17 Of The Most Important Equations You'll Ever Learn

IntroductionIn 2012, mathematician Ian Stewart wrote the book "In Pursuit of the Unknown: 17 Equations That Changed the World", aiming to translate these important equations to the general audience. In his book, he describes equations as being in two distinct groups- the first are equations that show relationships between mathematical quantities and the second one are equations that provide information about unknown quantities. 
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In this article, we lay out these 17 equations and provide a summary of each one. We'll explore the theory, importance, history and applications of each equation. Enjoy!
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Pythagoras's TheoremPythagoras's Theorem (or Pythagorean Theorem) relates the lengths of the sides of a right triangle. The square of the hypotenuse (c) length equals the sum of the squares of the other two sides (a and b). It is fundamental in geometry, physics, and engineering.
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Equation
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a2+b2=c2a^2 +b^2 =c^2a2+b2=c2
Importance
It's one of the foundational theorems in mathematics, used in various fields, from geometry to physics, providing a fundamental relationship between the lengths of the sides of a right triangle.
History
Attributed to the ancient Greek mathematician Pythagoras, this theorem has been used for millennia. It was known to Babylonians and Indians well before Pythagoras, but he and his followers may have been the first to prove it.
Applications
The Pythagorean Theorem is mostly used in architecture, construction, navigation and surveying [2]. 
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After more detail on Pythagoras? Check out our detailed write-up and explanation here!
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Logarithms Logarithms are the inverse of exponential functions. For a given base b, the logarithm function 🚨logbx🚨 gives the power to which b must be raised to produce x. It's vital in solving exponential equations.
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source: desmos.com
Equation
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log⁡(xy)=log⁡(x)+log⁡(y)\log(xy) = \log(x) + \log(y)log(xy)=log(x)+log(y)
Importance
Logarithms are used extensively in computer science, engineering, and economics, often for complexity analysis and growth modelling.
History
Invented independently by John Napier and Joost Bürgi in the early 17th century, logarithms dramatically simplified calculations before calculators, especially in astronomy and navigation.
Applications
Logarithms vastly reduce the time required in multiplying numbers with multiple digits [3] and are used in surveying, celestial navigation and calculation of noise [4]. Specifically, logarithms led to efficient methods for calculating astronomical phenomena and measure of time radioactivity. 
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CalculusCalculus deals with the concepts of derivatives and integrals. The derivative represents the slope of the tangent line to the function at a point, giving the function's rate of change. Integrals provide the accumulation of quantities.
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Equation
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dfdt=lim⁡h→0f(t+h)−f(t)h\frac{df}{dt} = \lim_{{h \to 0}} \frac{f(t + h) - f(t)}{h}dtdf​=limh→0​hf(t+h)−f(t)​
Importance
It's fundamental in physics, engineering, economics, and other fields for modelling change and motion.
History
The foundations of calculus were laid independently and simultaneously by Sir Isaac Newton and Gottfried Wilhelm Leibniz in the late 17th century. It led to a bitter dispute over priority.
Applications
Calculus has many uses in a variety of industries, mostly for optimization, summation and trend prediction. In business, it used in maximizing profits and minimizing costs [5]. It is also used in various engineering fields like fluid mechanics and determination of frictional forces[6]. 
Law of GravityNewton's law of universal gravitation states that every particle attracts every other particle with a force proportional to the product of their masses and inversely proportional to the square of the distance between them.
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Equation
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F=G⋅m1⋅m2r2F = \frac{{G \cdot m_1 \cdot m_2}}{{r^2}}F=r2G⋅m1​⋅m2​​
Importance
It's the foundation for classical mechanics, explaining phenomena from an apple's falling to planets' motion.
History
Formulated by Isaac Newton in 1687 in his work "Mathematical Principles of Natural Philosophy," this law revolutionized the understanding of motion and allowed the prediction of planetary movements.
Application
Newton's Law of Universal Gravitation explains the gravitational forces between two bodies [7] by approximating the gravitational strength between two objects [8]. It is used in discovering new planets and stars and predicting tides. It is also used in finding the masses and predicting the path of astronomical bodies [9]. 
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Learn more about orbital mechanics and calculate orbital velocity here!
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Wave EquationThe wave equation describes the propagation of waves, such as sound or light. It relates the second spatial and temporal partial derivatives of the wave function.
Equation
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∂2u∂t2=v2∂2u∂x2\frac{\partial^2 u}{\partial t^2} = v^2 \frac{\partial^2 u}{\partial x^2}∂t2∂2u​=v2∂x2∂2u​
Importance
It's essential in physics, particularly optics and acoustics, for understanding wave behaviour.
History
Jean d'Alembert derived the one-dimensional wave equation in 1746. This fundamental equation has since become crucial in studying physical phenomena involving waves.
Applications
The wave equation is used in various phenomena, including gravitational, light and sound waves and is also used to study the oscillations of string in string theory [7]. It can also help us understand string and fluid surface movement [10]. 
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The Square Root of Minus OneThe imaginary unit i is defined as the square root of -1. It forms the basis for complex numbers, extending real numbers to accommodate values like square roots of negative numbers.
Equation
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i2=−1i^2 = -1i2=−1
Importance
Complex numbers are central to advanced mathematics, physics, engineering, and signal processing.
History
Imaginary numbers were first coined in the 17th century. The concept was met with resistance but eventually became a fundamental part of complex analysis and engineering.
Applications
Imaginary numbers are used in signal processing, radar, or any parameter that relies on using the sine and cosine wave [11]. 
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Euler's Formula for PolyhedraEuler's polyhedron formula relates the number of vertices (V), edges (E), and faces (F) of a convex polyhedron. It is a topological invariant.
Equation
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V−E+F=2V - E + F = 2V−E+F=2
Importance
It's fundamental in geometry and topology, often used in network analysis and computer graphics.
History
Discovered by Leonhard Euler in the 18th century, this formula united geometry and topology and laid the foundation for graph theory.
Applications
A notable example of the application of Euler's Formula is in computer chip designing, to make sure that the number of intersections in the system are as low as possible [12]. 
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Normal DistributionThe normal distribution is a probability distribution that is symmetric about the mean μ and characterized by its standard deviation σ. It describes many natural phenomena.
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ϕ(x)=12πρ e−(x−μ)22ρ2\phi(x) = \frac{1}{ \sqrt{2\pi}\rho} \, e^{-\frac{(x - \mu)^2}{2\rho^2}}ϕ(x)=2π​ρ1​e−2ρ2(x−μ)2​
Importance
It's widely used in statistics, finance, and social sciences to model random variables.
History
The normal distribution is often called the Gaussian distribution, developed by Carl Friedrich Gauss in the early 19th century. It's fundamental in statistics and probability theory.
Applications
Applications of the normal distribution include quality control, sales forecasting and tasks that require statistical inference and data analysis [13]. 
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Fourier TransformThe Fourier Transform decomposes a function into constituent frequencies, bridging time and frequency domains.
Equation
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F(ω)=∫−∞∞f(x)e−2πixω dx F(\omega) = \int_{-\infty}^{\infty} f(x) e^{-2\pi i  x\omega} \,dx\ F(ω)=∫−∞∞​f(x)e−2πixωdx 
Importance
It's fundamental in signal processing, image analysis, and quantum mechanics.
History
Introduced by Joseph Fourier in the early 19th century, the Fourier Transform has been fundamental in fields such as signal processing and quantum physics.
Applications
In acoustics, the Fourier Transform is used to filter sound by taking signals and expressing them in terms of the wave frequencies that make up the signal [14]. In image processing, the Fourier Transform is used for de-blurring images via the Wiener filter [15]. 
Interested in learning more about Fourier Series and transforms? Check out our detailed write-up and calculator here!
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Navier-Stokes EquationThe Navier-Stokes equations describe the motion of fluid substances. They encompass both the conservation of momentum and the conservation of mass.
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ρ(∂v∂t+v⋅∇v)=−∇p+∇⋅T+f\rho \left(\frac{\partial \textbf{v}}{\partial t}+\textbf{v}\cdot\nabla v\right)=-\nabla p + \nabla\cdot\textbf{T}+\textbf{f}ρ(∂t∂v​+v⋅∇v)=−∇p+∇⋅T+f
Importance
Essential in fluid dynamics, aerodynamics, weather modelling, and more.
History
Named after Claude-Louis Navier and George Gabriel Stokes, these equations were developed in the 19th century and form the foundation of fluid dynamics.
Applications
The Navier-Stokes equations are used to describe fluid flow in its environment and has uses in various fields, including biomedical research and even animation [16]. 
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Maxwell's Equations Maxwell's equations unify electricity and magnetism, providing a complete description of electromagnetic phenomena, including how they generate each other and move through space.
Equations
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∇⋅E=0\nabla \cdot \mathbf{E} = 0∇⋅E=0
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∇×E=−1c∂H∂t  \nabla \times \mathbf{E} = -\frac{1}{c}\frac{\partial \mathbf{H}}{\partial t}\ \ ∇×E=−c1​∂t∂H​  
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∇⋅E=0\nabla \cdot \mathbf{E} = 0∇⋅E=0
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∇×E=1c∂H∂t  \nabla \times \mathbf{E} = \frac{1}{c}\frac{\partial \mathbf{H}}{\partial t}\ \ ∇×E=c1​∂t∂H​  
Importance
They form the foundation of classical electromagnetism, optics, and electric circuits.
History
James Clerk Maxwell formulated these equations in the mid-19th century. They revolutionized physics, leading to the understanding of electromagnetism.
Applications
Maxwell's equations are prominent in electronics advancement and used by electrical engineers to aid in designing electronic devices [17]. 
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Second Law of ThermodynamicsThe second law states that an isolated system's total entropy (or disorder) can never decrease over time. It leads to the definition of a thermodynamic arrow of time.
Equation
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dS≥0dS \geq 0dS≥0
Importance
It's foundational to thermodynamics, with wide-reaching implications in physics, chemistry, and engineering.
History
This law, developed in the 19th century by physicists like Rudolf Clausius and Lord Kelvin, underpins much of modern thermodynamics and statistical mechanics.
Applications
One application of the second law of thermodynamics is the heat engine model where the law dictates the amount of work a heat engine produces when changing temperatures [18]. The second law is used in assessing the theoretical performance limit of these heat engines [19]. 
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RelativityEinstein's famous equation relates energy (E) to mass (m) and the speed of light (c). It's a fundamental result of the theory of relativity.
Equation
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E = mc2E\ =\ mc^2E = mc2
Importance
It revolutionized physics, leading to the development of nuclear energy and a deeper understanding of the universe.
History
This equation was proposed by Albert Einstein in 1905 as part of his Special Theory of Relativity. This equation changed how we understand energy and mass.
Applications
Aside from time dilation, the theory of relativity can also be observed in electromagnetism and GPS navigation [20]. 
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Schrodinger's Equation This equation governs the behaviour of quantum mechanical systems. It describes how the quantum state of a physical system changes over time.
Equation
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iℏ∂∂t=H^Ψi\hbar \frac{\partial}{\partial t}  = \hat{H} \Psiiℏ∂t∂​=H^Ψ
Importance
It's the foundational equation of quantum mechanics, underlying much of modern physics and chemistry.
History
Erwin Schrödinger formulated this equation in 1926. It's a central result in quantum mechanics, describing how wave functions evolve.
Applications
Schrodinger's Equation is one of the most fundamental equations in non-relativistic quantum mechanics and is one of the basic equations in partial differential equations in the field of mathematics [21]. 
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Information TheoryInformation theory quantifies information in bits, using Shannon Entropy to measure uncertainty or surprise with random variables.
Equation
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H(X)=−∑p(x)log⁡p(x)H(X) = -\sum{p(x)} \log p(x)H(X)=−∑p(x)logp(x)
Importance
Information theory is fundamental in digital communication which is vital for efficient data transmission, data compression, and cryptography.
History
Introduced by Claude Shannon in 1948, this measure of uncertainty led to the digital revolution and has applications in data compression and cryptography.
Applications
Information Theory is applied in diverse fields like genomics and statistics, like in computational genomics and machine learning [22].  It can also be used in cryptology and linguistics [23]. 
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Chaos TheoryChaos theory deals with nonlinear systems showing complex and unpredictable behaviour. The Lorenz System, a set of differential equations, exemplifies this "butterfly effect".
Equation
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xt+1=kxt(1−xt)x_{t+1}=kx_t(1-x_t)xt+1​=kxt​(1−xt​)
Importance
Chaos theory has reshaped our understanding of complex systems and phenomena, influencing various fields, from meteorology to economics.
History
Edward Lorenz discovered chaos theory in the 1960s while studying weather patterns. It's a branch of mathematics that deals with systems that appear to be disordered but are governed by underlying patterns.
Applications
Chaos theory has an impact in many fields, including quantum mechanics and cosmetology [24]. Examples of its diverse use include the study of the turbulent flow of fluids, heartbeat irregularities and population dynamics [25].
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Black-Scholes EquationThe Black-Scholes Equation is a model that values European-style options, considering factors like underlying price, strike price, time until expiration, and volatility.
Equation
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12σ2S2∂2V∂S2+rS∂V∂S−rV=0  \frac{1}{2} \sigma^2 S^2 \frac{{\partial^2 V}}{{\partial S^2}} + rS \frac{{\partial V}}{{\partial S}} - rV = 021​σ2S2∂S2∂2V​+rS∂S∂V​−rV=0
Importance
Transforming financial markets by pricing derivatives and hedging risks, this equation is a cornerstone of modern finance and led to a Nobel Prize in Economics.
History
Formulated by Fischer Black, Myron Scholes, and Robert Merton in 1973, this equation plays a crucial role in modern finance, leading to options pricing and earning a Nobel Prize in Economics.
Applications
The Black-Scholes Equation is mostly used in finance by allowing risk management and portfolio optimization. It is also used in market efficiency and streamlined pricing [26]. 
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References
1.	The 17 Equations that Changed the World by Ian Stewart
2.	https://sciencing.com/construct-theodolite-8577736.html
3.	https://www.britannica.com/science/logarithm
4.	https://www.storyofmathematics.com/logarithm/
5.	https://compscicentral.com/what-is-calculus-used-for/r
6.	https://sciencing.com/info-8785081-use-calculus-engineering.html
7.	https://brilliant.org/wiki/wave-equation
8.	https://www.khanacademy.org/science/physics/centripetal-force-and-gravitation/gravity-newtonian/v/introduction-to-newton-s-law-of-gravitation#:~:text=Gravity%20is%20a%20force%20of,and%20the%20distance%20between%20them
9.	https://unacademy.com/content/nda/study-material/physics/universal-law-of-gravitation-physics/#:~:text=This%20law%20helps%20in%20discovering,predict%20the%20astronomical%20bodies'%20path.
10.	https://unacademy.com/content/neet-ug/study-material/physics/all-about-wave-equation/#:~:text=The%20wave%20equation%20is%20utilised,surfaces%20such%20as%20water%20waves.
11.	https://www.livescience.com/42748-imaginary-numbers.html
12.	https://iq.opengenus.org/eulers-polyhedron-formula/#:~:text=Application%20of%20Euler's%20Polyhedron%20Formula,-Euler's%20formula%20is&text=We%20use%20it%20to%20check,the%20connectivity%20of%20a%20graph.
13.	https://www.linkedin.com/pulse/normal-distribution-its-applications-quality-control-process-shaikh#:~:text=The%20normal%20distribution%20is%20widely,people%2C%20follow%20a%20normal%20distribution.
14.	https://www.askamathematician.com/2012/09/q-what-is-a-fourier-transform-what-is-it-used-for
15.	https://medium.com/analytics-vidhya/why-fourier-transform-is-so-important-cb7841733bb8
16.	https://oden.utexas.edu/news-and-events/news/caffarelli-explains-role-in-understanding-navier-stokes-equations/#:~:text=The%20Navier%2DStokes%20equations%20are,through%20a%20well%20or%20pipeline.
17.	https://www.iop.org/explore-physics/big-ideas-physics/maxwells-equations#gref
18.	https://www.britannica.com/question/What-are-some-applications-of-the-second-law-of-thermodynamics
19.	https://www.sfu.ca/~mbahrami/ENSC%20388/Notes/Second%20Law%20of%20Thermodynamics.pdf
20.	https://www.livescience.com/58245-theory-of-relativity-in-real-life.html
21.	https://www.math.ucla.edu/~tao/preprints/schrodinger.pdf#:~:text=In%20pure%20mathematics%2C%20the%20Schr%20%CC%88%20odinger%20equation,spectral%20and%20scattering%20theory%2C%20and%20to%20integrable%20systems.
22.	https://ee.stanford.edu/research/information-theory-and-applications
23.	https://www.britannica.com/science/information-theory/Applications-of-information-theory
24.	https://www.thoughtco.com/chaos-theory-3026621
25.	https://www.britannica.com/science/chaos-theory
26.	https://www.investopedia.com/terms/b/blackscholes.asp
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