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176 results for “ph”

Formulae (176)

Parabola Vertex

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

x-coordinate of the vertex of a quadratic.

Geometry & TrigonometryBasic

Volume of a Sphere

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

The volume enclosed by a sphere.

Geometry & TrigonometryBasic

Surface Area of a Sphere

A = 4\pi r^2

The total surface area of a sphere.

Geometry & TrigonometryAdvanced

Spherical Cap Volume

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

Volume of a slice cut from a sphere.

TrigonometryIntermediate

Sinusoidal Function

y = A\sin(Bx + C) + D

General sine wave with amplitude A, period 2π/B, phase C, shift D.

Data Science & MLAdvanced

Gradient Descent Update

\theta = \theta - \alpha \nabla J(\theta)

Iterative optimization step to minimize a cost function.

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

Gravitational Potential Energy

PE = mgh

Energy stored due to height in a gravitational field.

PhysicsBasic

Work Done

W = F d \cos\theta

Work done by a force over a distance.

PhysicsBasic

Power

P = \frac{W}{t}

Rate of doing work.

PhysicsBasic

Momentum

p = mv

Quantity of motion of an object.

PhysicsIntermediate

Impulse

J = F\Delta t = \Delta p

Change in momentum from a force over time.

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

Ohm's Law

V = IR

Voltage equals current times resistance.

PhysicsBasic

Electrical Power

P = VI = I^2R

Power dissipated in an electrical component.

PhysicsAdvanced

Mass–Energy Equivalence

E = mc^2

Energy equivalent of mass.

PhysicsBasic

Wave Speed

v = f\lambda

Speed of a wave from frequency and wavelength.

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.

PhysicsIntermediate

Hooke's Law

F = -kx

Restoring force of a spring.

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.

PhysicsBasic

Heat Energy

Q = mc\Delta T

Heat needed to change temperature of a substance.

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

Weight

W = mg

The gravitational force on a mass.

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.

PhysicsIntermediate

Energy of a Photon

E = hf

Energy of a photon from its frequency.

PhysicsBasic

Weight

W = mg

Gravitational force on a mass near Earth.

PhysicsBasic

Hooke Law

F = -kx

Restoring force of a spring proportional to displacement.

PhysicsIntermediate

Impulse

J = F\Delta t = \Delta p

Change in momentum from a force over time.

PhysicsIntermediate

Torque

\tau = rF\sin\theta

Rotational effect of a force about a pivot.

PhysicsIntermediate

Angular Momentum

L = I\omega

Rotational analog of linear momentum.

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

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 & HeatIntermediate

Latent Heat

Q = mL

Heat for a phase change at constant temperature.

Thermodynamics & HeatIntermediate

Linear Thermal Expansion

\Delta L = \alpha L_0 \Delta T

Length change of a solid when heated.

Optics & WavesIntermediate

Photon Energy

E = hf

Energy of a single photon.

Optics & WavesIntermediate

Photon Energy (Wavelength)

E = \frac{hc}{\lambda}

Photon energy from its wavelength.

Electricity & MagnetismAdvanced

Gauss's Law

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

Electric flux through a closed surface.

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.

Fluid MechanicsAdvanced

Stokes' Drag

F = 6\pi\mu r v

Drag on a small sphere in viscous flow.

Modern & Quantum PhysicsIntermediate

Mass-Energy Equivalence

E = mc^2

Energy contained in mass.

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 PhysicsIntermediate

Planck Relation

E = hf

Quantised energy of radiation.

Modern & Quantum PhysicsAdvanced

Photoelectric Effect

K_{max} = hf - \phi

Kinetic energy of ejected electrons.

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

Binding Energy

E_b = \Delta m\,c^2

Energy holding a nucleus together.

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.

Modern & Quantum PhysicsAdvanced

Schrödinger (Time-Independent)

\hat{H}\psi = E\psi

Fundamental equation of quantum mechanics.

Electronics & CircuitsAdvanced

AC Real Power

P = VI\cos\phi

Real power with a phase angle between V and I.

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.

ChemistryBasic

pH and pOH

pH + pOH = 14

Relationship between acidity and basicity at 25°C.

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 SciencesIntermediate

Osmolarity

\text{Osm} = n \times C

Solute particle concentration in solution.

Biology & Life SciencesBasic

Monohybrid Cross Ratio

3 : 1

Phenotype ratio from a heterozygous monohybrid cross.

Biology & Life SciencesIntermediate

Dihybrid Cross Ratio

9 : 3 : 3 : 1

Phenotype ratio from a dihybrid cross.

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

Respiratory Quotient

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

Ratio revealing which fuel is being metabolized.

Health & FitnessAdvanced

Creatinine Clearance (Cockcroft-Gault)

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

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

Astronomy & SpaceAdvanced

Photon Sphere Radius

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

Radius where gravity bends light into a circular orbit.

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.

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 & CivilIntermediate

Linear Thermal Expansion

\Delta L = \alpha L_0 \Delta T

Change in length due to temperature change.

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

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

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

Great Circle Distance

d = R\arccos(\sin\phi_1\sin\phi_2 + \cos\phi_1\cos\phi_2\cos\Delta\lambda)

Shortest distance between two points on the Earth for long-range navigation.

Aviation & AeronauticsAdvanced

Turn Radius

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

Radius of a coordinated banked turn.

Aviation & AeronauticsAdvanced

Density Altitude

DA = PA + 120(OAT - ISA)

Pressure altitude corrected for temperature.

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 & EarthBasic

Degrees to Distance

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

Approximate ground distance per degree of latitude.

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

Earthquake Energy (Richter)

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

Energy released by a quake of Richter magnitude M.

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

Lithostatic Pressure

P = \rho g h

Pressure from the weight of overlying rock.

Geography & EarthIntermediate

Geothermal Gradient

\Delta T = G_T \, d

Temperature rise with depth into the Earth.

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 ScienceIntermediate

Solar Panel Output

P = A \times r \times H \times PR

Energy output of a photovoltaic array.

Environmental ScienceIntermediate

pH

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

Acidity from hydrogen ion concentration.

Environmental ScienceIntermediate

Albedo

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

Fraction of solar radiation reflected by a surface.

Unit ConversionsBasic

km/h to mph

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

Convert speed from km/h to mph.

Unit ConversionsBasic

Atmospheres to Pascals

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

Convert atmospheres to pascals.

Unit ConversionsBasic

MPH to KM/H

km/h = mph \times 1.6093

Convert miles per hour to kilometers per hour.

Unit ConversionsBasic

Atmospheres to Pascals

Pa = atm \times 101325

Convert atmospheres to pascals.

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

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.