point.

Electricity & MagnetismIntermediate

Field of a Point Charge

E = k\frac{q}{r^2}

Electric field around a point charge.

Electricity & MagnetismIntermediate

Electric Potential

V = k\frac{q}{r}

Potential due to a point charge.

Electricity & MagnetismIntermediate

Electric Potential Energy

U = k\frac{q_1 q_2}{r}

Energy of two interacting charges.

Electricity & MagnetismAdvanced

Gauss's Law

\Phi_E = \frac{Q_{enc}}{\varepsilon_0}

Electric flux through a closed surface.

Electricity & MagnetismAdvanced

Parallel-Plate Capacitance

C = \frac{\varepsilon_0 A}{d}

Capacitance from plate area and separation.

Electricity & MagnetismAdvanced

Field of a Straight Wire

B = \frac{\mu_0 I}{2\pi r}

Magnetic field around a long straight wire.

Electricity & MagnetismAdvanced

Faraday's Law

\mathcal{E} = -N\frac{d\Phi_B}{dt}

Induced EMF from changing magnetic flux.

Electricity & MagnetismIntermediate

Magnetic Flux

\Phi_B = BA\cos\theta

Flux through a surface in a magnetic field.

Electricity & MagnetismAdvanced

Cyclotron Frequency

f = \frac{qB}{2\pi m}

Rotation frequency of a charge in a magnetic field.

Electricity & MagnetismAdvanced

EM Energy Density

u = \tfrac{1}{2}\varepsilon_0 E^2 + \frac{B^2}{2\mu_0}

Energy stored per unit volume in EM fields.

Electricity & MagnetismAdvanced

Speed of Light (EM)

c = \frac{1}{\sqrt{\mu_0 \varepsilon_0}}

Light speed from vacuum constants.

Fluid MechanicsBasic

Pressure

P = \frac{F}{A}

Force per unit area.

Fluid MechanicsIntermediate

Archimedes' Principle

F_b = \rho g V

Buoyant force equals weight of displaced fluid.

Fluid MechanicsAdvanced

Bernoulli's Equation

P + \tfrac{1}{2}\rho v^2 + \rho g h = \text{const}

Energy conservation along a streamline.

Fluid MechanicsAdvanced

Reynolds Number

Re = \frac{\rho v L}{\mu}

Predicts laminar vs turbulent flow.

Fluid MechanicsAdvanced

Poiseuille's Law

Q = \frac{\pi r^4 \Delta P}{8\mu L}

Laminar flow rate through a pipe.

Fluid MechanicsIntermediate

Surface Tension

\gamma = \frac{F}{L}

Force per unit length along a liquid surface.

Fluid MechanicsIntermediate

Hydraulic Press (Pascal)

\frac{F_1}{A_1} = \frac{F_2}{A_2}

Pressure transmits equally in a hydraulic system.

Fluid MechanicsAdvanced

Drag Force

F_D = \tfrac{1}{2}\rho v^2 C_D A

Aerodynamic drag on a moving body.

Fluid MechanicsBasic

Specific Gravity

SG = \frac{\rho}{\rho_{water}}

Density relative to water.

Modern & Quantum PhysicsAdvanced

Time Dilation

\Delta t = \frac{\Delta t_0}{\sqrt{1 - v^2/c^2}}

Moving clocks run slow.

Modern & Quantum PhysicsAdvanced

Length Contraction

L = L_0\sqrt{1 - v^2/c^2}

Moving objects shorten along motion.

Modern & Quantum PhysicsAdvanced

Relativistic Momentum

p = \frac{mv}{\sqrt{1 - v^2/c^2}}

Momentum at speeds near light.

Modern & Quantum PhysicsAdvanced

Lorentz Factor

\gamma = \frac{1}{\sqrt{1 - v^2/c^2}}

Scaling factor in special relativity.

Modern & Quantum PhysicsAdvanced

de Broglie Wavelength

\lambda = \frac{h}{p}

Wavelength associated with a particle.

Modern & Quantum PhysicsAdvanced

Uncertainty Principle

\Delta x\,\Delta p \geq \frac{\hbar}{2}

Limit on knowing position and momentum together.

Modern & Quantum PhysicsAdvanced

Bohr Energy Levels

E_n = -\frac{13.6}{n^2}\ \text{eV}

Energy of the nth hydrogen orbit.

Modern & Quantum PhysicsIntermediate

Radioactive Decay

N = N_0 e^{-\lambda t}

Number of nuclei remaining over time.

Modern & Quantum PhysicsIntermediate

Half-Life

t_{1/2} = \frac{\ln 2}{\lambda}

Time for half a sample to decay.

Modern & Quantum PhysicsIntermediate

Radioactive Activity

A = \lambda N

Decays per second in a sample.

Modern & Quantum PhysicsAdvanced

Wien's Displacement Law

\lambda_{max} = \frac{b}{T}

Peak emission wavelength of a hot body.

Modern & Quantum PhysicsAdvanced

Compton Shift

\Delta\lambda = \frac{h}{m_e c}(1 - \cos\theta)

Wavelength change of scattered photons.

Electronics & CircuitsBasic

Power (V²/R)

P = \frac{V^2}{R}

Power from voltage and resistance.

Electronics & CircuitsIntermediate

Resistors in Parallel

\frac{1}{R_{eq}} = \frac{1}{R_1} + \frac{1}{R_2} + \cdots

Reciprocals of parallel resistances add.

Electronics & CircuitsIntermediate

Capacitors in Parallel

C_{eq} = C_1 + C_2 + \cdots

Parallel capacitances add.

Electronics & CircuitsIntermediate

Capacitors in Series

\frac{1}{C_{eq}} = \frac{1}{C_1} + \frac{1}{C_2} + \cdots

Reciprocals of series capacitances add.

Electronics & CircuitsBasic

Capacitance

C = \frac{Q}{V}

Charge stored per unit voltage.

Electronics & CircuitsIntermediate

Capacitor Energy

E = \tfrac{1}{2}CV^2

Energy stored in a capacitor.

Electronics & CircuitsIntermediate

Inductor Energy

E = \tfrac{1}{2}LI^2

Energy stored in a magnetic field.

Electronics & CircuitsAdvanced

Capacitive Reactance

X_C = \frac{1}{2\pi f C}

Opposition of a capacitor to AC.

Electronics & CircuitsAdvanced

Inductive Reactance

X_L = 2\pi f L

Opposition of an inductor to AC.

Electronics & CircuitsAdvanced

Impedance (RLC)

Z = \sqrt{R^2 + (X_L - X_C)^2}

Total opposition in an AC circuit.

Electronics & CircuitsAdvanced

Resonant Frequency

f_0 = \frac{1}{2\pi\sqrt{LC}}

Frequency at which an LC circuit resonates.

Electronics & CircuitsIntermediate

RC Time Constant

\tau = RC

Time to charge a capacitor to ~63%.

Electronics & CircuitsIntermediate

RL Time Constant

\tau = \frac{L}{R}

Time constant of an inductor-resistor circuit.

Electronics & CircuitsIntermediate

Voltage Divider

V_{out} = V_{in}\frac{R_2}{R_1 + R_2}

Output voltage from two series resistors.

Electronics & CircuitsAdvanced

Current Divider

I_1 = I\frac{R_2}{R_1 + R_2}

Current split between two parallel resistors.

Electronics & CircuitsIntermediate

RMS Voltage

V_{rms} = \frac{V_{peak}}{\sqrt{2}}

Effective value of a sinusoidal voltage.

Electronics & CircuitsAdvanced

AC Real Power

P = VI\cos\phi

Real power with a phase angle between V and I.

Electronics & CircuitsIntermediate

Decibel (Power)

dB = 10\log_{10}\frac{P_2}{P_1}

Power ratio expressed in decibels.

Electronics & CircuitsIntermediate

Decibel (Voltage)

dB = 20\log_{10}\frac{V_2}{V_1}

Voltage ratio in decibels.

Electronics & CircuitsIntermediate

Transformer Turns Ratio

\frac{V_p}{V_s} = \frac{N_p}{N_s}

Voltage ratio equals turns ratio.

Electronics & CircuitsAdvanced

Inverting Amplifier Gain

A_v = -\frac{R_f}{R_{in}}

Gain of an inverting op-amp.

Electronics & CircuitsAdvanced

Non-Inverting Gain

A_v = 1 + \frac{R_f}{R_{in}}

Gain of a non-inverting op-amp.

Electronics & CircuitsBasic

LED Series Resistor

R = \frac{V_s - V_f}{I}

Resistor to safely drive an LED.

Electronics & CircuitsAdvanced

Drift Velocity

v_d = \frac{I}{nAq}

Average velocity of charge carriers.

Electronics & CircuitsIntermediate

Wire Resistance

R = \frac{\rho L}{A}

Resistance from resistivity, length and area.

Electronics & CircuitsAdvanced

Apparent Power

S = \sqrt{P^2 + Q^2}

Combination of real and reactive power.

Electronics & CircuitsAdvanced

Resonant Frequency

f_0 = \frac{1}{2\pi\sqrt{LC}}

Frequency where an LC circuit resonates.

Electronics & CircuitsAdvanced

Quality Factor

Q = \frac{f_0}{\Delta f}

Sharpness of a resonant peak.

Electronics & CircuitsIntermediate

Capacitor Energy

E = \frac{1}{2}CV^2

Energy stored in a capacitor.

Electronics & CircuitsIntermediate

Inductor Energy

E = \frac{1}{2}LI^2

Energy stored in an inductor magnetic field.

Electronics & CircuitsBasic

Capacitor Charge

Q = CV

Charge stored on a capacitor.

Electronics & CircuitsBasic

Current from Charge

I = \frac{Q}{t}

Current as the rate of charge flow.

Electronics & CircuitsIntermediate

LED Series Resistor

R = \frac{V_s - V_f}{I}

Resistor to safely drive an LED.

Electronics & CircuitsAdvanced

Wheatstone Bridge Balance

\frac{R_1}{R_2} = \frac{R_3}{R_4}

Condition for a balanced measurement bridge.

Electronics & CircuitsBasic

Voltage Divider Output

V_{out} = V_{in}\frac{R_2}{R_1 + R_2}

Output of a two-resistor divider.

ChemistryBasic

Molarity

M = \frac{n}{V}

Concentration in moles of solute per litre of solution.

ChemistryBasic

Moles from Mass

n = \frac{m}{M}

Number of moles from mass and molar mass.

ChemistryBasic

pH

pH = -\log_{10}[H^+]

Acidity from hydrogen-ion concentration.

ChemistryAdvanced

Henderson–Hasselbalch

pH = pK_a + \log\frac{[A^-]}{[HA]}

pH of a buffer solution.

ChemistryIntermediate

Rate Law

r = k[A]^m[B]^n

Reaction rate as a function of reactant concentrations.

ChemistryAdvanced

Arrhenius Equation

k = A e^{-E_a/RT}

Temperature dependence of the reaction rate constant.

ChemistryAdvanced

Gibbs Free Energy

\Delta G = \Delta H - T\Delta S

Predicts spontaneity of a reaction.

ChemistryAdvanced

Nernst Equation

E = E^\circ - \frac{RT}{nF}\ln Q

Cell potential under non-standard conditions.

ChemistryAdvanced

Equilibrium Constant

K = \frac{[C]^c[D]^d}{[A]^a[B]^b}

Ratio of product to reactant concentrations at equilibrium.

ChemistryIntermediate

Combined Gas Law

\frac{P_1 V_1}{T_1} = \frac{P_2 V_2}{T_2}

Relates two states of a fixed amount of gas.

ChemistryIntermediate

Percent Yield

\% = \frac{\text{actual}}{\text{theoretical}} \times 100

Efficiency of a chemical reaction.

ChemistryBasic

Density

\rho = \frac{m}{V}

Mass per unit volume of a substance.

ChemistryBasic

Percent Composition

\% = \frac{\text{mass of element}}{\text{molar mass}} \times 100

Mass percentage of an element in a compound.

ChemistryIntermediate

Charles's Law

\frac{V_1}{T_1} = \frac{V_2}{T_2}

Volume is proportional to temperature at constant pressure.

ChemistryBasic

Charles Law

\frac{V_1}{T_1} = \frac{V_2}{T_2}

At constant pressure, volume is proportional to temperature.

ChemistryBasic

Gay-Lussac Law

\frac{P_1}{T_1} = \frac{P_2}{T_2}

At constant volume, pressure is proportional to temperature.

ChemistryAdvanced

Gas Density

\rho = \frac{PM}{RT}

Density of an ideal gas.

ChemistryAdvanced

Rate Law

rate = k[A]^m[B]^n

Reaction rate as a function of reactant concentrations.

ChemistryAdvanced

First-Order Half-Life

t_{1/2} = \frac{0.693}{k}

Half-life of a first-order reaction.

ChemistryBasic

Mass Percent

\% = \frac{m_{solute}}{m_{solution}}\times 100

Concentration as a percentage by mass.

ChemistryIntermediate

Molality

b = \frac{n_{solute}}{m_{solvent}}

Moles of solute per kilogram of solvent.

ChemistryIntermediate

Combined Gas Law

\frac{P_1 V_1}{T_1} = \frac{P_2 V_2}{T_2}

Combines Boyle, Charles, and Gay-Lussac laws.

ChemistryIntermediate

Enthalpy Change

\Delta H = H_{products} - H_{reactants}

Heat absorbed or released at constant pressure.

ChemistryBasic

pH and pOH

pH + pOH = 14

Relationship between acidity and basicity at 25°C.

ChemistryAdvanced

Water Ion Product

K_w = [H^+][OH^-] = 10^{-14}

Self-ionization constant of water at 25°C.

Biology & Life SciencesAdvanced

Logistic Growth

\frac{dN}{dt} = rN\left(1 - \frac{N}{K}\right)

Population growth limited by carrying capacity.

Biology & Life SciencesIntermediate

Population Doubling Time

t_d = \frac{\ln 2}{r}

Time for a population to double.

Biology & Life SciencesBasic

Surface Area to Volume

SA:V = \frac{\text{surface area}}{\text{volume}}

Ratio governing cellular exchange efficiency.

Biology & Life SciencesBasic

Microscope Magnification

M = \frac{\text{image size}}{\text{actual size}}

How much larger an image appears than the object.

Biology & Life SciencesIntermediate

GC Content

\%GC = \frac{G + C}{A + T + G + C} \times 100

Proportion of G and C bases in DNA.

Biology & Life SciencesIntermediate

Lincoln-Petersen Estimate

N = \frac{M \times C}{R}

Population size from mark-and-recapture sampling.

Biology & Life SciencesIntermediate

Cardiac Output

CO = HR \times SV

Blood pumped by the heart per minute.

Biology & Life SciencesIntermediate

Respiratory Quotient

RQ = \frac{CO_2\ produced}{O_2\ consumed}

Ratio revealing which fuel is being metabolised.

Biology & Life SciencesAdvanced

Michaelis-Menten Kinetics

v = \frac{V_{max}[S]}{K_m + [S]}

Rate of an enzyme-catalysed reaction.

Biology & Life SciencesBasic

Cell Dilution Factor

DF = \frac{V_{final}}{V_{sample}}

Factor by which a sample is diluted.

Biology & Life SciencesIntermediate

Bacterial Generations

n = \frac{\log N_t - \log N_0}{\log 2}

Number of divisions over a growth period.

Biology & Life SciencesBasic

Bacterial Growth Count

N_t = N_0 \times 2^n

Cell count after n binary fissions.

Biology & Life SciencesAdvanced

Recombination Frequency

RF = \frac{\text{recombinants}}{\text{total offspring}} \times 100

Genetic map distance in centimorgans.

Biology & Life SciencesIntermediate

Enzyme Activity Unit

U = \frac{\mu mol\ product}{min}

One unit converts 1 µmol of substrate per minute.

Biology & Life SciencesIntermediate

Ecological Efficiency

E = \frac{\text{energy at level } n+1}{\text{energy at level } n} \times 100

Energy transfer between trophic levels (~10%).

Biology & Life SciencesAdvanced

Biochemical Oxygen Demand

BOD = \frac{D_1 - D_2}{P}

Oxygen consumed by microbes decomposing organic matter.

Biology & Life SciencesIntermediate

Surface-to-Volume Ratio

SA:V = \frac{\text{surface area}}{\text{volume}}

Ratio governing cell exchange efficiency.

Biology & Life SciencesIntermediate

Population Growth Rate

r = \frac{B - D}{N}

Per-capita growth from births and deaths.

Biology & Life SciencesIntermediate

Bacterial Doubling

N = N_0\times 2^{t/g}

Exponential bacterial growth by generations.

Biology & Life SciencesAdvanced

Respiratory Quotient

RQ = \frac{CO_2\text{ produced}}{O_2\text{ consumed}}

Ratio revealing which fuel is being metabolized.

Biology & Life SciencesAdvanced

Enzyme Turnover Number

k_{cat} = \frac{V_{max}}{[E]}

Reactions catalyzed per enzyme per second.

Health & FitnessBasic

Body Mass Index (BMI)

BMI = \frac{m}{h^2}

Weight-to-height ratio used to classify body weight.

Health & FitnessBasic

BMI (Imperial)

BMI = 703 \times \frac{m}{h^2}

BMI using pounds and inches.

Health & FitnessIntermediate

Total Daily Energy (TDEE)

TDEE = BMR \times \text{activity factor}

Daily calories burned including activity.

Health & FitnessAdvanced

Body Fat % (Navy, Men)

BF = \frac{495}{1.0324 - 0.19077\log_{10}(w-n) + 0.15456\log_{10}h} - 450

US Navy body-fat estimate for men from tape measurements.

Health & FitnessBasic

Waist-to-Hip Ratio

WHR = \frac{\text{waist}}{\text{hip}}

Fat-distribution health indicator.

Health & FitnessIntermediate

One-Rep Max (Epley)

1RM = w\left(1 + \frac{r}{30}\right)

Estimated maximum single-lift weight.

Health & FitnessAdvanced

VO₂ Max (Cooper)

VO_2 = \frac{d - 504.9}{44.73}

Aerobic fitness from a 12-minute run distance (metres).

Health & FitnessBasic

Protein from Calories

g = \frac{0.30 \times C}{4}

Grams of protein for a 30% protein diet (4 kcal/g).

Health & FitnessBasic

Running Pace

\text{pace} = \frac{t}{d}

Time taken to cover one unit of distance.

Health & FitnessAdvanced

Fat-Free Mass Index

FFMI = \frac{LBM}{h^2}

Muscularity index normalised for height.

Health & FitnessAdvanced

Corrected QT (Bazett)

QTc = \frac{QT}{\sqrt{RR}}

Heart-rate-corrected QT interval from an ECG.

Health & FitnessIntermediate

Mean Arterial Pressure

MAP = DBP + \tfrac{1}{3}(SBP - DBP)

Average blood pressure during a cardiac cycle.

Health & FitnessAdvanced

Creatinine Clearance (Cockcroft-Gault)

CrCl = \frac{(140 - a)\,m}{72 \times S_{cr}}

Kidney function estimate (men; ×0.85 for women).

Health & FitnessIntermediate

Body Surface Area (Mosteller)

BSA = \sqrt{\frac{h \times m}{3600}}

Total skin surface area of the body.

Health & FitnessBasic

Waist-to-Height Ratio

WHtR = \frac{\text{waist}}{\text{height}}

Central-fat health indicator.

Health & FitnessBasic

BMI Prime

BMI\ prime = \frac{BMI}{25}

Your BMI relative to the upper healthy limit.

Health & FitnessIntermediate

Ponderal Index

PI = \frac{m}{h^3}

A leanness index that scales better for extreme heights.

Health & FitnessIntermediate

One-Rep Max (Brzycki)

1RM = w \times \frac{36}{37 - r}

Estimated one-rep max, an alternative to Epley.

Health & FitnessBasic

Max Heart Rate (Tanaka)

HR_{max} = 208 - 0.7\,a

A more accurate maximum heart rate estimate.

Health & FitnessBasic

Fat Mass

FM = m \times \frac{BF\%}{100}

Total mass of body fat.

Health & FitnessIntermediate

MET to Oxygen Uptake

VO_2 = 3.5 \times MET

Oxygen consumption from a MET activity level.

Health & FitnessIntermediate

Fat Loss from Deficit

\Delta m = \frac{\text{deficit}}{7700}

Body-fat lost from a total calorie deficit.

Health & FitnessBasic

Speed from Pace

v = \frac{60}{\text{pace}}

Running speed in km/h from pace in min/km.

Health & FitnessAdvanced

Corrected QT (Fridericia)

QTc = \frac{QT}{\sqrt[3]{RR}}

Heart-rate-corrected QT interval using a cube root.

Health & FitnessBasic

Blood Glucose Conversion

\text{mmol/L} = \frac{\text{mg/dL}}{18}

Converts blood-sugar units between countries.

Nutrition & FoodBasic

Food Energy (Atwater)

E = 4P + 4C + 9F + 7A

Total food energy in kilocalories from its macronutrients using the Atwater factors.

Nutrition & FoodBasic

Calories from Protein

E_P = 4P

Energy contributed by protein at 4 kcal per gram.

Nutrition & FoodBasic

Calories from Carbohydrate

E_C = 4C

Energy from carbohydrate at 4 kcal per gram.

Nutrition & FoodBasic

Calories from Fat

E_F = 9F

Energy from fat at 9 kcal per gram — the most energy-dense macronutrient.

Nutrition & FoodIntermediate

Caloric (Energy) Density

D = \frac{E}{m}

Calories per gram of a food — a key idea in filling, lower-calorie eating.

Nutrition & FoodBasic

Macro % of Calories

\%_M = \frac{E_M}{E_{total}} \times 100

Share of total energy coming from a given macronutrient.

Nutrition & FoodBasic

Protein Grams from Calories

g = \frac{r \times C}{4}

Grams of protein for a target protein ratio of the diet.

Nutrition & FoodAdvanced

Glycemic Load

GL = \frac{GI \times C}{100}

Impact of a food portion on blood sugar, combining quality (GI) and quantity.

Nutrition & FoodIntermediate

Net Carbohydrates

C_{net} = C_{total} - F - \tfrac{A}{2}

Digestible carbs after subtracting fibre and half of sugar alcohols.

Nutrition & FoodBasic

Percent Daily Value (%DV)

\%DV = \frac{a}{r} \times 100

How much a nutrient in a serving contributes to daily needs.

Nutrition & FoodIntermediate

Weight Change from Energy Balance

\Delta m = \frac{E_{deficit}}{7700}

Approximate body-fat change (kg) from a cumulative calorie deficit or surplus.

Nutrition & FoodIntermediate

Baker's Percentage

\%_i = \frac{m_i}{m_{flour}} \times 100

Every ingredient expressed relative to flour weight — the backbone of bread recipes.

Nutrition & FoodIntermediate

Dough Hydration

H = \frac{m_{water}}{m_{flour}} \times 100

Water as a percentage of flour — controls how wet and open a dough is.

Nutrition & FoodBasic

Recipe Scaling Factor

m_{new} = m_{old} \times \frac{n_{want}}{n_{have}}

Rescale each ingredient for a different number of servings.

Nutrition & FoodIntermediate

Brine Salinity

S = \frac{m_{salt}}{m_{salt}+m_{water}} \times 100

Salt concentration of a brine or curing solution by mass.

Nutrition & FoodIntermediate

Yield Percentage

Y = \frac{m_{EP}}{m_{AP}} \times 100

Edible portion left after trimming, relative to the amount purchased.

Nutrition & FoodIntermediate

Food Cost Percentage

FC = \frac{c_{ingredients}}{p_{selling}} \times 100

Ingredient cost as a share of menu price — a core restaurant metric.

Nutrition & FoodBasic

Cost per Serving

c_s = \frac{c_{total}}{n}

Total recipe cost divided across the number of servings.

Nutrition & FoodBasic

Caffeine Concentration

c = \frac{m_{caffeine}}{V}

Caffeine amount per unit volume of a drink.

Nutrition & FoodAdvanced

Water Activity

a_w = \frac{p}{p_0}

Ratio of a food's vapour pressure to pure water — governs microbial spoilage.

Astronomy & SpaceIntermediate

Escape Velocity

v_e = \sqrt{\frac{2GM}{r}}

Minimum speed to escape a body's gravity.

Astronomy & SpaceIntermediate

Orbital Velocity

v = \sqrt{\frac{GM}{r}}

Speed of a circular orbit.

Astronomy & SpaceAdvanced

Kepler's Third Law

T^2 = \frac{4\pi^2}{GM}a^3

Relates orbital period to semi-major axis.

Astronomy & SpaceAdvanced

Schwarzschild Radius

r_s = \frac{2GM}{c^2}

Event horizon radius of a black hole.

Astronomy & SpaceAdvanced

Hubble's Law

v = H_0 d

Recession velocity of galaxies with distance.

Astronomy & SpaceAdvanced

Stefan–Boltzmann Law

L = 4\pi R^2 \sigma T^4

Luminosity of a star from radius and temperature.

Astronomy & SpaceAdvanced

Redshift

z = \frac{\lambda_{obs} - \lambda_{emit}}{\lambda_{emit}}

Fractional wavelength shift of light.

Astronomy & SpaceAdvanced

Parallax Distance

d = \frac{1}{p}

Distance in parsecs from parallax angle in arcseconds.

Astronomy & SpaceIntermediate

Surface Gravity

g = \frac{GM}{r^2}

Gravitational acceleration at a body's surface.

Astronomy & SpaceAdvanced

Apparent Brightness

b = \frac{L}{4\pi d^2}

Observed brightness from luminosity and distance.

Astronomy & SpaceAdvanced

Distance Modulus

m - M = 5\log_{10}(d) - 5

Relates apparent and absolute magnitude to distance.

Astronomy & SpaceBasic

Light-Year

d = c \times t

Distance light travels in a given time.

Astronomy & SpaceIntermediate

Wien's Displacement Law

\lambda_{max} = \frac{b}{T}

Peak emission wavelength of a blackbody.

Astronomy & SpaceIntermediate

Newton's Law of Gravitation

F = G\frac{m_1 m_2}{r^2}

The attractive force between any two masses.

Astronomy & SpaceAdvanced

Gravitational Potential Energy

U = -\frac{GMm}{r}

Energy of a mass in another body's gravitational field.

Astronomy & SpaceAdvanced

Mass–Luminosity Relation

\frac{L}{L_\odot} \approx \left(\frac{M}{M_\odot}\right)^{3.5}

How a main-sequence star's brightness scales with its mass.

Astronomy & SpaceAdvanced

Eddington Luminosity

L_{Edd} = \frac{4\pi G M m_p c}{\sigma_T}

The maximum steady luminosity before radiation blows a star apart.

Astronomy & SpaceAdvanced

Chandrasekhar Limit

M_{Ch} \approx 1.44\,M_\odot

The maximum mass of a stable white dwarf.

Astronomy & SpaceAdvanced

Hawking Temperature

T = \frac{\hbar c^3}{8\pi G M k_B}

Temperature of the radiation a black hole emits.

Astronomy & SpaceAdvanced

Photon Sphere Radius

r_{ph} = \frac{3GM}{c^2}

Radius where gravity bends light into a circular orbit.

Astronomy & SpaceIntermediate

Synodic Period

\frac{1}{S} = \left|\frac{1}{P_1} - \frac{1}{P_2}\right|

Time between repeated alignments of two orbiting bodies.

Astronomy & SpaceAdvanced

Gravitational Redshift

z \approx \frac{GM}{rc^2}

Fractional stretching of light climbing out of gravity.

Astronomy & SpaceAdvanced

Hill Sphere Radius

r_H \approx a\left(\frac{m}{3M}\right)^{1/3}

The zone where a body's gravity dominates over its host's.

EngineeringBasic

Stress

\sigma = \frac{F}{A}

Internal force per unit area in a material.

EngineeringBasic

Strain

\varepsilon = \frac{\Delta L}{L_0}

Relative deformation of a material.

EngineeringIntermediate

Young's Modulus

E = \frac{\sigma}{\varepsilon}

Stiffness of a material (stress over strain).

EngineeringAdvanced

Bernoulli's Equation

P + \frac{1}{2}\rho v^2 + \rho g h = \text{const}

Conservation of energy in fluid flow.

EngineeringAdvanced

Reynolds Number

Re = \frac{\rho v L}{\mu}

Predicts laminar vs turbulent flow.

EngineeringAdvanced

Second Moment of Area (Rect.)

I = \frac{bh^3}{12}

Resistance of a rectangular section to bending.

EngineeringAdvanced

Bending Stress

\sigma = \frac{My}{I}

Stress in a beam due to bending.

EngineeringBasic

Efficiency

\eta = \frac{P_{out}}{P_{in}} \times 100\%

Ratio of useful output to input power.

EngineeringIntermediate

Resistors in Parallel

\frac{1}{R_{eq}} = \frac{1}{R_1} + \frac{1}{R_2} + \cdots

Total resistance of resistors in parallel.

EngineeringIntermediate

Factor of Safety

FoS = \frac{\sigma_{ult}}{\sigma_{allow}}

Ratio of ultimate strength to allowable stress.

EngineeringBasic

Gear Ratio

GR = \frac{N_{out}}{N_{in}}

Ratio of teeth (or speeds) between gears.

EngineeringBasic

Mechanical Advantage

MA = \frac{F_{out}}{F_{in}}

Force amplification of a machine.

EngineeringBasic

Efficiency

\eta = \frac{P_{out}}{P_{in}}\times 100

Ratio of useful output to input power.

EngineeringAdvanced

Conductive Heat Transfer

Q = \frac{kA\Delta T}{d}

Heat conducted through a material.

EngineeringIntermediate

Strain Energy

U = \frac{1}{2}F\delta

Energy stored in an elastically deformed body.

EngineeringIntermediate

Power Factor

PF = \frac{P}{S}

Ratio of real power to apparent power.

EngineeringIntermediate

Transformer Ratio

\frac{V_p}{V_s} = \frac{N_p}{N_s}

Voltage ratio equals the turns ratio.

EngineeringBasic

Factor of Safety

FoS = \frac{\text{failure load}}{\text{working load}}

Ratio of failure capacity to actual load.

EngineeringIntermediate

RC Time Constant

\tau = RC

Charging time constant of a resistor-capacitor circuit.

EngineeringAdvanced

Synchronous Speed

n_s = \frac{120 f}{P}

Rotational speed of an AC motor field.

Materials & CivilBasic

Stress

\sigma = \frac{F}{A}

Internal force per unit cross-sectional area.

Materials & CivilBasic

Strain

\varepsilon = \frac{\Delta L}{L_0}

Fractional deformation under load.

Materials & CivilIntermediate

Young Modulus

E = \frac{\sigma}{\varepsilon}

Ratio of stress to strain; measures material stiffness.

Materials & CivilIntermediate

Factor of Safety

FoS = \frac{\sigma_{ult}}{\sigma_{allow}}

Ratio of a material strength to the actual applied stress.

Materials & CivilAdvanced

Cantilever Beam Deflection

\delta = \frac{F L^3}{3 E I}

Maximum deflection of a cantilever with an end load.

Materials & CivilAdvanced

Rectangular Area Moment of Inertia

I = \frac{b h^3}{12}

Second moment of area for a rectangular cross-section.

Materials & CivilAdvanced

Bending Stress

\sigma = \frac{M c}{I}

Stress in a beam due to a bending moment.

Materials & CivilBasic

Uniformly Distributed Load

w = \frac{W}{L}

Load per unit length of a beam.

Materials & CivilAdvanced

Max Moment (Simple Beam, UDL)

M_{max} = \frac{w L^2}{8}

Maximum bending moment for a simply supported beam under uniform load.

Materials & CivilBasic

Density

\rho = \frac{m}{V}

Mass per unit volume of a material.

Materials & CivilAdvanced

Slenderness Ratio

\lambda = \frac{L_e}{r}

Ratio governing column buckling.

Materials & CivilAdvanced

Euler Buckling Load

P_{cr} = \frac{\pi^2 E I}{(K L)^2}

Critical axial load at which a column buckles.

Materials & CivilIntermediate

Hydraulic Radius

R_h = \frac{A}{P}

Cross-sectional flow area divided by wetted perimeter.

Materials & CivilIntermediate

Water-Cement Ratio

w/c = \frac{m_{water}}{m_{cement}}

Ratio controlling concrete strength and workability.

Aviation & AeronauticsBasic

True Airspeed (rule of thumb)

TAS \approx IAS + (IAS \times 0.02 \times \tfrac{alt}{1000})

Quick estimate: true airspeed increases about 2% per 1000 ft of altitude above indicated airspeed.

Aviation & AeronauticsBasic

Flight Time

t = \dfrac{d}{GS}

Time en route equals distance divided by ground speed.

Aviation & AeronauticsBasic

Rate of Climb

ROC = \dfrac{\Delta h}{t}

Vertical speed is the altitude gained per unit time (ft per minute).

Aviation & AeronauticsBasic

3-Degree Descent Rate

ROD \approx GS \times 5

On a standard 3° glide path, required descent rate (fpm) is roughly ground speed (kt) times 5.

Aviation & AeronauticsBasic

Wind Components

HW = W\cos\theta,\; XW = W\sin\theta

Resolve wind into headwind and crosswind components relative to runway/track.

Aviation & AeronauticsBasic

Fuel Uplift

V_{up} = \dfrac{F_{req} - F_{OB}}{\rho}

Fuel to be loaded: the required ramp/block fuel minus fuel already on board, converted from mass to volume using fuel density.

Aviation & AeronauticsIntermediate

Lift Equation

L = \tfrac{1}{2}\rho v^2 S C_L

Aerodynamic lift produced by a wing depends on air density, speed squared, area and lift coefficient.

Aviation & AeronauticsIntermediate

Drag Equation

D = \tfrac{1}{2}\rho v^2 S C_D

Aerodynamic drag force on the aircraft, analogous to lift with a drag coefficient.

Aviation & AeronauticsIntermediate

Stall Speed

V_s = \sqrt{\dfrac{2W}{\rho S C_{L,max}}}

Minimum speed for level flight, set by maximum lift coefficient.

Aviation & AeronauticsIntermediate

Load Factor in a Turn

n = \dfrac{1}{\cos\phi}

The g-load in a level, coordinated turn depends only on the bank angle.

Aviation & AeronauticsIntermediate

Stall Speed vs Load Factor

V_{s,n} = V_s\sqrt{n}

Accelerated stall speed rises with the square root of the load factor.

Aviation & AeronauticsIntermediate

Turn Radius

r = \dfrac{v^2}{g\tan\phi}

Radius of a level coordinated turn from speed and bank angle.

Aviation & AeronauticsIntermediate

Rate of Turn

\omega = \dfrac{g\tan\phi}{v}

Angular rate of a coordinated turn (radians/s; a standard rate turn is 3°/s).

Aviation & AeronauticsIntermediate

Glide Ratio

GR = \dfrac{d}{h} = \dfrac{L}{D}

Horizontal distance traveled per unit of altitude lost equals the lift-to-drag ratio.

Aviation & AeronauticsIntermediate

Mach Number

M = \dfrac{v}{a}

Ratio of true airspeed to the local speed of sound.

Aviation & AeronauticsIntermediate

Speed of Sound

a = 38.94\sqrt{T}

Local speed of sound (kt) as a function of absolute air temperature (Kelvin).

Aviation & AeronauticsIntermediate

Specific Range

SR = \dfrac{GS}{\dot{f}}

Distance flown per unit of fuel; maximizing it gives best fuel economy.

Aviation & AeronauticsIntermediate

Point of No Return

PNR = \dfrac{E \cdot GS_{out} \cdot GS_{back}}{GS_{out}+GS_{back}}

Farthest distance an aircraft can fly and still return with available endurance.

Aviation & AeronauticsAdvanced

Reynolds Number

Re = \dfrac{\rho v L}{\mu}

Ratio of inertial to viscous forces; characterises airflow over an airfoil.

Aviation & AeronauticsAdvanced

Lift Coefficient

C_L = \dfrac{2L}{\rho v^2 S}

Non-dimensional lift, obtained by rearranging the lift equation.

Aviation & AeronauticsAdvanced

Induced Drag Coefficient

C_{D,i} = \dfrac{C_L^2}{\pi e AR}

Drag due to lift, decreasing with higher aspect ratio and span efficiency.

Aviation & AeronauticsAdvanced

Aspect Ratio

AR = \dfrac{b^2}{S}

Ratio of wingspan squared to wing area; higher AR reduces induced drag.

Aviation & AeronauticsAdvanced

Thrust Required (level flight)

T = \dfrac{W}{L/D}

In steady level flight, thrust must equal drag, which is weight divided by L/D.

Aviation & AeronauticsAdvanced

Breguet Range Equation

R = \dfrac{V}{c}\dfrac{L}{D}\ln\dfrac{W_i}{W_f}

Maximum range of a jet aircraft from speed, efficiency, L/D and fuel fraction.

Aviation & AeronauticsAdvanced

Maximum Endurance (jet)

E = \dfrac{1}{c}\dfrac{L}{D}\ln\dfrac{W_i}{W_f}

Longest time aloft for a jet, maximized at maximum L/D.

Aviation & AeronauticsAdvanced

ISA Temperature Lapse

T = 15 - 1.98\left(\dfrac{h}{1000}\right)

Standard atmosphere temperature decreases about 2°C per 1000 ft up to the tropopause.

Aviation & AeronauticsAdvanced

Barometric Altitude

h = \dfrac{RT}{g}\ln\dfrac{P_0}{P}

Altitude derived from the ratio of sea-level to measured pressure.

Aviation & AeronauticsAdvanced

Takeoff Ground Roll

s = \dfrac{V_{LO}^2}{2a}

Approximate ground-roll distance assuming constant acceleration to lift-off speed.

Aviation & AeronauticsAdvanced

Climb Gradient

\gamma = \dfrac{T - D}{W}

Climb angle (as a fraction) equals excess thrust divided by weight.

Aviation & AeronauticsIntermediate

Glide Ratio

GR = \frac{\text{horizontal distance}}{\text{altitude lost}}

Distance traveled forward per unit of height lost.

Aviation & AeronauticsIntermediate

Load Factor

n = \frac{L}{W}

Ratio of lift to weight, felt as g-force.

Aviation & AeronauticsAdvanced

Turn Radius

r = \frac{v^2}{g\tan\phi}

Radius of a coordinated banked turn.

Aviation & AeronauticsAdvanced

Stall Speed

V_s = \sqrt{\frac{2W}{\rho S C_{L,max}}}

Minimum speed to maintain lift.

Aviation & AeronauticsIntermediate

Mach Number

M = \frac{v}{a}

Ratio of speed to the local speed of sound.

Aviation & AeronauticsIntermediate

Rate of Climb

ROC = \frac{\text{excess power}}{W}

Vertical speed from excess engine power.

Aviation & AeronauticsIntermediate

Wing Loading

WL = \frac{W}{S}

Aircraft weight per unit wing area.

Aviation & AeronauticsBasic

Endurance

E = \frac{\text{fuel}}{\text{fuel flow}}

Time an aircraft can stay airborne.

Aviation & AeronauticsAdvanced

True Airspeed (approx)

TAS \approx IAS\left(1 + \frac{h}{60000}\right)

Airspeed corrected for altitude.

Finance & InterestBasic

Compound Interest

A = P\left(1 + \frac{r}{n}\right)^{nt}

Value of principal compounded n times per year.

Finance & InterestBasic

Present Value

PV = \frac{FV}{(1 + r)^n}

Present worth of a future sum.

Finance & InterestAdvanced

Future Value of Annuity

FV = P\frac{(1+r)^n - 1}{r}

Future value of regular equal payments.

Finance & InterestAdvanced

Present Value of Annuity

PV = P\frac{1 - (1+r)^{-n}}{r}

Present value of a series of equal payments.

Finance & InterestAdvanced

Loan EMI

EMI = \frac{P r (1+r)^n}{(1+r)^n - 1}

Equated monthly installment on a loan.

Finance & InterestAdvanced

CAGR

CAGR = \left(\frac{V_f}{V_i}\right)^{1/n} - 1

Compound annual growth rate of an investment.

Finance & InterestBasic

Return on Investment

ROI = \frac{V_f - V_i}{V_i} \times 100\%

Percentage gain or loss on an investment.

Finance & InterestBasic

Rule of 72

t \approx \frac{72}{r}

Approximate years to double an investment.

Finance & InterestAdvanced

Net Present Value

NPV = \sum_{t=0}^{n}\frac{C_t}{(1+r)^t}

Present value of a stream of cash flows.

Finance & InterestAdvanced

Perpetuity

PV = \frac{C}{r}

Present value of an infinite constant cash flow.

Finance & InterestBasic

Straight-Line Depreciation

D = \frac{C - S}{n}

Annual depreciation of an asset.

Finance & InterestAdvanced

Effective Annual Rate

EAR = \left(1 + \frac{r}{n}\right)^n - 1

True annual rate accounting for compounding.

Finance & InterestIntermediate

Break-Even Point

Q = \frac{FC}{P - VC}

Units to sell to cover all costs.

Finance & InterestBasic

Profit

\text{Profit} = \text{Revenue} - \text{Cost}

The money left after subtracting cost from revenue.

Finance & InterestBasic

Profit Margin

\text{Margin} = \frac{\text{Profit}}{\text{Revenue}} \times 100\%

Profit as a percentage of revenue.

Finance & InterestBasic

Markup

\text{Markup} = \frac{\text{Price} - \text{Cost}}{\text{Cost}} \times 100\%

Increase over cost as a percentage.

Finance & InterestIntermediate

Price-to-Earnings Ratio

PE = \frac{\text{Price}}{\text{EPS}}

Share price relative to earnings per share.

Finance & InterestBasic

Earnings Per Share

EPS = \frac{\text{Net Income}}{\text{Shares}}

Company profit allocated to each share.

Finance & InterestBasic

Dividend Yield

Y = \frac{\text{Dividend}}{\text{Price}}\times 100

Annual dividend as a percentage of share price.

Finance & InterestIntermediate

Break-Even Point

Q = \frac{FC}{P - VC}

Units to sell to cover all costs.

Finance & InterestBasic

Gross Margin

GM = \frac{Revenue - COGS}{Revenue}\times 100

Percentage of revenue left after cost of goods.

Finance & InterestIntermediate

Current Ratio

CR = \frac{\text{Current Assets}}{\text{Current Liabilities}}

Liquidity measure of short-term solvency.

Finance & InterestIntermediate

Debt-to-Equity Ratio

D/E = \frac{\text{Total Debt}}{\text{Equity}}

Leverage measure comparing debt to equity.

Finance & InterestIntermediate

Return on Equity

ROE = \frac{\text{Net Income}}{\text{Equity}}\times 100

Profit generated per dollar of equity.

Finance & InterestIntermediate

Return on Assets

ROA = \frac{\text{Net Income}}{\text{Total Assets}}\times 100

Profit generated per dollar of assets.

Finance & InterestAdvanced

Real Rate of Return

r = \frac{1+n}{1+i} - 1

Return adjusted for inflation.

Finance & InterestBasic

Payback Period

PP = \frac{\text{Initial Investment}}{\text{Annual Cash Flow}}

Time to recover an investment.

AccountingBasic

Accounting Equation

Assets = Liabilities + Equity

The foundation of double-entry bookkeeping: everything a business owns is financed either by debt or by owners’ capital.

AccountingBasic

Cost of Goods Sold

COGS = Opening\ Inventory + Purchases - Closing\ Inventory

The direct cost of the inventory actually sold during a period.

AccountingBasic

Gross Profit

GP = Revenue - COGS

What remains from sales after subtracting the direct cost of the goods sold, before operating expenses.

AccountingBasic

Net Income

NI = Revenue - Expenses

The “bottom line” — total earnings after all expenses, interest and taxes are deducted from revenue.

AccountingBasic

Retained Earnings

RE_1 = RE_0 + NI - Dividends

Cumulative profit kept in the business rather than paid out to shareholders.

AccountingIntermediate

Current Ratio

CR = \dfrac{Current\ Assets}{Current\ Liabilities}

A liquidity measure of whether a firm can cover its short-term obligations with short-term assets.

AccountingIntermediate

Quick Ratio (Acid Test)

QR = \dfrac{Current\ Assets - Inventory}{Current\ Liabilities}

A stricter liquidity test that excludes inventory, which can be slow to convert to cash.

AccountingIntermediate

Working Capital

WC = Current\ Assets - Current\ Liabilities

The short-term capital available to run day-to-day operations.

AccountingIntermediate

Debt-to-Equity Ratio

D/E = \dfrac{Total\ Debt}{Total\ Equity}

Measures financial leverage by comparing borrowed funds to owner-supplied funds.

AccountingIntermediate

Inventory Turnover

IT = \dfrac{COGS}{Average\ Inventory}

How many times inventory is sold and replaced over a period.

AccountingIntermediate

Return on Assets (ROA)

ROA = \dfrac{Net\ Income}{Total\ Assets}

How efficiently a company turns its assets into profit.

AccountingIntermediate

Return on Equity (ROE)

ROE = \dfrac{Net\ Income}{Shareholder\ Equity}

The profit generated for each dollar of shareholders’ equity.

AccountingIntermediate

Net Profit Margin

NPM = \dfrac{Net\ Income}{Revenue}\times 100\%

The percentage of revenue that survives as profit after all costs.

AccountingAdvanced

EBITDA

EBITDA = NI + Interest + Taxes + Depreciation + Amortization

Earnings before interest, taxes, depreciation and amortization — a proxy for core operating cash generation.

AccountingAdvanced

Double-Declining Balance

D = \dfrac{2}{n}\times Book\ Value

An accelerated depreciation method charging more expense in an asset’s early years.

AccountingAdvanced

Book Value per Share

BVPS = \dfrac{Equity - Preferred}{Shares\ Outstanding}

The net asset value backing each common share.

AccountingBasic

Accounting Equation

Assets = Liabilities + Equity

The fundamental balance-sheet identity.

AccountingBasic

Net Income

NI = Revenue - Expenses

Profit after all expenses.

AccountingIntermediate

Retained Earnings

RE = RE_0 + NI - Dividends

Accumulated profit kept in the business.

AccountingIntermediate

Cost of Goods Sold

COGS = Beginning + Purchases - Ending

Direct cost of goods sold in a period.

AccountingIntermediate

Quick Ratio

QR = \frac{Current Assets - Inventory}{Current Liabilities}

Stringent liquidity measure excluding inventory.

AccountingIntermediate

Inventory Turnover

IT = \frac{COGS}{\text{Avg Inventory}}

How many times inventory is sold per period.

AccountingAdvanced

Double Declining Depreciation

D = \frac{2}{n}\times BV

Accelerated depreciation of an asset.

AccountingBasic

Book Value

BV = Cost - \text{Accumulated Depreciation}

Asset value on the balance sheet.

AccountingIntermediate

Operating Margin

OM = \frac{\text{Operating Income}}{Revenue}\times 100

Profit from core operations as a percent of revenue.

AccountingIntermediate

Contribution Margin

CM = Price - \text{Variable Cost}

Revenue per unit left to cover fixed costs.

Business & CommerceIntermediate

Contribution Margin

CM = Price - Variable\ Cost

The amount each unit sold contributes toward covering fixed costs and profit.

Business & CommerceIntermediate

Contribution Margin Ratio

CMR = \dfrac{Price - Variable\ Cost}{Price}

Contribution margin expressed as a fraction of the selling price.

Business & CommerceIntermediate

Break-Even (Units)

Q = \dfrac{Fixed\ Costs}{Price - Variable\ Cost}

The number of units that must be sold to exactly cover all costs.

Business & CommerceIntermediate

Margin of Safety

MoS = \dfrac{Sales - Break\text{-}even}{Sales}\times 100\%

How far sales can fall before the business hits its break-even point.

Business & CommerceIntermediate

Economic Order Quantity (EOQ)

EOQ = \sqrt{\dfrac{2DS}{H}}

The order size that minimizes total inventory ordering and holding costs.

Business & CommerceIntermediate

Customer Acquisition Cost (CAC)

CAC = \dfrac{Sales\ \&\ Marketing\ Cost}{New\ Customers}

The average cost to win one new customer.

Business & CommerceIntermediate

Customer Lifetime Value (CLV)

CLV = \dfrac{ARPU \times Gross\ Margin}{Churn\ Rate}

The total profit expected from a customer over the whole relationship.

Business & CommerceIntermediate

Churn Rate

Churn = \dfrac{Customers\ Lost}{Customers\ at\ Start}\times 100\%

The percentage of customers who stop buying over a period.

Business & CommerceIntermediate

Conversion Rate

CR = \dfrac{Conversions}{Visitors}\times 100\%

The share of visitors or leads that take the desired action.

Business & CommerceAdvanced

Month-over-Month Growth

g = \left(\dfrac{V_1}{V_0}\right) - 1

The percentage change in a metric from one period to the next.

EconomicsBasic

GDP (Expenditure Approach)

GDP = C + I + G + (X - M)

National output measured as total spending by households, firms, government and net exports.

EconomicsBasic

GDP Growth Rate

g = \dfrac{GDP_1 - GDP_0}{GDP_0}\times 100\%

The percentage change in real output between two periods.

EconomicsBasic

Inflation Rate

\pi = \dfrac{CPI_1 - CPI_0}{CPI_0}\times 100\%

The rate at which the general price level rises over time.

EconomicsBasic

Unemployment Rate

u = \dfrac{Unemployed}{Labor\ Force}\times 100\%

The share of the labor force that is jobless and actively seeking work.

EconomicsIntermediate

Real GDP

Real\ GDP = \dfrac{Nominal\ GDP}{Deflator}\times 100

Output adjusted for inflation, so growth reflects real production, not rising prices.

EconomicsIntermediate

GDP Deflator

Deflator = \dfrac{Nominal\ GDP}{Real\ GDP}\times 100

A broad price index covering all goods in GDP, not just a consumer basket.

EconomicsIntermediate

Price Elasticity of Demand

E_d = \dfrac{\%\Delta Q_d}{\%\Delta P}

How responsive quantity demanded is to a change in price.

EconomicsIntermediate

Price Elasticity of Supply

E_s = \dfrac{\%\Delta Q_s}{\%\Delta P}

How responsive quantity supplied is to a change in price.

EconomicsIntermediate

Marginal Cost

MC = \dfrac{\Delta TC}{\Delta Q}

The added cost of producing one more unit of output.

EconomicsIntermediate

Marginal Revenue

MR = \dfrac{\Delta TR}{\Delta Q}

The added revenue from selling one more unit of output.

EconomicsIntermediate

Marginal Propensity to Consume

MPC = \dfrac{\Delta C}{\Delta Y}

The fraction of an extra dollar of income that is spent rather than saved.

EconomicsIntermediate

Spending Multiplier

k = \dfrac{1}{1 - MPC}

How much total output rises from an initial change in spending.

EconomicsAdvanced

Consumer Surplus

CS = \tfrac{1}{2}\times \Delta P \times \Delta Q

The benefit consumers gain when they pay less than the maximum they were willing to pay.

EconomicsIntermediate

GDP (Expenditure)

GDP = C + I + G + (X - M)

Gross domestic product by spending.

EconomicsIntermediate

Price Elasticity of Demand

E_d = \frac{\%\Delta Q}{\%\Delta P}

Responsiveness of demand to price changes.

EconomicsIntermediate

Consumer Price Index

CPI = \frac{\text{Basket now}}{\text{Basket base}}\times 100

Measures price levels relative to a base year.

EconomicsBasic

Inflation Rate

\pi = \frac{CPI_1 - CPI_0}{CPI_0}\times 100

Percentage change in the price index.

EconomicsBasic

Unemployment Rate

u = \frac{\text{Unemployed}}{\text{Labor Force}}\times 100

Fraction of the labor force without work.

EconomicsAdvanced

Spending Multiplier

k = \frac{1}{1 - MPC}

Total impact of spending given marginal consumption.

EconomicsIntermediate

Real GDP

Real = \frac{Nominal}{Deflator}\times 100

GDP adjusted for inflation.

EconomicsIntermediate

Marginal Utility

MU = \frac{\Delta U}{\Delta Q}

Extra satisfaction from one more unit.

EconomicsBasic

Opportunity Cost

OC = \frac{\text{what you give up}}{\text{what you gain}}

Value of the next best alternative foregone.

EconomicsBasic

GDP per Capita

\frac{GDP}{Population}

Average economic output per person.

Currency & ForexBasic

Inverse Exchange Rate

Rate_{B/A} = \dfrac{1}{Rate_{A/B}}

Flip a quoted rate to express it from the other currency’s point of view.

Currency & ForexBasic

Currency % Change

\%\Delta = \dfrac{Rate_1 - Rate_0}{Rate_0}\times 100\%

The percentage move in an exchange rate between two points in time.

Currency & ForexIntermediate

Spread Percentage

Spread\% = \dfrac{Ask - Bid}{Ask}\times 100\%

The bid-ask spread expressed as a percentage of the ask price.

Currency & ForexIntermediate

Pip Value

Pip\ Value = \dfrac{Pip\ Size}{Rate}\times Lot

The monetary value of a one-pip move for a given trade size.

Currency & ForexIntermediate

Appreciation / Depreciation

App\% = \dfrac{Rate_1 - Rate_0}{Rate_0}\times 100\%

The percentage by which a currency strengthens (positive) or weakens (negative).

Currency & ForexAdvanced

Purchasing Power Parity

S = \dfrac{P_{domestic}}{P_{foreign}}

The exchange rate that would equalize the price of an identical basket of goods across countries.

Currency & ForexAdvanced

Real Exchange Rate

RER = S \times \dfrac{P_{foreign}}{P_{domestic}}

The nominal rate adjusted for price levels, measuring relative purchasing power.

Currency & ForexAdvanced

Forward Exchange Rate

F = S \times \dfrac{1 + i_d}{1 + i_f}

The agreed future exchange rate implied by interest-rate differentials (covered interest parity).

Currency & ForexAdvanced

Triangular Arbitrage

Profit = \left(\dfrac{1}{R_{AB}} \times R_{AC} \times R_{CB}\right) - 1

A riskless profit captured when three currencies’ cross rates are momentarily inconsistent.

Everyday MathBasic

Unit Price

\text{Unit Price} = \frac{\text{Total Price}}{\text{Quantity}}

Price per single unit to compare value between products.

Everyday MathBasic

Fuel Economy (mpg)

\text{mpg} = \frac{\text{Miles}}{\text{Gallons}}

Distance traveled per gallon of fuel.

Everyday MathBasic

Trip Fuel Cost

\text{Cost} = \frac{D}{E} \times p

Total fuel cost for a trip.

Everyday MathBasic

Average Speed

v = \frac{d}{t}

Distance covered per unit time.

Everyday MathBasic

Travel Time

t = \frac{d}{v}

Time needed to cover a distance at a given speed.

Everyday MathBasic

Recipe Scaling

Q_{new} = Q_{old} \times \frac{S_{new}}{S_{old}}

Adjust ingredient quantities for a different number of servings.

Everyday MathBasic

Paint Needed

\text{Gallons} = \frac{A}{C}

Amount of paint required for a wall area.

Everyday MathBasic

Tiles Needed

N = \frac{A_{floor}}{A_{tile}}

Number of tiles to cover a floor.

Everyday MathBasic

Celsius to Fahrenheit

F = \frac{9}{5}C + 32

Convert temperature from Celsius to Fahrenheit.

Everyday MathBasic

Body Mass Index

BMI = \frac{m}{h^2}

Weight-to-height ratio used as a health screening tool.

Everyday MathBasic

Percentage Change

\%\Delta = \frac{new - old}{old}\times 100

Relative change between two values.

Everyday MathBasic

Split the Bill

\text{Each} = \frac{T(1+r)}{n}

Per-person share of a bill including tip.

Everyday MathIntermediate

Monthly Loan Payment

M = P\frac{r(1+r)^n}{(1+r)^n - 1}

Fixed monthly payment for an amortizing loan.

Everyday MathIntermediate

Weight Loss Estimate

\Delta w = \frac{\text{deficit} \times days}{7700}

Approximate body-mass loss from a calorie deficit (kg).

Everyday MathBasic

Aspect Ratio Scaling

h = w \times \frac{r_h}{r_w}

Find a height that preserves an aspect ratio.

Time & CalendarAdvanced

Zeller's Congruence

h = \left(q + \left\lfloor\tfrac{13(m+1)}{5}\right\rfloor + K + \left\lfloor\tfrac{K}{4}\right\rfloor + \left\lfloor\tfrac{J}{4}\right\rfloor + 5J\right) \bmod 7

Finds the day of the week for any Gregorian date.

Time & CalendarIntermediate

Day of the Week from a Birthdate

W = \left(d + \left\lfloor 2.6m - 0.2 \right\rfloor + y + \left\lfloor\tfrac{y}{4}\right\rfloor + \left\lfloor\tfrac{c}{4}\right\rfloor - 2c\right) \bmod 7

Gauss's weekday formula — find which day of the week any birthdate fell on.

Time & CalendarBasic

Age in Years

A = \left\lfloor \dfrac{JDN_{today} - JDN_{birth}}{365.2425} \right\rfloor

Whole years elapsed since a birth date.

Time & CalendarAdvanced

Sidereal Day

T_{sid} = \dfrac{T_{sol}}{1 + T_{sol}/T_{yr}}

Earth's rotation period measured against the stars, not the Sun.

Time & CalendarAdvanced

ISO Week Number

W = \left\lfloor \dfrac{d_o - w_d + 10}{7} \right\rfloor

ISO-8601 week of the year (weeks start on Monday).

Time & CalendarBasic

Period & Frequency

T = \dfrac{1}{f}

A repeating event's period is the reciprocal of its frequency.

Time & CalendarIntermediate

Ordinal Day of Year

N = \left\lfloor \dfrac{275M}{9} \right\rfloor - k\left\lfloor \dfrac{M+9}{12} \right\rfloor + D - 30

Day-of-year number (1–366) for a given date.

Geography & EarthAdvanced

Haversine Distance

d = 2r\arcsin\sqrt{\sin^2\frac{\Delta\phi}{2} + \cos\phi_1\cos\phi_2\sin^2\frac{\Delta\lambda}{2}}

Great-circle distance between two lat/long points.

Geography & EarthBasic

Map Scale

\text{Scale} = \frac{\text{map distance}}{\text{ground distance}}

Ratio of map distance to real-world distance.

Geography & EarthBasic

Slope / Gradient

\text{gradient} = \frac{\text{rise}}{\text{run}} \times 100\%

Steepness of terrain as a percentage.

Geography & EarthAdvanced

Earth Curvature Drop

h \approx \frac{d^2}{2R}

Approximate drop of the horizon over distance d.

Geography & EarthAdvanced

Barometric Formula

P = P_0 e^{-\frac{Mgh}{RT}}

Atmospheric pressure as a function of altitude.

Geography & EarthIntermediate

Richter Magnitude

M = \log_{10}\frac{A}{A_0}

Earthquake magnitude from seismograph amplitude.

Geography & EarthBasic

Population Density

D = \frac{P}{A}

Number of people per unit area.

Geography & EarthBasic

Time Zone Difference

\Delta t = \frac{\Delta \text{longitude}}{15^\circ}

Each 15 degrees of longitude equals one hour.

Geography & EarthAdvanced

Moment Magnitude

M_w = \tfrac{2}{3}\log_{10} M_0 - 10.7

Modern magnitude from the seismic moment of a quake.

Geography & EarthAdvanced

Seismic Wave Speed

v = \sqrt{\frac{E}{\rho}}

Speed of an elastic wave through rock.

Geography & EarthIntermediate

Tsunami Wave Speed

v = \sqrt{g\,d}

Speed of a shallow-water wave across the ocean.

Geography & EarthAdvanced

Radiometric Dating Age

t = \frac{1}{\lambda}\ln\!\left(1 + \frac{D}{P}\right)

Age of a rock from parent and daughter isotopes.

Geography & EarthAdvanced

Schumann Resonance

f_1 \approx \frac{c}{2\pi R_E}

Fundamental electromagnetic resonance of the Earth.

Geography & EarthAdvanced

Isostatic Root Depth

r = \frac{\rho_c\,h}{\rho_m - \rho_c}

Depth of a mountain's crustal root floating in the mantle.

Geography & EarthAdvanced

Darcy's Law

Q = -KA\frac{\Delta h}{L}

Rate of groundwater flow through porous rock.

Environmental ScienceBasic

Carbon Footprint

CO_2 = E \times f

Emissions from energy use and an emission factor.

Environmental ScienceAdvanced

Wind Turbine Power

P = \frac{1}{2}\rho A v^3 C_p

Power extracted from wind by a turbine.

Environmental ScienceAdvanced

Betz Limit

C_{p,max} = 0.593

Maximum fraction of wind energy a turbine can capture.

Environmental ScienceIntermediate

Radioactive Decay

N = N_0\left(\frac{1}{2}\right)^{t/t_{1/2}}

Amount remaining after a number of half-lives.

Environmental ScienceIntermediate

pH

pH = -\log_{10}[H^+]

Acidity from hydrogen ion concentration.

Environmental ScienceAdvanced

Logistic Growth

\frac{dN}{dt} = rN\left(1 - \frac{N}{K}\right)

Population growth that slows as it nears carrying capacity.

Environmental ScienceBasic

Water Use Intensity

WUI = \frac{V_{water}}{Output}

Water consumed per unit of product or service.

Environmental ScienceIntermediate

Albedo

\alpha = \frac{\text{reflected}}{\text{incident}}

Fraction of solar radiation reflected by a surface.

Environmental ScienceIntermediate

Energy Payback Ratio

EPR = \frac{E_{generated}}{E_{invested}}

Energy produced over a lifetime divided by energy to build it.

Environmental ScienceIntermediate

Sound Level

L = 10\log_{10}\frac{I}{I_0}

Sound intensity level in decibels.

Unit ConversionsBasic

Celsius to Fahrenheit

°F = °C \times \frac{9}{5} + 32

Convert a temperature from Celsius to Fahrenheit.

Unit ConversionsBasic

Fahrenheit to Celsius

°C = (°F - 32) \times \frac{5}{9}

Convert a temperature from Fahrenheit to Celsius.

Unit ConversionsBasic

Acres to Square Meters

\text{m}^2 = \text{acre} \times 4046.86

Convert acres to square meters.

Unit ConversionsBasic

Hectares to Acres

\text{acre} = \text{ha} \times 2.47105

Convert hectares to acres.

Unit ConversionsBasic

Degrees to Radians

\text{rad} = \text{deg} \times \frac{\pi}{180}

Convert an angle from degrees to radians.

Unit ConversionsBasic

Radians to Degrees

\text{deg} = \text{rad} \times \frac{180}{\pi}

Convert an angle from radians to degrees.

Unit ConversionsBasic

Radians to Degrees

\deg = rad \times \frac{180}{\pi}

Convert radians to degrees.

Unit ConversionsBasic

Acres to Square Meters

m^2 = acre \times 4046.9

Convert acres to square meters.

Number TheoryAdvanced

Euler's Totient

\phi(n) = n\prod_{p|n}\left(1 - \frac{1}{p}\right)

Counts integers up to n coprime to n.

Number TheoryAdvanced

Prime Number Theorem

\pi(x) \sim \frac{x}{\ln x}

Approximates the number of primes below x.

Number TheoryBasic

LCM from GCD

\text{lcm}(a,b) = \frac{|a b|}{\gcd(a,b)}

Least common multiple via the greatest common divisor.

Number TheoryBasic

Sum of First n Integers

\sum_{k=1}^{n} k = \frac{n(n+1)}{2}

Closed form for the sum of the first n natural numbers.

Number TheoryIntermediate

Sum of First n Squares

\sum_{k=1}^{n} k^2 = \frac{n(n+1)(2n+1)}{6}

Closed form for the sum of the first n squares.

Number TheoryIntermediate

Sum of First n Cubes

\sum_{k=1}^{n} k^3 = \left(\frac{n(n+1)}{2}\right)^2

The sum of the first n cubes equals the square of the sum.

Number TheoryAdvanced

Number of Divisors

d(n) = \prod (a_i + 1)

Count of divisors from prime factorization exponents.

Number TheoryAdvanced

Sum of Divisors

\sigma(n) = \prod \frac{p_i^{a_i+1}-1}{p_i - 1}

Sum of all divisors from prime factorization.

Number TheoryBasic

Triangular Number

T_n = \frac{n(n+1)}{2}

Sum of the first n natural numbers.

Number TheoryIntermediate

Pentagonal Number

P_n = \frac{n(3n-1)}{2}

The nth pentagonal figurate number.

Number TheoryBasic

Fibonacci Recurrence

F_n = F_{n-1} + F_{n-2}

Each term is the sum of the two before it.

Number TheoryAdvanced

Binet Formula

F_n = \frac{\varphi^n - \psi^n}{\sqrt{5}}

Closed form for the nth Fibonacci number.

Number TheoryIntermediate

Golden Ratio

\varphi = \frac{1+\sqrt{5}}{2} \approx 1.618

Ratio where the whole is to the larger as the larger is to the smaller.

Computer ScienceIntermediate

XOR Operation

a \oplus b = (a \lor b) \land \lnot(a \land b)

Exclusive OR: true when exactly one input is true.

Computer ScienceAdvanced

Exponential Time

O(2^n)

Runtime that doubles with each added element.

Computer ScienceBasic

Data Transfer Time

t = \frac{S}{R}

Time to transfer data of a given size at a rate.

Computer ScienceIntermediate

Address Space

N = 2^n

Number of addressable locations for n bits.

Computer ScienceAdvanced

Shannon Channel Capacity

C = B\log_2(1 + SNR)

Maximum error-free data rate of a noisy channel.

Computer ScienceAdvanced

Average Memory Access Time

AMAT = H + M \times P

Effective access time from hit time, miss rate, and penalty.

Computer ScienceAdvanced

Amdahl Law

S = \frac{1}{(1-p) + p/s}

Maximum speedup from parallelizing a fraction of a task.

Computer ScienceBasic

Throughput

X = \frac{N}{T}

Number of operations completed per unit time.

Math Tricks (13)

Nikhilam (Multiply Near Base)

A Vedic sutra for multiplying numbers close to a base (10, 100, 1000, etc.). Works by finding how far each number is from the base.

View trick

Subtract from 1000 Instantly

Subtract each digit from 9 except the last, which you subtract from 10. Works for any power of 10.

View trick

Common Fraction-Decimal Pairs

Memorize common fraction-to-decimal conversions for instant calculation.

View trick

Multiply by 9, 99, 999 (Nikhilam Special)

Multiplying any number by a string of nines is instant: subtract 1 from the number for the left part, then take the complement for the right part.

View trick

Multiply by 9 the Easy Way

Multiply by 10 and subtract the original number — much faster than multiplying by 9 directly.

View trick

Doubling and Halving

When one factor is even, halve it and double the other to reach an easier multiplication.

View trick

Add Left to Right

Add the big place values first (left to right) instead of the traditional right-to-left carrying — easier to hold in your head.

View trick

Sum of First N Odd Numbers = N²

The sum of the first n odd numbers is always a perfect square, exactly n².

View trick

Difference of Squares Shortcut

a² − b² = (a + b)(a − b). Turns a subtraction of squares into an easy product.

View trick

Product of Numbers Around a Square

Two numbers equally spaced around a middle value m multiply to m² − d², using the difference of squares.

View trick

Quick Celsius ↔ Fahrenheit

A fast approximation: to go C→F, double and add 30. For exact values use ×9/5 + 32.

View trick

Miles ↔ Kilometres Fast

Multiply miles by 1.6 for km, or divide km by 1.6 for miles. Consecutive Fibonacci numbers are handy approximations.

View trick

Estimate a Square Root

Bracket the number between two perfect squares, then refine with a quick average step.

View trick