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190 results for “em”

Formulae (184)

Order of Operations (PEMDAS)

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

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

Arithmetic & BasicsBasic

Even and Odd

n \bmod 2 = 0 \Rightarrow \text{even}

A number is even if it divides by 2 with no remainder.

AlgebraIntermediate

Binomial Theorem

(a+b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k

Expands any power of a binomial into a sum of terms.

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.

Geometry & TrigonometryBasic

Pythagorean Theorem

a^2 + b^2 = c^2

Relates the sides of a right triangle.

Geometry & TrigonometryIntermediate

Area of a Regular Polygon

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

Area from apothem and perimeter.

Geometry & TrigonometryIntermediate

Area of Regular Polygon

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

Area from apothem and perimeter.

Geometry & TrigonometryBasic

Cofunction Identity

\sin\theta = \cos(90\degree - \theta)

Sine and cosine of complementary angles are equal.

TrigonometryIntermediate

Heron's Formula

\text{Area} = \sqrt{s(s-a)(s-b)(s-c)}

Triangle area from the three side lengths, where s is the semi-perimeter.

TrigonometryAdvanced

De Moivre's Theorem

(\cos\theta + i\sin\theta)^n = \cos n\theta + i\sin n\theta

Powers of complex numbers in trig form.

CalculusAdvanced

Fundamental Theorem of Calculus

\int_a^b f(x)\,dx = F(b) - F(a)

Connects differentiation and integration.

CalculusAdvanced

Mean Value Theorem

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

Guarantees a point where the tangent equals the average slope.

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.

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

Permutations with Repetition

n^r

Ordered arrangements of r items from n when repetition is allowed.

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

Complement Rule

P(A') = 1 - P(A)

Probability that an event does not occur.

Probability & CombinatoricsAdvanced

Bayes Theorem

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

Reverses conditional probabilities using prior knowledge.

Probability & CombinatoricsIntermediate

Pigeonhole Principle

\lceil n/k \rceil

If n items go into k containers, some container holds at least this many.

Probability & CombinatoricsAdvanced

Stars and Bars

\binom{n+k-1}{k-1}

Ways to distribute n identical items into k distinct groups.

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 & MLIntermediate

Gini Impurity

Gini = 1 - \sum_{i=1}^{n} p_i^2

Probability of misclassifying a random element; used in decision trees.

Linear AlgebraAdvanced

Rank-Nullity Theorem

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

Relates rank and null space dimension to columns.

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.

PhysicsBasic

Heat Energy

Q = mc\Delta T

Heat needed to change temperature of a substance.

PhysicsBasic

Hooke Law

F = -kx

Restoring force of a spring proportional to displacement.

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

Conservation of Momentum

\sum p_i = \sum p_f

Total momentum of an isolated system stays constant.

Physics Laws & TheoremsBasic

Conservation of Energy

E_{total} = \text{const}

Energy cannot be created or destroyed, only transformed.

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 & TheoremsBasic

Hooke's Law

F = -kx

The restoring force of a spring is proportional to its extension.

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

Kepler's Third Law

T^2 \propto a^3

The square of the orbital period is proportional to the cube of the semi-major axis.

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 & TheoremsIntermediate

First Law of Thermodynamics

\Delta U = Q - W

Energy conservation: change in internal energy equals heat added minus work done.

Physics Laws & TheoremsAdvanced

Second Law of Thermodynamics

\Delta S_{universe} \ge 0

The entropy of an isolated system never decreases.

Physics Laws & TheoremsBasic

Boyle's Law

P_1 V_1 = P_2 V_2

At constant temperature, pressure and volume of a gas are inversely proportional.

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 & TheoremsAdvanced

Ampère's Law

\oint \vec{B} \cdot d\vec{l} = \mu_0 I

A magnetic field circulates around an electric current.

Physics Laws & TheoremsIntermediate

Kirchhoff's Current Law

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

Current into a junction equals current out of it (charge conservation).

Physics Laws & TheoremsIntermediate

Kirchhoff's Voltage Law

\sum V = 0

The sum of voltages around any closed loop is zero.

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 & TheoremsIntermediate

Mass–Energy Equivalence

E = mc^2

Mass and energy are equivalent and interconvertible.

Physics Laws & TheoremsAdvanced

Planck–Einstein Relation

E = hf

The energy of a photon is proportional to its frequency.

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

Conservation of Energy

E_{total} = \text{constant}

Energy cannot be created or destroyed, only transformed.

Physics Laws & TheoremsIntermediate

Conservation of Momentum

\sum p_i = \sum p_f

Total momentum is conserved without external forces.

Physics Laws & TheoremsIntermediate

Conservation of Charge

\sum q = \text{constant}

Total electric charge in an isolated system is conserved.

Physics Laws & TheoremsIntermediate

First Law of Thermodynamics

\Delta U = Q - W

Change in internal energy equals heat added minus work done.

Physics Laws & TheoremsAdvanced

Second Law of Thermodynamics

\Delta S_{universe} \geq 0

Total entropy of an isolated system never decreases.

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

Lenz Law

\text{induced current opposes change}

Induced currents oppose the change producing them.

Physics Laws & TheoremsIntermediate

Archimedes Principle

F_b = \rho V g

Buoyant force equals the weight of displaced fluid.

Physics Laws & TheoremsIntermediate

Pascal Principle

\Delta P = \text{transmitted equally}

Pressure applied to a confined fluid transmits undiminished.

Thermodynamics & HeatBasic

Heat Energy

Q = mc\Delta T

Heat needed to change a substance’s temperature.

Thermodynamics & HeatIntermediate

Latent Heat

Q = mL

Heat for a phase change at constant temperature.

Thermodynamics & HeatIntermediate

Ideal Gas Law

PV = nRT

Relates pressure, volume, moles and temperature.

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

First Law of Thermodynamics

\Delta U = Q - W

Energy conservation for a thermodynamic system.

Thermodynamics & HeatAdvanced

Entropy Change

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

Reversible entropy change at temperature T.

Thermodynamics & HeatBasic

Celsius to Fahrenheit

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

Convert Celsius to Fahrenheit.

Thermodynamics & HeatBasic

Celsius to Kelvin

K = C + 273.15

Convert Celsius to Kelvin.

Electricity & MagnetismAdvanced

Faraday's Law

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

Induced EMF from changing magnetic flux.

Electricity & MagnetismAdvanced

Motional EMF

\mathcal{E} = BLv

EMF from a conductor moving through a 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 MechanicsIntermediate

Hydraulic Press (Pascal)

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

Pressure transmits equally in a hydraulic system.

Modern & Quantum PhysicsIntermediate

Radioactive Decay

N = N_0 e^{-\lambda t}

Number of nuclei remaining over time.

Modern & Quantum PhysicsAdvanced

Wien's Displacement Law

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

Peak emission wavelength of a hot body.

Electronics & CircuitsAdvanced

Wheatstone Bridge Balance

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

Condition for a balanced measurement bridge.

ChemistryIntermediate

Ideal Gas Law

PV = nRT

Relates pressure, volume, moles and temperature of a gas.

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.

ChemistryBasic

pOH

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

Basicity from hydroxide-ion concentration.

ChemistryAdvanced

Henderson–Hasselbalch

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

pH of a buffer solution.

ChemistryBasic

Dilution Equation

C_1 V_1 = C_2 V_2

Relates concentration and volume before and after dilution.

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

Beer–Lambert Law

A = \varepsilon l c

Absorbance of light by a solution.

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

Avogadro's Number

N = n \times N_A

Number of particles from moles.

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

Boyle's Law

P_1 V_1 = P_2 V_2

Pressure and volume are inversely related at constant temperature.

ChemistryIntermediate

Charles's Law

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

Volume is proportional to temperature at constant pressure.

ChemistryIntermediate

Ideal Gas Law

PV = nRT

Relates pressure, volume, moles, and temperature of a gas.

ChemistryBasic

Boyle Law

P_1 V_1 = P_2 V_2

At constant temperature, pressure and volume are inversely related.

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.

ChemistryBasic

Avogadro Number

N_A = 6.022\times 10^{23}

Number of particles in one mole.

ChemistryBasic

Number of Particles

N = n\times N_A

Particles from moles and Avogadro number.

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.

ChemistryAdvanced

Hess Law

\Delta H = \sum \Delta H_{steps}

Total enthalpy change is the sum of step changes.

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

Michaelis-Menten Kinetics

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

Rate of an enzyme-catalysed reaction.

Biology & Life SciencesAdvanced

Hemocytometer Cell Count

C = \bar{n} \times DF \times 10^4

Cells per mL from a counting-chamber average.

Biology & Life SciencesIntermediate

Enzyme Activity Unit

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

One unit converts 1 µmol of substrate per minute.

Biology & Life SciencesAdvanced

Biochemical Oxygen Demand

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

Oxygen consumed by microbes decomposing organic matter.

Biology & Life SciencesIntermediate

Net Primary Productivity

NPP = GPP - R

Energy available to an ecosystem after respiration.

Biology & Life SciencesAdvanced

Enzyme Turnover Number

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

Reactions catalyzed per enzyme per second.

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 & FitnessIntermediate

Ponderal Index

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

A leanness index that scales better for extreme heights.

Nutrition & FoodBasic

Protein Requirement (RDA)

P = 0.8 \times m

Recommended daily protein for a healthy adult (grams).

Nutrition & FoodAdvanced

Glycemic Load

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

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

Astronomy & SpaceAdvanced

Kepler's Third Law

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

Relates orbital period to semi-major axis.

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 & SpaceIntermediate

Wien's Displacement Law

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

Peak emission wavelength of a blackbody.

Astronomy & SpaceAdvanced

Hawking Temperature

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

Temperature of the radiation a black hole emits.

EngineeringIntermediate

Linear Thermal Expansion

\Delta L = \alpha L_0 \Delta T

Change in length due to temperature change.

EngineeringAdvanced

Electrical Power (3-phase)

P = \sqrt{3}\,V I \cos\phi

Real power in a balanced three-phase system.

Materials & CivilBasic

Concrete Mix Volume

V = V_c + V_s + V_g

Total volume from cement, sand, and gravel parts.

Materials & CivilIntermediate

Linear Thermal Expansion

\Delta L = \alpha L_0 \Delta T

Change in length due to temperature change.

Materials & CivilIntermediate

Water-Cement Ratio

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

Ratio controlling concrete strength and workability.

Aviation & AeronauticsIntermediate

Density Altitude

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

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

Aviation & AeronauticsIntermediate

Speed of Sound

a = 38.94\sqrt{T}

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

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

Density Altitude

DA = PA + 120(OAT - ISA)

Pressure altitude corrected for temperature.

Finance & InterestAdvanced

Loan EMI

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

Equated monthly installment on a loan.

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.

AccountingIntermediate

Net Profit Margin

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

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

EconomicsBasic

Unemployment Rate

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

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

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 Demand

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

Responsiveness of demand to price changes.

EconomicsBasic

Unemployment Rate

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

Fraction of the labor force without work.

Everyday MathBasic

Celsius to Fahrenheit

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

Convert temperature from Celsius to Fahrenheit.

Geography & EarthIntermediate

Geothermal Gradient

\Delta T = G_T \, d

Temperature rise with depth into the Earth.

Environmental ScienceBasic

Carbon Footprint

CO_2 = E \times f

Emissions from energy use and an emission factor.

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 ScienceBasic

Dilution Equation

C_1 V_1 = C_2 V_2

Relates concentration and volume before and after dilution.

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

Celsius to Kelvin

K = °C + 273.15

Convert Celsius to the absolute Kelvin scale.

Unit ConversionsBasic

Kelvin to Celsius

°C = K - 273.15

Convert Kelvin to Celsius.

Unit ConversionsBasic

Celsius to Kelvin

K = C + 273.15

Convert Celsius to Kelvin.

Number TheoryBasic

Modular Congruence

a \equiv b \pmod{m}

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

Number TheoryAdvanced

Fermat's Little Theorem

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

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

Number TheoryAdvanced

Prime Number Theorem

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

Approximates the number of primes below x.

Number TheoryAdvanced

Wilson Theorem

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

A primality criterion for p.

Computer ScienceAdvanced

Master Theorem (form)

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

Analyzes running time of divide-and-conquer algorithms.

Computer ScienceBasic

Decimal to Binary

N = \text{repeated division by } 2

Convert a decimal number to binary by dividing by 2 and reading remainders.

Computer ScienceAdvanced

Two's Complement

-N = \lnot N + 1

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

Computer ScienceAdvanced

Exponential Time

O(2^n)

Runtime that doubles with each added element.

Computer ScienceAdvanced

Master Theorem Form

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

General form for analyzing divide-and-conquer recurrences.

Computer ScienceIntermediate

Address Space

N = 2^n

Number of addressable locations for n bits.

Computer ScienceBasic

Hexadecimal to Decimal

d = \sum h_i \times 16^i

Convert a hex number to decimal by positional weights.

Computer ScienceAdvanced

Average Memory Access Time

AMAT = H + M \times P

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

Math Tricks (6)

Vinculum Numbers

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

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 4 and 8 by Doubling

Multiplying by 4 is doubling twice; by 8 is doubling three times. No memorised tables needed.

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