There are many important formulas used in science, math, chemistry, and engineering. Some of the most important equations that have changed the course of humanity include the Pythagorean Theorem, the logarithm and its identities, the fundamental theorem of calculus, Newton’s universal law of gravitation, the origin of complex numbers, Euler’s formula for polyhedra, the normal distribution, and the wave equation[1][2]. Other important equations include the first Friedmann equation, which connects the matter and energy that’s present to the expansion rate of the universe today, in the past, and in the future[3]. In physics, some of the most important equations include Einstein’s energy-mass equivalence, Newton’s second law, the Schrödinger equation, the Maxwell-Faraday equation, and Planck’s equation[4]. Finally, in a TED talk, mathematician David Sumpter discusses the 10 equations that rule the world, including the Kelly criterion, which is used to determine the optimal size of a series of bets, and the Black-Scholes equation, which is used to price options[5].
Here are our picks for humanity’s greatest formula hits, perhaps, you might want to have around if you ever need to reboot humanity.
- Einstein’s Mass-Energy Equivalence: E = mc²
Symbol: E (energy), m (mass), c (speed of light);
Units: Energy (joules), mass (kilograms), speed of light (meters per second).
Example: Calculating the energy released in a nuclear reaction. (relating energy and mass) - Newton’s Second Law of Motion: F = ma
Symbol: F (force), m (mass), a (acceleration);
Units: Force (newtons), mass (kilograms), acceleration (meters per second squared).
Example: Determining the force required to accelerate an object. Force is mass times acceleration. - Pythagorean theorem: a² + b² = c²
Symbol: a, b, c (lengths of sides in a right-angled triangle);
Units: Length (generic units).
Example: Finding the third side of a right triangle with known sides. - Newton’s law of universal gravitation: F = G * (m₁ * m₂) / r²
Symbol: F (gravitational force), G (constant of gravitation), m₁, m₂ (masses of interacting objects), r (separation distance);
Units: Force (newtons), mass (kilograms), separation distance (meters), constant of gravitation (newton-meter squared per kilogram squared).
Example: Calculating gravitational force between two planets. (relating gravitational force between two masses) - Einstein’s field equations: Rμν – ½R * gμν = 8πG/c⁴ * Tμν
Symbol: Rμν (Ricci curvature tensor), R (scalar curvature), gμν (metric tensor), G (gravitational constant), c (speed of light), Tμν (energy-momentum tensor);
Units: Curvature (generic units), constant of gravitation (newton-meter squared per kilogram squared), speed of light (meters per second), energy-momentum tensor (generic units). - Ohm’s law: V = I * R
Symbol: V (voltage), I (current), R (resistance);
Units: Voltage (volts), current (amperes), resistance (ohms).
Example: Calculating the current flowing through a resistor. (relating voltage, current, and resistance in an electrical circuit) - Gauss’s law for electric fields: ∮ E · dA = Q/ε₀
Symbol: E (electric field), dA (infinitesimal area vector), Q (total electric charge), ε₀ (vacuum permittivity);
Units: Electric field (newtons per coulomb), area (square meters), electric charge (coulombs), vacuum permittivity (farads per meter). - Schrödinger equation: ĤΨ = EΨ
Symbol: Ψ (wavefunction), Ĥ (Hamiltonian operator), E (energy);
Units: Wavefunction (generic), energy (joules).
Example: Finding the allowed energy levels of an electron in an atom. (describing quantum wave functions and energy levels) - Boltzmann’s entropy formula: S = k * ln(W)
Symbol: S (entropy), k (Boltzmann constant), ln (natural logarithm), W (number of microstates);
Units: Entropy (joules per kelvin), Boltzmann constant (joules per kelvin). - Coulomb’s law: F = k * (q₁ * q₂) / r²
Symbol: F (electrostatic force), k (Coulomb constant), q₁, q₂ (electric charges), r (separation distance);
Units: Electrostatic force (newtons), electric charge (coulombs), separation distance (meters), Coulomb constant (newton-meter squared per coulomb squared).
Example: Calculating the force between two charged particles. (relating electrostatic force between two charged objects) - Planck’s law: E = hf
Symbol: E (energy), h (Planck’s constant), f (frequency);
Units: Energy (joules), Planck’s constant (joule-seconds), frequency (hertz). - Maxwell’s equations
Here are Maxwell’s equations, a set of four differential equations that describe electromagnetic phenomena, with the units for each symbol:
1. Gauss’s law for electric fields:
– ∇ · E = ρ/ε₀
– ∇ · E is the divergence of the electric field (V/m^2).
– ρ is the charge density (C/m^3).
– ε₀ is the permittivity of free space (F/m).
2. Gauss’s law for magnetic fields:
– ∇ · B = 0
– ∇ · B is the divergence of the magnetic field (T/m).
– Here, there is no specific symbol for it being zero.
3. Faraday’s law of electromagnetic induction:
– ∇ × E = -∂B/∂t
– ∇ × E is the curl of the electric field (V/m·s).
– -∂B/∂t is the negative rate of change of magnetic field with respect to time (T/s).
4. Ampère’s law with Maxwell’s addition:
– ∇ × B = μ₀J + μ₀ε₀∂E/∂t
– ∇ × B is the curl of the magnetic field (T/m·s).
– μ₀ is the permeability of free space (N/A^2).
– J is the current density (A/m^2).
– μ₀ε₀∂E/∂t represents the term related to displacement current (A/m^2).
Note: The units mentioned above are the SI (International System of Units) units commonly used for Maxwell’s equations. Various symbols and units related to electric and magnetic fields and their interactions, such as electric field (volts per meter), magnetic field (teslas), electric charge (coulombs), etc.
Example: Describing the behavior of electric and magnetic fields. - Fourier transform
Various symbols and units related to transforming a function between the time and frequency domains, such as function values (generic), time (seconds), frequency (hertz), etc.
Example: Analyzing the frequency content of a signal. - Navier-Stokes equations
Various symbols and units related to describing fluid flow, such as velocity (meters per second), pressure (pascals), viscosity (pascal-seconds), etc. - Henderson-Hasselbalch equation: pH = pKa + log([A⁻]/[HA])
Symbol: pH (acidity or basicity), pKa (dissociation constant), [A⁻] (concentration of base), [HA] (concentration of acid), log (base-10 logarithm);
Units: Acidity/basicity (generic), concentration (moles per liter). - Snell’s law: n₁ * sin(θ₁) = n₂ * sin(θ₂)
Symbol: n₁, n₂ (refractive indices), θ₁, θ₂ (angles of incidence and refraction);
Units: Refractive index (generic), angle (radians).
Example: Calculating the angle of refraction when light passes from one medium to another. (relating the angles and indices of refraction in optics) - Law of conservation of mass
Symbol and units depend on the specific system being considered but usually related to mass (kilograms), time (seconds), and balanced chemical equations.
Example: The law of conservation of mass states that in a chemical reaction, the total mass of the reactants will be equal to the total mass of the products. The burning of propane where there is oxygen results in carbon dioxide and water ( C₃H₈ + 5O₂ → 3CO₂ + 4H₂O ) and the numbers in front of the symbols (5, 3, and 4 in this case) represent the stoichiometric coefficients, indicating the ratio of each compound involved in the reaction. - Bernoulli’s principle: P₁ + ½ρv₁² + ρgh₁ = P₂ + ½ρv₂² + ρgh₂
Symbol: P (pressure), ρ (density), v (velocity), g (acceleration due to gravity), h (height difference);
Units: Pressure (pascals), density (kilograms per cubic meter), velocity (meters per second), acceleration due to gravity (meters per second squared), height (meters). - Pythagorean theorem for trigonometric functions: sin²(θ) + cos²(θ) = 1
Symbol: θ (angle);
Units: Angle (radians). - Stefan-Boltzmann law: P = σAT⁴
Symbol: P (power emitted), σ (Stefan-Boltzmann constant), A (surface area), T (temperature);
Units: Power (watts), Stefan-Boltzmann constant (watt per square meter per kelvin to the fourth power), surface area (square meters), temperature (kelvin). - Bayes’ theorem: P(A|B) = [P(B|A) * P(A)] / P(B)
Symbol: P(A|B) (probability of A given B), P(B|A) (probability of B given A), P(A) (probability of A), P(B) (probability of B);
Units: Probability (dimensionless).
Example: Updating the probability of an event based on new evidence. (relating conditional probabilities) - Le Chatelier’s principle
Symbol and units depend on the specific equilibrium reaction being considered, but generally related to concentrations (moles per liter) and temperature (kelvin). - Second law of thermodynamics
Various symbols and units related to entropy change, heat transfer, and efficiency, such as entropy (joules per kelvin), heat (joules), efficiency (percent), etc. - Fisher’s fundamental theorem of natural selection
Various symbols and units related to population genetics and evolutionary biology, such as allele frequencies (dimensionless), selection coefficients (dimensionless), etc. - Drake equation: N = R* × fp × ne × fl × fi × fc × L
Symbol: N (estimated number of intelligent civilizations), R* (rate of star formation), fp (fraction of stars with planets), ne (average number of planets that could support life per star), fl (fraction of planets where life evolves), fi (fraction of life that develops intelligence), fc (fraction of intelligent civilizations that communicate), L (length of time they communicate);
Units: Estimated number (dimensionless), rates and fractions (dimensionless), length of time (years). - Boyle’s Law: P₁V₁ = P₂V₂
Symbol: P₁: Initial pressure (typically measured in units of pressure such as atmospheres, pascals, or torr). V₁: Initial volume (usually measured in units of volume like liters or cubic meters). P₂: Final pressure (also measured in units of pressure like atmospheres, pascals, or torr). V₂: Final volume (measured in units of volume like liters or cubic meters).
The product of the initial pressure and volume is equal to the product of the final pressure and volume in a given system, assuming the temperature and amount of gas remain constant.
Example: Determining the final volume of a gas when pressure is changed. (relating pressure and volume of a gas at constant temperature) - Exponential Growth: A = P * (1 + r/n)^(nt)
Symbols:
A – represents the future value of the investment or population at time t
P – represents the principal or initial value of the investment or population
r – represents the interest rate or growth rate (expressed as a decimal)
n – represents the number of times the interest is compounded per time unit
t – represents the time period (in years, months, etc.)
The units will depend on the context.
For financial investments, P could be in dollars ($), A would also be in dollars ($), r would be in percentage (%), n may be the number of times interest is compounded per year, and t would be in years. For population growth, P can represent the initial population size, A would be the future population size, and t could be in years. The units will vary based on the specific application.
Example: Calculating the future value of an investment with compound interest. (relating compound interest over time) - Quadratic Formula: x = (-b ± √(b² – 4ac)) / (2a)
Example: Finding the roots of a quadratic equation. (solving quadratic equations) - Law of Conservation of Energy: E₁ + E₂ = E₃
Example: Evaluating the total energy before and after a collision. (energy is conserved in a closed system) - The Golden Ratio: φ = (1 + √5) / 2
Example: Finding the ratio between the length of a line segment and its longer part. (relating two segments in a ratio) - Archimedes’ Principle: Fb = ρ * V * g
– Fb represents the buoyant force, which is the upward force exerted on an object submerged in a fluid.
– ρ (rho) represents the density of the fluid.
– V represents the volume of the fluid displaced by the submerged object.
– g represents the acceleration due to gravity.
Units:
– Fb: The buoyant force is typically measured in Newtons (N).
– ρ: Density is measured in kilograms per cubic meter (kg/m³).
– V: Volume is measured in cubic meters (m³).
– g: Acceleration due to gravity is measured in meters per second squared (m/s²).
Example: Calculating the buoyant force on an object submerged in a liquid. (relating buoyant force, volume, density, and gravity) - Hooke’s Law: F = k * Δx
– F represents the force (in newtons, N).
– k represents the spring constant (in newtons per meter, N/m).
– Δx represents the displacement or change in length of the spring (in meters, m).
Example: Determining the force required to stretch or compress a spring. (relating the force exerted by a spring and its displacement) - Doppler Effect: f’ = f * (c ± v) / (c ± vs)
f’ – frequency observed by the observer (Hz)
f – frequency of the source (Hz)
c – speed of sound (m/s)
v – velocity of the observer or source relative to the medium (m/s)
vs – speed of the source relative to the medium (m/s)
Example: Determining the observed frequency of a moving sound source. (relating the frequency of sound waves to the relative motion of the source and observer) - Bernoulli’s Equation: P₁ + ½ρv₁² + ρgh₁ = P₂ + ½ρv₂² + ρgh₂
P₁ and P₂: Pressure (Pascal or Pa)
ρ: Density of the fluid (kilograms per cubic meter or kg/m³)
v₁ and v₂: Velocity of the fluid (meters per second or m/s)
g: Acceleration due to gravity (meters per second squared or m/s²)
h₁ and h₂: Height of the fluid column (meters or m)
Example: Calculating the pressure difference along a streamline in a fluid flow. (relating pressure, velocity, and height in fluid dynamics) - Shannon’s Entropy: H(X) = -Σ(p(x) * log₂(p(x)))
Example: Quantifying the level of randomness in a coin toss. (relating the randomness or uncertainty of a probability distribution) - First Law of Thermodynamics: ΔU = Q – W
ΔU: change in internal energy (Joules)
Q: heat transferred to the system (Joules)
W: work done by the system (Joules)
Example: Evaluating the change in internal energy during a heating process. (relating the internal energy change of a system, heat transfer, and work done) - Calculus Fundamental Theorem: ∫(dF(x)/dx)dx = F(x)
Example: Finding the function when given its derivative or integral. (relating the derivative and integral of a function) - Central Limit Theorem:
states that the sum or average of a large number of independent and identically distributed random variables will follow a nearly normal distribution
Example: Approximating the distribution of test scores in a large population.
Formula of Humanity
The “Formula of Humanity” is a key concept in the ethical philosophy of Immanuel Kant. It is part of his categorical imperative and emphasizes the inherent value of human beings. The formula states: “Act in such a way that you always treat humanity, whether in your own person or in the person of any other, never simply as a means, but always at the same time as an end”[3]. This principle is derived from Kant’s idea that rational beings should be treated as ends in themselves, not merely as a means to an end. The formula of humanity is discussed in various philosophical works, including those by Stephen Darwall, Christine M. Korsgaard, and Onora O’Neill[2]. The concept is also examined in academic publications such as “The Argument for the Humanity Formula” and “The Formula of Humanity (FH)”[4][5]. These sources provide in-depth analysis and discussion of the importance of the formula of humanity in Kant’s ethical theory.
Citations:
[1] https://www.thejournal.ie/equations-changed-course-of-humanity-764429-Jan2013/
[2] https://eighteeneight.com/blog/the-17-equations-that-changed-the-course-of/amp/
[3] https://www.forbes.com/sites/startswithabang/2018/04/17/the-most-important-equation-in-the-universe/
[4] https://blog.praxilabs.com/2019/05/06/3-most-important-physics-equations-in-history/
[5] https://youtube.com/watch?v=-81rDqPBrGs
[6] https://myweb.ecu.edu/mccartyr/gw/FormulaOfHumanity.asp
[7] https://ndpr.nd.edu/reviews/means-ends-and-persons-the-meaning-and-psychological-dimensions-of-kants-humanity-formula/
[8] https://plato.stanford.edu/entries/persons-means/
[9] https://academic.oup.com/book/9330/chapter-abstract/156102435?redirectedFrom=fulltext
[10] https://academic.oup.com/book/11513/chapter-abstract/160268625?redirectedFrom=fulltext
