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Mass-Energy Equivalence

E = mc^2

Energy contained in mass.

Time Dilation

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

Moving clocks run slow.

Length Contraction

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

Moving objects shorten along motion.

Relativistic Momentum

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

Momentum at speeds near light.

Lorentz Factor

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

Scaling factor in special relativity.

de Broglie Wavelength

\lambda = \frac{h}{p}

Wavelength associated with a particle.

Uncertainty Principle

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

Limit on knowing position and momentum together.

Bohr Energy Levels

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

Energy of the nth hydrogen orbit.

Radioactive Decay

N = N_0 e^{-\lambda t}

Number of nuclei remaining over time.

Half-Life

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

Time for half a sample to decay.

Radioactive Activity

A = \lambda N

Decays per second in a sample.

Ohm's Law

V = IR

Voltage equals current times resistance.

Electrical Power (VI)

P = VI

Power dissipated by a component.

Power (I²R)

P = I^2 R

Power lost as heat in a resistor.

Power (V²/R)

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

Power from voltage and resistance.

Resistors in Series

R_{eq} = R_1 + R_2 + \cdots

Series resistances add directly.

Resistors in Parallel

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

Reciprocals of parallel resistances add.

Capacitors in Parallel

C_{eq} = C_1 + C_2 + \cdots

Parallel capacitances add.

Capacitors in Series

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

Reciprocals of series capacitances add.

Capacitance

C = \frac{Q}{V}

Charge stored per unit voltage.

Capacitor Energy

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

Energy stored in a capacitor.

Inductor Energy

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

Energy stored in a magnetic field.

Capacitive Reactance

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

Opposition of a capacitor to AC.

Inductive Reactance

X_L = 2\pi f L

Opposition of an inductor to AC.

Impedance (RLC)

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

Total opposition in an AC circuit.

Resonant Frequency

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

Frequency at which an LC circuit resonates.

RC Time Constant

\tau = RC

Time to charge a capacitor to ~63%.

RL Time Constant

\tau = \frac{L}{R}

Time constant of an inductor-resistor circuit.

Voltage Divider

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

Output voltage from two series resistors.

Current Divider

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

Current split between two parallel resistors.

Kirchhoff's Voltage Law

\sum V = 0

Voltages around any closed loop sum to zero.

Kirchhoff's Current Law

\sum I_{in} = \sum I_{out}

Current into a node equals current out.

RMS Voltage

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

Effective value of a sinusoidal voltage.

AC Real Power

P = VI\cos\phi

Real power with a phase angle between V and I.

Decibel (Power)

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

Power ratio expressed in decibels.

Decibel (Voltage)

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

Voltage ratio in decibels.

Transformer Turns Ratio

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

Voltage ratio equals turns ratio.

Inverting Amplifier Gain

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

Gain of an inverting op-amp.

Non-Inverting Gain

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

Gain of a non-inverting op-amp.

LED Series Resistor

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

Resistor to safely drive an LED.

Charge from Current

Q = It

Charge transferred by a steady current.

Drift Velocity

v_d = \frac{I}{nAq}

Average velocity of charge carriers.

Wire Resistance

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

Resistance from resistivity, length and area.

Nyquist Bit Rate

C = 2B\log_2 M

Maximum data rate of a noiseless channel.

AND Gate

Y = A \cdot B

Output is 1 only when both inputs are 1.

OR Gate

Y = A + B

Output is 1 if either input is 1.

XOR Gate

Y = A \oplus B

Output is 1 when inputs differ.

Apparent Power

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

Combination of real and reactive power.

Resonant Frequency

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

Frequency where an LC circuit resonates.

Quality Factor

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

Sharpness of a resonant peak.

Capacitor Energy

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

Energy stored in a capacitor.

Inductor Energy

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

Energy stored in an inductor magnetic field.

Capacitor Charge

Q = CV

Charge stored on a capacitor.

Current from Charge

I = \frac{Q}{t}

Current as the rate of charge flow.

LED Series Resistor

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

Resistor to safely drive an LED.

Wheatstone Bridge Balance

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

Condition for a balanced measurement bridge.

Voltage Divider Output

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

Output of a two-resistor divider.

Molarity

M = \frac{n}{V}

Concentration in moles of solute per litre of solution.

pH

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

Acidity from hydrogen-ion concentration.

pOH

pOH = -\log_{10}[OH^-]

Basicity from hydroxide-ion concentration.

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.

Density

\rho = \frac{m}{V}

Mass per unit volume of a substance.

Percent Composition

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

Mass percentage of an element in a compound.

ChemistryAdvanced

Gas Density

\rho = \frac{PM}{RT}

Density of an ideal gas.

ChemistryIntermediate

Molality

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

Moles of solute per kilogram of solvent.

pH and pOH

pH + pOH = 14

Relationship between acidity and basicity at 25°C.

Exponential Population Growth

N_t = N_0 e^{rt}

Unlimited growth of a population over time.

Biology & Life SciencesBasic

Logistic Growth

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

Population growth limited by carrying capacity.

Per Capita Growth Rate

r = b - d

Net growth rate from birth and death rates.

Shannon Diversity Index

H = -\sum p_i \ln p_i

Measures species diversity in a community.

Simpson Diversity Index

D = 1 - \sum p_i^2

Probability two random individuals differ in species.

Kleiber's Law

q_0 = 70\,M^{0.75}

Metabolic rate scales with body mass to the ¾ power.

Osmolarity

\text{Osm} = n \times C

Solute particle concentration in solution.

Enzyme Activity Unit

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

One unit converts 1 µmol of substrate per minute.

Net Primary Productivity

NPP = GPP - R

Energy available to an ecosystem after respiration.

Population Growth Rate

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

Per-capita growth from births and deaths.

Body Mass Index (BMI)

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

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

BMI (Imperial)

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

BMI using pounds and inches.

BMR (Mifflin, Men)

BMR = 10m + 6.25h - 5a + 5

Basal metabolic rate — daily calories at rest for men.

BMR (Mifflin, Women)

BMR = 10m + 6.25h - 5a - 161

Basal metabolic rate for women.

Total Daily Energy (TDEE)

TDEE = BMR \times \text{activity factor}

Daily calories burned including activity.

Max Heart Rate

HR_{max} = 220 - a

Estimated maximum heart rate by age.

Target Heart Rate (Karvonen)

HR_{target} = (HR_{max} - HR_{rest})\,I + HR_{rest}

Training heart rate at intensity I.

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.

Waist-to-Hip Ratio

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

Fat-distribution health indicator.

Ideal Body Weight (Devine)

IBW = 50 + 2.3(h - 60)

Devine formula for men (kg), height in inches; women use base 45.5.

Calories Burned (MET)

C = MET \times m \times t

Energy used during exercise.

One-Rep Max (Epley)

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

Estimated maximum single-lift weight.

VO₂ Max (Cooper)

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

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

Daily Water Intake

W = 0.033 \times m

Rough daily water need in litres.

Lean Body Mass (Boer, Men)

LBM = 0.407m + 0.267h - 19.2

Body mass excluding fat for men.

Protein from Calories

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

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

Running Pace

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

Time taken to cover one unit of distance.

Step Length from Height

L \approx 0.415 \times h

Estimated walking step length from height.

Fat-Free Mass Index

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

Muscularity index normalised for height.

Corrected QT (Bazett)

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

Heart-rate-corrected QT interval from an ECG.

Mean Arterial Pressure

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

Average blood pressure during a cardiac cycle.

Creatinine Clearance (Cockcroft-Gault)

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

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

Due Date (Naegele)

\text{EDD} = LMP + 280\ \text{days}

Estimated delivery date from last menstrual period.

Waist-to-Hip Ratio

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

Body-shape ratio linked to health risk.

Ideal Body Weight (Devine)

IBW = 50 + 2.3(h - 60)

Estimated ideal weight in kg for men (h in inches).

Daily Water Intake

W = m \times 0.033

Approximate daily water need in liters.

Daily Protein Need

P = m \times 0.8

Baseline daily protein in grams per kg.

Target Heart Rate (Karvonen)

THR = (HR_{max} - HR_{rest})\times I + HR_{rest}

Training heart rate at a given intensity.

Running Pace

pace = \frac{time}{distance}

Time per unit distance.

Lean Body Mass

LBM = m(1 - BF\%)

Body mass excluding fat.

Calories Burned (MET)

C = MET \times m \times t

Energy expenditure from activity intensity.

BMR (Katch-McArdle)

BMR = 370 + 21.6\,LBM

Resting calories from lean body mass.

RMR (Cunningham)

RMR = 500 + 22\,LBM

Resting metabolic rate favoured for athletes.

Body Surface Area (Mosteller)

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

Total skin surface area of the body.

Waist-to-Height Ratio

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

Central-fat health indicator.

BMI Prime

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

Your BMI relative to the upper healthy limit.

Ponderal Index

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

A leanness index that scales better for extreme heights.

One-Rep Max (Brzycki)

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

Estimated one-rep max, an alternative to Epley.

Max Heart Rate (Tanaka)

HR_{max} = 208 - 0.7\,a

A more accurate maximum heart rate estimate.

Heart Rate Reserve

HRR = HR_{max} - HR_{rest}

The usable range between rest and maximum heart rate.

Fat Mass

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

Total mass of body fat.

MET to Oxygen Uptake

VO_2 = 3.5 \times MET

Oxygen consumption from a MET activity level.

Fat Loss from Deficit

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

Body-fat lost from a total calorie deficit.

Daily Protein Target

P = 1.6 \times m

Grams of protein per day to build muscle.

Speed from Pace

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

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

Corrected QT (Fridericia)

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

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

Blood Glucose Conversion

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

Converts blood-sugar units between countries.

Food Energy (Atwater)

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

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

Calories from Protein

E_P = 4P

Energy contributed by protein at 4 kcal per gram.

Calories from Carbohydrate

E_C = 4C

Energy from carbohydrate at 4 kcal per gram.

Calories from Fat

E_F = 9F

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

Calories from Alcohol

E_A = 7A

Energy from pure alcohol (ethanol) at 7 kcal per gram.

Kilocalories to Kilojoules

kJ = kcal \times 4.184

Convert dietary Calories to the SI energy unit used on many food labels.

Caloric (Energy) Density

D = \frac{E}{m}

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

Macro % of Calories

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

Share of total energy coming from a given macronutrient.

Protein Requirement (RDA)

P = 0.8 \times m

Recommended daily protein for a healthy adult (grams).

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.

Net Carbohydrates

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

Digestible carbs after subtracting fibre and half of sugar alcohols.

Percent Daily Value (%DV)

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

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

Weight Change from Energy Balance

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

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

Baker's Percentage

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

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

Dough Hydration

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

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

Recipe Scaling Factor

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

Rescale each ingredient for a different number of servings.

Brine Salinity

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

Salt concentration of a brine or curing solution by mass.

Yield Percentage

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

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

Food Cost Percentage

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

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

Cost per Serving

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

Total recipe cost divided across the number of servings.

Nutrition & FoodAdvanced

Alcohol by Volume (ABV)

ABV = (OG - FG) \times 131.25

Estimated alcohol content of beer or wine from gravity readings.

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.

Escape Velocity

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

Minimum speed to escape a body's gravity.

Orbital Velocity

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

Speed of a circular orbit.

Kepler's Third Law

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

Relates orbital period to semi-major axis.

Schwarzschild Radius

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

Event horizon radius of a black hole.

Hubble's Law

v = H_0 d

Recession velocity of galaxies with distance.

Stefan–Boltzmann Law

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

Luminosity of a star from radius and temperature.

Redshift

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

Fractional wavelength shift of light.

Surface Gravity

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

Gravitational acceleration at a body's surface.

Apparent Brightness

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

Observed brightness from luminosity and distance.

Distance Modulus

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

Relates apparent and absolute magnitude to distance.

Surface Gravity

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

Gravitational acceleration at a body surface.

Kepler Third Law (General)

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

Orbital period from semi-major axis.

Distance Modulus

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

Relates apparent and absolute magnitudes to distance.

Density from Orbit

\rho = \frac{3\pi}{GT^2}

Mean density inferred from a close orbital period.

Tidal Force

F_t \propto \frac{2GMm r}{d^3}

Differential gravitational force across a body.

Newton's Law of Gravitation

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

The attractive force between any two masses.

Gravitational Potential Energy

U = -\frac{GMm}{r}

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

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.

Eddington Luminosity

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

The maximum steady luminosity before radiation blows a star apart.

Chandrasekhar Limit

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

The maximum mass of a stable white dwarf.

Hawking Temperature

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

Temperature of the radiation a black hole emits.

Photon Sphere Radius

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

Radius where gravity bends light into a circular orbit.

Synodic Period

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

Time between repeated alignments of two orbiting bodies.

Gravitational Redshift

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

Fractional stretching of light climbing out of gravity.

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.

Stress

\sigma = \frac{F}{A}

Internal force per unit area in a material.

Young's Modulus

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

Stiffness of a material (stress over strain).

Electrical Power

P = VI

Power in an electrical circuit.

Resistors in Series

R_{eq} = R_1 + R_2 + \cdots

Total resistance of resistors in series.

Resistors in Parallel

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

Total resistance of resistors in parallel.

Volumetric Flow Rate

Q = Av

Volume of fluid passing per unit time.

Continuity Equation

A_1 v_1 = A_2 v_2

Conservation of mass in incompressible flow.

Factor of Safety

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

Ratio of failure capacity to actual load.

Materials & CivilAdvanced

RC Time Constant

\tau = RC

Charging time constant of a resistor-capacitor circuit.

Materials & CivilBasic

Stress

\sigma = \frac{F}{A}

Internal force per unit cross-sectional area.

Cantilever Beam Deflection

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

Maximum deflection of a cantilever with an end load.

Materials & CivilBasic

Uniformly Distributed Load

w = \frac{W}{L}

Load per unit length of a beam.

Materials & CivilBasic

Density

\rho = \frac{m}{V}

Mass per unit volume of a material.

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

Water-Cement Ratio

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

Ratio controlling concrete strength and workability.

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.

Rate of Climb

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

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

Pressure Altitude

PA = (29.92 - QNH)\times 1000 + h

Altitude referenced to standard pressure; correct field elevation for the altimeter setting.

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.

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.

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.

Stall Speed vs Load Factor

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

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

Crosswind Component

XW = V_w\sin\theta

Crosswind component determines whether landing is within aircraft limits.

Density Altitude

DA = PA + 120(OAT - ISA_{temp})

Pressure altitude corrected for non-standard temperature; higher DA reduces performance.

Glide Ratio

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

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

Mach Number

M = \dfrac{v}{a}

Ratio of true airspeed to the local speed of sound.

Specific Range

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

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

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.

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.

Barometric Altitude

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

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

Glide Ratio

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

Distance traveled forward per unit of height lost.

Wing Loading

WL = \frac{W}{S}

Aircraft weight per unit wing area.

True Airspeed (approx)

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

Airspeed corrected for altitude.

Density Altitude

DA = PA + 120(OAT - ISA)

Pressure altitude corrected for temperature.

Continuous Compounding

A = P e^{rt}

Value with interest compounded continuously.

Future Value of Annuity

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

Future value of regular equal payments.

Present Value of Annuity

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

Present value of a series of equal payments.

Perpetuity

PV = \frac{C}{r}

Present value of an infinite constant cash flow.

Break-Even Point

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

Units to sell to cover all costs.

Profit

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

The money left after subtracting cost from revenue.

Profit Margin

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

Profit as a percentage of revenue.

Earnings Per Share

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

Company profit allocated to each share.

Break-Even Point

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

Units to sell to cover all costs.

Gross Margin

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

Percentage of revenue left after cost of goods.

Current Ratio

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

Liquidity measure of short-term solvency.

Debt-to-Equity Ratio

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

Leverage measure comparing debt to equity.

Return on Equity

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

Profit generated per dollar of equity.

Return on Assets

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

Profit generated per dollar of assets.

Payback Period

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

Time to recover an investment.

Working Capital

WC = \text{Current Assets} - \text{Current Liabilities}

Short-term financial health of a business.

Black-Scholes (Call Option)

C = S_0\,N(d_1) - K e^{-rT} N(d_2)

Prices a European call option, where d₁ and d₂ depend on the stock price, strike, volatility, rate and time. N(·) is the standard normal CDF.

Accounting Equation

Assets = Liabilities + Equity

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

Gross Profit

GP = Revenue - COGS

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

Net Income

NI = Revenue - Expenses

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

Retained Earnings

RE_1 = RE_0 + NI - Dividends

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

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.

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.

Working Capital

WC = Current\ Assets - Current\ Liabilities

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

Debt-to-Equity Ratio

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

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

Inventory Turnover

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

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

Return on Assets (ROA)

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

How efficiently a company turns its assets into profit.

Return on Equity (ROE)

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

The profit generated for each dollar of shareholders’ equity.

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

Book Value per Share

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

The net asset value backing each common share.

Accounting Equation

Assets = Liabilities + Equity

The fundamental balance-sheet identity.

Net Income

NI = Revenue - Expenses

Profit after all expenses.

Retained Earnings

RE = RE_0 + NI - Dividends

Accumulated profit kept in the business.

Quick Ratio

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

Stringent liquidity measure excluding inventory.

Inventory Turnover

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

How many times inventory is sold per period.

Operating Margin

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

Profit from core operations as a percent of revenue.

Contribution Margin

CM = Price - \text{Variable Cost}

Revenue per unit left to cover fixed costs.

Business & CommerceBasic

Selling Price from Markup

SP = Cost \times (1 + m)

The price to charge after adding a percentage markup to unit cost.

Business & CommerceBasic

Revenue

Revenue = Price \times Quantity

Total income from sales before any costs are deducted.

Contribution Margin

CM = Price - Variable\ Cost

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

Break-Even (Units)

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

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

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.

Economic Order Quantity (EOQ)

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

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

Customer Acquisition Cost (CAC)

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

The average cost to win one new customer.

Customer Lifetime Value (CLV)

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

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

Conversion Rate

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

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

GDP (Expenditure Approach)

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

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

Price Elasticity of Demand

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

How responsive quantity demanded is to a change in price.

Price Elasticity of Supply

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

How responsive quantity supplied is to a change in price.

Marginal Cost

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

The added cost of producing one more unit of output.

Marginal Revenue

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

The added revenue from selling one more unit of output.

Marginal Propensity to Consume

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

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

Spending Multiplier

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

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

EconomicsAdvanced

Quantity Theory of Money

M \times V = P \times Q

Links the money supply and its velocity to the price level and real output.

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.

GDP (Expenditure)

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

Gross domestic product by spending.

Price Elasticity of Demand

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

Responsiveness of demand to price changes.

Unemployment Rate

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

Fraction of the labor force without work.

Marginal Utility

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

Extra satisfaction from one more unit.

Opportunity Cost

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

Value of the next best alternative foregone.

Currency & ForexIntermediate

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.

Cross Rate

Rate_{A/C} = Rate_{A/B} \times Rate_{B/C}

Derive the exchange rate between two currencies via a common third currency.

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

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.

Unit Price

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

Price per single unit to compare value between products.

Tip Amount

\text{Tip} = B \times r

Gratuity based on a bill amount.

Average Speed

v = \frac{d}{t}

Distance covered per unit time.

Recipe Scaling

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

Adjust ingredient quantities for a different number of servings.

Simple Interest

I = P r t

Interest earned or owed without compounding.

Celsius to Fahrenheit

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

Convert temperature from Celsius to Fahrenheit.

Everyday MathIntermediate

Split the Bill

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

Per-person share of a bill including tip.

Weight Loss Estimate

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

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

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 & CalendarBasic

Period & Frequency

T = \dfrac{1}{f}

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

Time & CalendarAdvanced

Easter Date (Computus)

d = (19a + b) \bmod 30,\quad a = y \bmod 19

Core step of the Gregorian algorithm locating the Paschal full moon.

Barometric Formula

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

Atmospheric pressure as a function of altitude.

Degrees to Distance

1^\circ \text{lat} \approx 111\ \text{km}

Approximate ground distance per degree of latitude.

Richter Magnitude

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

Earthquake magnitude from seismograph amplitude.

Population Density

D = \frac{P}{A}

Number of people per unit area.

Time Zone Difference

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

Each 15 degrees of longitude equals one hour.

Population Density

D = \frac{P}{A}

People per unit area.

Time Zone Offset

\Delta t = \frac{\Delta\text{longitude}}{15\degree}

Hours difference from longitude difference.

Relative Humidity

RH = \frac{e}{e_s}\times 100

Water vapor as a fraction of the saturation amount.

Coriolis Parameter

f = 2\Omega\sin\phi

Strength of the Coriolis effect by latitude.

Dry Adiabatic Lapse Rate

\Gamma \approx 9.8\,\degree C/km

Rate air cools with altitude when rising dry.

Barometric Altitude

h = 44330\left(1 - \left(\frac{P}{P_0}\right)^{0.19}\right)

Altitude estimated from pressure ratio.

Earthquake Energy (Richter)

\log_{10} E = 4.8 + 1.5\,M

Energy released by a quake of Richter magnitude M.

Moment Magnitude

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

Modern magnitude from the seismic moment of a quake.

Lithostatic Pressure

P = \rho g h

Pressure from the weight of overlying rock.

Environmental ScienceBasic

Geothermal Gradient

\Delta T = G_T \, d

Temperature rise with depth into the Earth.

Carbon Footprint

CO_2 = E \times f

Emissions from energy use and an emission factor.

Environmental ScienceAdvanced

Betz Limit

C_{p,max} = 0.593

Maximum fraction of wind energy a turbine can capture.

Environmental ScienceIntermediate

pH

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

Acidity from hydrogen ion concentration.

Environmental ScienceBasic

Energy Consumption

E = P \times t

Electrical energy used over time.

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

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.

Environmental ScienceAdvanced

Richter Magnitude Energy

\log_{10}E = 4.8 + 1.5M

Approximate energy released by an earthquake.

Celsius to Fahrenheit

°F = °C \times \frac{9}{5} + 32

Convert a temperature from Celsius to Fahrenheit.

Fahrenheit to Celsius

°C = (°F - 32) \times \frac{5}{9}

Convert a temperature from Fahrenheit to Celsius.

Celsius to Kelvin

K = °C + 273.15

Convert Celsius to the absolute Kelvin scale.

Kelvin to Celsius

°C = K - 273.15

Convert Kelvin to Celsius.

Kilometers to Miles

\text{mi} = \text{km} \times 0.621371

Convert kilometers to miles.

Miles to Kilometers

\text{km} = \text{mi} \times 1.60934

Convert miles to kilometers.

Meters to Feet

\text{ft} = \text{m} \times 3.28084

Convert meters to feet.

Feet to Meters

\text{m} = \text{ft} \times 0.3048

Convert feet to meters.

Inches to Centimeters

\text{cm} = \text{in} \times 2.54

Convert inches to centimeters.

Nautical Miles to Kilometers

\text{km} = \text{nmi} \times 1.852

Convert nautical miles to kilometers.

Square Meters to Square Feet

\text{ft}^2 = \text{m}^2 \times 10.7639

Convert area from square meters to square feet.

Acres to Square Meters

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

Convert acres to square meters.

Hectares to Acres

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

Convert hectares to acres.

Square Km to Square Miles

\text{mi}^2 = \text{km}^2 \times 0.386102

Convert area from square kilometers to square miles.

Liters to US Gallons

\text{gal} = \text{L} \times 0.264172

Convert liters to US gallons.

US Gallons to Liters

\text{L} = \text{gal} \times 3.78541

Convert US gallons to liters.

Cubic Meters to Cubic Feet

\text{ft}^3 = \text{m}^3 \times 35.3147

Convert volume from cubic meters to cubic feet.

Milliliters to Fluid Ounces

\text{fl oz} = \text{mL} \times 0.033814

Convert milliliters to US fluid ounces.

Kilograms to Pounds

\text{lb} = \text{kg} \times 2.20462

Convert kilograms to pounds.

Pounds to Kilograms

\text{kg} = \text{lb} \times 0.453592

Convert pounds to kilograms.

Grams to Ounces

\text{oz} = \text{g} \times 0.035274

Convert grams to ounces.

Metric Tonnes to Pounds

\text{lb} = \text{t} \times 2204.62

Convert metric tonnes to pounds.

km/h to mph

\text{mph} = \text{km/h} \times 0.621371

Convert speed from km/h to mph.

m/s to km/h

\text{km/h} = \text{m/s} \times 3.6

Convert speed from meters per second to km/h.

Calories to Joules

\text{J} = \text{cal} \times 4.184

Convert calories to joules.

kWh to Joules

\text{J} = \text{kWh} \times 3.6\times10^6

Convert kilowatt-hours to joules.

Atmospheres to Pascals

\text{Pa} = \text{atm} \times 101325

Convert atmospheres to pascals.

Bar to PSI

\text{psi} = \text{bar} \times 14.5038

Convert bar to pounds per square inch.

Degrees to Radians

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

Convert an angle from degrees to radians.

Radians to Degrees

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

Convert an angle from radians to degrees.

Gigabytes to Megabytes

\text{MB} = \text{GB} \times 1024

Convert gigabytes to megabytes (binary).

Kilometers to Miles

\text{mi} = km \times 0.6214

Convert kilometers to miles.

Miles to Kilometers

km = mi \times 1.6093

Convert miles to kilometers.

Kilograms to Pounds

lb = kg \times 2.2046

Convert kilograms to pounds.

Pounds to Kilograms

kg = lb \times 0.4536

Convert pounds to kilograms.

Inches to Centimeters

cm = in \times 2.54

Convert inches to centimeters.

Liters to US Gallons

gal = L \times 0.2642

Convert liters to US gallons.

Celsius to Kelvin

K = C + 273.15

Convert Celsius to Kelvin.

MPH to KM/H

km/h = mph \times 1.6093

Convert miles per hour to kilometers per hour.

Calories to Joules

J = cal \times 4.184

Convert calories to joules.

Atmospheres to Pascals

Pa = atm \times 101325

Convert atmospheres to pascals.

Horsepower to Watts

W = hp \times 745.7

Convert horsepower to watts.

Bytes to Bits

bits = bytes \times 8

Convert bytes to bits.

Radians to Degrees

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

Convert radians to degrees.

Acres to Square Meters

m^2 = acre \times 4046.9

Convert acres to square meters.

Feet to Meters

m = ft \times 0.3048

Convert feet to meters.

Euclidean Algorithm

\gcd(a,b) = \gcd(b, a \bmod b)

Recursively finds the greatest common divisor.

Number TheoryAdvanced

Fermat's Little Theorem

a^{p-1} \equiv 1 \pmod{p}

Holds when p is prime and gcd(a,p)=1.

Divisibility Rule for 3

n \text{ divisible by } 3 \iff \text{digit sum divisible by } 3

A number is divisible by 3 if the sum of its digits is.

Divisibility Rule for 9

n \text{ divisible by } 9 \iff \text{digit sum divisible by } 9

A number is divisible by 9 if the sum of its digits is.

Even and Odd Numbers

n = 2k \;(\text{even}), \quad n = 2k+1 \;(\text{odd})

Every integer is either even or odd.

Fibonacci Recurrence

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

Each term is the sum of the two before it.

Number TheoryAdvanced

Wilson Theorem

(p-1)! \equiv -1 \pmod{p}

A primality criterion for p.

Shannon Entropy

H = -\sum_i p_i \log_2 p_i

Average information content of a random source (bits).

Master Theorem (form)

T(n) = aT(n/b) + f(n)

Analyzes running time of divide-and-conquer algorithms.

Hamming Distance

d(x,y) = \sum_i [x_i \ne y_i]

Number of positions at which two strings differ.

Bits to Represent n

b = \lceil \log_2(n) \rceil

Minimum number of bits to represent n distinct values.

XOR Operation

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

Exclusive OR: true when exactly one input is true.

Two's Complement

-N = \lnot N + 1

Represents negative integers in binary by inverting bits and adding one.

Common Time Complexities

O(1) < O(\log n) < O(n) < O(n\log n) < O(n^2) < O(2^n)

Ordering of common algorithmic growth rates.

Logarithmic Time

O(\log n)

Runtime that grows logarithmically with input size.

Linearithmic Time

O(n\log n)

Runtime of efficient comparison sorts.

Quadratic Time

O(n^2)

Runtime that grows with the square of input size.

Exponential Time

O(2^n)

Runtime that doubles with each added element.

Master Theorem Form

T(n) = aT(n/b) + f(n)

General form for analyzing divide-and-conquer recurrences.

Address Space

N = 2^n

Number of addressable locations for n bits.

Hexadecimal to Decimal

d = \sum h_i \times 16^i

Convert a hex number to decimal by positional weights.

Parity Bit

p = b_1 \oplus b_2 \oplus \cdots \oplus b_n

XOR of data bits used for error detection.

Nyquist Sampling Rate

f_s \geq 2f_{max}

Minimum sampling rate to capture a signal without aliasing.

Shannon Channel Capacity

C = B\log_2(1 + SNR)

Maximum error-free data rate of a noisy channel.

Hamming Code Bits

2^r \geq m + r + 1

Parity bits r needed to protect m data bits.

Average Memory Access Time

AMAT = H + M \times P

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

Throughput

X = \frac{N}{T}

Number of operations completed per unit time.

Gradient Descent Update

\theta := \theta - \alpha\,\nabla_\theta J(\theta)

Iteratively moves parameters in the direction that most reduces the loss.

ReLU Activation

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

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

Softmax

\sigma(z)_i = \dfrac{e^{z_i}}{\sum_{j} e^{z_j}}

Turns a vector of scores into a probability distribution that sums to 1.

Binary Cross-Entropy

L = -\big[y\log\hat{y} + (1-y)\log(1-\hat{y})\big]

Loss for binary classification comparing predicted probability to the true label.

Categorical Cross-Entropy

L = -\sum_{i} y_i \log \hat{y}_i

Multi-class loss summing the log-probability of the correct class.

Logistic Regression

\hat{y} = \dfrac{1}{1+e^{-(\mathbf{w}^\top \mathbf{x}+b)}}

Applies the sigmoid to a linear model to output a class probability.

Cosine Similarity

\cos\theta = \dfrac{\mathbf{a}\cdot\mathbf{b}}{\lVert\mathbf{a}\rVert\,\lVert\mathbf{b}\rVert}

Measures the angle between two vectors; 1 means identical direction.

Entropy (Information)

H(S) = -\sum_i p_i \log_2 p_i

Measures impurity/uncertainty of a set; used to split decision trees.

Gini Impurity

G = 1 - \sum_i p_i^2

Probability of misclassifying a random sample; the default split metric in CART trees.

Information Gain

IG = H(S) - \sum_k \dfrac{|S_k|}{|S|}H(S_k)

Reduction in entropy achieved by splitting a node on a feature.

Naive Bayes Posterior

P(y\mid \mathbf{x}) \propto P(y)\prod_i P(x_i\mid y)

Classifies by assuming features are conditionally independent given the class.

L2 Regularization (Ridge)

J = \text{MSE} + \lambda\sum_j w_j^2

Adds a penalty on squared weights to reduce overfitting.

L1 Regularization (Lasso)

J = \text{MSE} + \lambda\sum_j |w_j|

Penalizes absolute weights, driving many to exactly zero for feature selection.

Hinge Loss (SVM)

L = \max(0,\,1 - y\,\hat{y})

Loss for support vector machines; zero once a point is correctly classified with margin.

SGD with Momentum

v := \beta v + (1-\beta)\nabla J;\;\; \theta := \theta - \alpha v

Accelerates gradient descent by accumulating a velocity of past gradients.

TF-IDF

\text{tfidf} = tf \cdot \log\dfrac{N}{df}

Weights a word by its frequency in a document against its rarity across the corpus.

PCA (Covariance)

\mathbf{C} = \tfrac{1}{n}\mathbf{X}^\top\mathbf{X},\;\; \mathbf{C}\mathbf{v}=\lambda\mathbf{v}

Finds directions of maximum variance via eigenvectors of the covariance matrix.

Positional Encoding

PE_{(pos,2i)}=\sin\!\left(\dfrac{pos}{10000^{2i/d}}\right)

Injects word-order information into a Transformer that otherwise ignores position.

Layer Normalization

\hat{x}=\dfrac{x-\mu}{\sqrt{\sigma^2+\epsilon}}\,\gamma+\beta

Normalizes across features within a single sample; standard in Transformers.

KL Divergence

D_{KL}(P\Vert Q)=\sum_i P(i)\log\dfrac{P(i)}{Q(i)}

Measures how one probability distribution differs from a reference distribution.

Saturation Vapour Pressure

e_s = 0.6108\,\exp\!\left(\frac{17.27\,T}{T + 237.3}\right)

The maximum water-vapour pressure air can hold at a given temperature (Tetens formula).

Apparent Temperature (Heat Index)

AT = T_a + 0.33\,e - 0.70\,w_s - 4.00

How hot the air actually feels once humidity and wind are taken into account.

Beaufort Wind Scale

v = 0.836\,B^{3/2}

Converts a Beaufort force number into an approximate wind speed.

Wind Pressure (Dynamic)

P = \tfrac{1}{2}\rho v^2

The pressure a moving airmass exerts on a surface such as a wall or roof.

Radiative Forcing from CO₂

\Delta F = 5.35\,\ln\!\left(\frac{C}{C_0}\right)

Extra energy trapped by the atmosphere as carbon-dioxide concentration rises.

Planetary Equilibrium Temperature

T_e = \left[\frac{S(1-\alpha)}{4\sigma}\right]^{1/4}

The temperature a planet would settle at from sunlight alone, before greenhouse warming.

Global Warming Potential (GWP)

GWP = \frac{\int_0^{TH} a_x\, C_x(t)\, dt}{\int_0^{TH} a_r\, C_r(t)\, dt}

Warming caused by a gas over a time horizon, relative to the same mass of CO₂.

Sea-Level Rise (Thermal Expansion)

\Delta h = h\,\beta\,\Delta T

Rise in sea level as ocean water warms and expands.

Flood Return Period

T = \frac{n+1}{m}

The average interval between floods of a given size (Weibull plotting position).

Flood Encounter Probability

P = 1 - \left(1 - \frac{1}{T}\right)^{n}

Chance of experiencing at least one flood of return period T over n years.

Rational Method (Peak Runoff)

Q = C\,i\,A

Estimates peak stormwater discharge from a catchment.

Weather & CalamitiesBasic

Manning’s Equation (Channel Flow)

V = \frac{1}{n}\,R^{2/3}\,S^{1/2}

Average velocity of water flowing in an open channel or river.

Stream Discharge

Q = A\,v

Volume of water flowing past a point per unit time.

Moment Magnitude (Mw)

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

The modern earthquake magnitude scale, based on the seismic moment released.

Tsunami Wave Speed

v = \sqrt{g\,d}

Speed of a tsunami in the open ocean (a shallow-water wave relative to its wavelength).

Target Rating Points (TRP)

TRP = \frac{\text{Impressions among target}}{\text{Target population}} \times 100

Advertising weight counted only within the specific target audience.

Cost Per Mille (CPM)

CPM = \frac{\text{Cost}}{\text{Impressions}} \times 1000

The cost of reaching one thousand impressions.

Cost Per Rating Point (CPP)

CPP = \frac{\text{Cost}}{GRP}

How much it costs to buy one rating point of audience.

Click-Through Rate (CTR)

CTR = \frac{\text{Clicks}}{\text{Impressions}} \times 100

Percentage of ad views that result in a click.

Cost Per Click (CPC)

CPC = \frac{\text{Cost}}{\text{Clicks}}

Average amount paid for each click on an ad.

Cost Per Acquisition (CPA)

CPA = \frac{\text{Cost}}{\text{Conversions}}

Average cost to gain one customer or completed action.

Return on Ad Spend (ROAS)

ROAS = \frac{\text{Revenue from ads}}{\text{Ad spend}}

Revenue generated for every unit of currency spent on advertising.

Conversion Rate

CR = \frac{\text{Conversions}}{\text{Visitors}} \times 100

Percentage of visitors who complete a desired action.

Engagement Rate

ER = \frac{\text{Engagements}}{\text{Followers}} \times 100

How actively an audience interacts with content.

Power Usage Effectiveness (PUE)

PUE = \frac{\text{Total facility energy}}{\text{IT equipment energy}}

How efficiently a data centre uses energy, versus overhead like cooling.

Data-Centre Infrastructure Efficiency

DCiE = \frac{1}{PUE} \times 100

The efficiency of a data centre expressed as a percentage.

Availability (Uptime)

A = \frac{MTBF}{MTBF + MTTR} \times 100

Percentage of time a system is operational.

Media & DataAdvanced

Bandwidth-Delay Product

BDP = \text{Bandwidth} \times RTT

Amount of data "in flight" on a network link at any instant.

Compression Ratio

CR = \frac{\text{Uncompressed size}}{\text{Compressed size}}

How much smaller data becomes after compression.

Media & DataAdvanced

Little’s Law

L = \lambda\,W

Relates the number of items in a system to arrival rate and waiting time.

AvionicsBasic

Radio Wavelength

\lambda = \dfrac{c}{f}

Wavelength of a radio signal is the speed of light divided by its frequency; it sets antenna length and dish sizing.

AvionicsBasic

Mode C Altitude Reporting

FL = \dfrac{PA}{100}

A Mode C/S transponder reports pressure altitude referenced to 1013.25 hPa; dividing by 100 gives the flight level, independent of the local altimeter setting.

Free-Space Path Loss

FSPL_{dB} = 20\log_{10}d + 20\log_{10}f + 32.44

Signal power lost simply by spreading through free space, growing with both distance and frequency.

Effective Isotropic Radiated Power

EIRP_{dBW} = P_t + G_t - L_c

Apparent power a transmitter would need if it radiated equally in all directions; combines transmit power, antenna gain and cable loss.

Radar Maximum Unambiguous Range

R_{max} = \dfrac{c}{2\,PRF}

Farthest range at which a pulse echo returns before the next pulse is sent; beyond it, targets appear at false (aliased) ranges.

Radar Range Resolution

\Delta R = \dfrac{c\tau}{2}

Smallest separation at which two targets appear as distinct echoes, set by the transmitted pulse width.

Radar Doppler Shift

f_d = \dfrac{2v_r}{\lambda}

Frequency shift of an echo from a moving target; the factor of two accounts for the round trip to the target and back.

Nyquist Sampling Rate

f_s \geq 2 f_{max}

To digitise an analogue avionics signal without aliasing, the sampling rate must be at least twice the highest frequency present.

Parabolic Antenna Gain

G = \eta\left(\dfrac{\pi D}{\lambda}\right)^2

Gain of a dish antenna rises with aperture area and falls with wavelength, scaled by aperture efficiency.

Radar Range Equation

P_r = \dfrac{P_t G^2 \lambda^2 \sigma}{(4\pi)^3 R^4}

Received echo power for a monostatic radar, combining transmit power, antenna gain, target size and range.

GPS Pseudorange

\rho = c\,(t_r - t_t) + c\,\delta t

Measured distance to a GPS satellite from signal travel time, biased by the receiver clock error — hence “pseudo” range.

GPS Position Error (DOP)

\sigma_{pos} = GDOP \times \sigma_{UERE}

Overall positioning error is the geometry factor (dilution of precision) multiplied by the per-satellite ranging error.

Math Tricks (22)

Squaring Numbers Near 50

Quick method to square any number near 50 using the identity (50±d)² = 2500 ± 100d + d².

Ekadhikena (Squaring Numbers Ending in 5)

Instantly square any number ending in 5. Multiply the tens digit by (tens digit + 1), then append 25.

Vinculum Numbers

Transform numbers with large digits (6-9) into "vinculum" form using complements from 10, making calculations easier.

Multiply Any 2-Digit Number by 11

Instantly multiply any two-digit number by 11 by splitting and adding the digits.

Percentage Flip Trick

x% of y = y% of x. When one direction is hard, flip it! This works because (x/100)×y = (y/100)×x.

Subtract from 1000 Instantly

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

Divisibility Quick Checks

Quickly check if a number is divisible by 2 through 12 without dividing.

Square Any Number (a±b Method)

Use the identity (a+b)(a-b) + b² = a² to square numbers quickly by going to the nearest easy number.

Breaking Down Large Multiplications

Split large multiplications into simpler parts using the distributive property.

Casting Out Nines (Verification)

A quick way to verify multiplication and addition results by checking digit sums modulo 9.

Duplex Method for Squaring

The Dwandwa (duplex) method squares any number by summing duplex values digit by digit — a general Vedic squaring technique.

Doubling and Halving

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

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.

Multiply Two Numbers Just Above 100

For numbers a little above 100, use the surplus from 100 to multiply almost instantly.

Percentages with Building Blocks

Any percentage can be built from 10%, 5%, and 1% — quick to find and easy to combine.

Square Numbers Ending in 1

Use (a+1)² = a² + 2a + 1. Square the round number below, add twice it, add 1.

Quick Tip Calculation

Find 10% first, then build 15% or 20% from it in your head.

Rule of 72 (Doubling Money)

Estimate how many years it takes an investment to double: divide 72 by the annual interest rate.

Quick Celsius ↔ Fahrenheit

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

Compare Unit Prices

To find the better deal, divide price by quantity to get the price per unit and compare.

Estimate a Square Root

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

Split a Bill Evenly

Round the total up to a friendly number, divide by the number of people, then adjust down slightly.