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Essential Mathematical Equations

From Robin, published on 27/06/2017 We Love Prof - UK > Academia > Maths > 10 Maths Formulas that Changed the World

Mathematics is always around us, everywhere we go.

Whether it’s the construction of your house, the layout of streets in your neighbourhood, the simple act of starting your car or turning on your dishwasher, when you do DIY or play the piano, maths is truly everywhere.

There is not an object in existence which is not somehow the outcome of maths in action.

Complex equations with many unknowns, mathematical theorems dating back to antiquity, to late twentieth century discoveries, have all shaped our world.

And with each new concept, our understanding of the physical world around us grows.

In 2013, renowned British mathematician and scientist Ian Stewart published a book entitled “The 17 Equations That Changed the World” (Ed. Robert Laffont).

If you wonder why maths is so important, and the impact that each major equation has wrought, read on to discover 10 revolutionary formulas to take your maths tuition to another level.


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Pythagoras’ Theorem

This is surely one of the best-known theorems. Even years after your last maths class, its name springs easily to mind.

A theorem of Euclidean geometry A proof from Euclid’s Elements (Source: Wikipedia.org)

You may know it by heart, but let’s quickly recap: In a right-angled triangle, the square of the hypotenuse is equal to the sum of the square roots of the lengths of the other two sides.

This theorem, which dates back to 530 BCE, is one of the foundations of maths to this day, and has contributed to the history of maths ever since its discovery.

This equation is essential to an understanding of geometry and trigonometry, and indeed has shaped our understanding of those branches of mathematics.

It is said that we have moved from Euclidean geometry to non-Euclidean geometry.

The central endpoint of the "seahorse tail" is also a Misiurewicz point When Pythagoras’ theorem meets art (Source: commons.wikimedia.org)

Since then, thanks to Pythagoras and his famous equation, it’s now easy to calculate lengths, angles and to demonstrate that a given triangle is right-angled.

This concept is often to be found in the realms of construction and architecture.


Logarithms, popularised by John Napier in 1610, combine inverse and exponential functions, and opposites.

Logarithms are common in formulas used in science, to measure the complexity of algorithms and fractals, and appear in formulas for counting prime numbers.

The logarithm is the inverse operation to exponentiation The logarithm of a product is the sum of the logarithms of the factors (Source: Wikipedia.org)

Until the development of the modern computer, calculating with logarithms was the most usual way of multiplying large numbers together, and made possible faster calculations, but above all helped make leaps and bounds in the fields of maths, physics, engineering and astronomy.

There are 3 types of logarithm:

  • Natural logarithms are the fundamental basis in mathematical analysis
  • Decimal logarithms are used in mathematical calculations
  • Binary logarithms are used in computational theory and for applied calculations

The logarithm of a number is the exponent to which another fixed number, the base, must be raised to produce that number.

For example, in the case of base 10, the logarithm (log) is: Log (1) = 0, log (10) = 1, log (100) = 2.

Such calculations are useful in, for example, poker, and in solving puzzles.

The Law of Gravity

Who has never heard of Isaac Newton’s famous law of gravity? You know the story of the apple which fell on the great thinker’s head while he pondered the moon in the night sky, in the year 1687.

The law of gravity, or the law of universal attraction, discovered by Isaac Newton The law of gravity in its modern form (Source: Wikipedia.org )

It was by drawing a connection between these two bodies (the moon and the apple) that Newton then wondered: Why does the moon not fall from the sky?

The answer is obvious – now: It is “held up” by a gravitational force.

The famous tree that gave rise to Newton’s idea? Newton’s Tree, Trinity College, Cambridge (Source: Geograph Britain and Ireland project – N Chadwick)

Thus was born Newton’s famous law of gravity: “Astral bodies attract each other with a force that is directly proportional to the product of their masses and inversely proportional to the square of the distance between their centres.”

200 years after Newton, Einstein replaced this theory of gravitation by his theory of relativity.

The Theory of Relativity

Whether one is versed in mathematics or physics, or knows nothing of the vocabulary of maths, everyone knows Albert Einstein’s famous formula: E = mc².


Mass–energy equivalence E represents energy, m the mass of a body and c the speed of light (Source: publicdomainpictures.net – Daniele Pellati)

This formula, which illustrates the theory of relativity (restricted relativity and general relativity) revolutionised our understanding of physics up to that point.

It remains crucial to this day, as it shows that matter can be converted into energy and vice versa.

Restricted relativity introduced the idea that the speed of light was a universal constant that did not change, and that the passage of time was not the same for bodies moving at different speeds.

Einstein’s general relativity describes the gravity in which space and time are curved and folded: A major change in our understanding following Newton’s law of gravity.

Einstein was the first to propose that the equivalence of mass and energy is a general principle and a consequence of the symmetries of space and time. Albert Einstein: Genius and fashion icon (Source: Pixbay.com – janeb13)

Even today, Einstein’s theory of relativity remains essential in our understanding of the origin, structure and destiny of our Universe.

Maths helps us better understand the world around us, and is an omnipresent force in our daily lives.

Chaos Theory

Chaos theory has shown us that it’s impossible to predict with certainty what will happen in the future. It is the study of the behaviour of dynamic systems.

This theory proves that no really existing processes may be predicted with certainty. Robert May’s theory is more recent, dating from 1975. It describes a process that is constantly evolving over time.

In his formula, May wanted to explain that chaotic behaviour (like climate, which experiences numerous changes in weather from moment to moment) can lead to changes in other completely different systems a few days later.

The best-known illustration is the so-called “butterfly effect” which shows that the beating of a butterfly’s wings in Brazil can lead to a hurricane or tornado in Asia.

In other words, the most insignificant of things can have unsuspected effects on our environment, near and far.

Studies of the critical point beyond which a system creates turbulence were important for chaos theory Turbulence in the tip vortex from an aeroplane wing (Source: National Aeronautics and Space Administration (NASA))

It is the multiplicity of factors related to an event that makes it unpredictable.

Euler’s Identity

Euler’s identity is considered to be “the finest of equations” in maths classes because it describes an unlikely combination of five mathematical constants.

Euler's identity is named after the Swiss mathematician Leonhard Euler Euler’s identity is the equality where e is Euler’s number, the base of natural logarithms, i is the imaginary unit, which satisfies i2 = −1, and π is pi, the ratio of the circumference of a circle to its diameter (Source: Wikipedia.org)

Euler’s equation (published by Leonhard Euler in 1755) applies in the case of a perfect fluid.

Why does this equation matter? Because it makes use of three fundamental operations in arithmetic: addition, multiplication and exponentiation.

The five constants represented are “0”, the additive identity; “1”, the multiplicative identity; the fabulous pi; “e” which is the base of natural logarithms and a number which widely occurs in mathematical analyses; and “i”, the imaginary unit of the complex numbers found in equations with 3 unknowns.

This equation, which decorates the Palais de la Découverte in Paris, paved the way for the development of topology, a branch of modern maths.

The Fourier Transform

The Fourier transform divides time into several frequencies and simple waves, just as a prism splits light in its constituent colours.

The Fourier transform decomposes a function of time into the frequencies that make it up. The Fourier transform (Source: Wikipedia.org )

The Fourier transform allows us to deal with non-periodic functions.

Another example could be a magnetic or an acoustic field that is defined as a signal. The Fourier transform is its spectrum, in that it deconstructs such a field.

The famous album cover by Hipgnosis and George Hardie The cover of Pink Floyd’s “Dark Side of the Moon” (Source: Flickr.com – El Silver)

This theory was so earth-shaking because, suddenly, it was possible to understand the structure of more complex waves, such as human speech.

Today this theory, which dates back to 1822, goes to the heart of modern signal processing and analysis, as well as data processing.

Maxwell’s Equations

Maxwell’s equations describe how electric charges interact, as well as explaining electric currents and magnetic fields.

Maxwell’s equations, also called Maxwell-Lorentz equations, are fundamental laws of physics.

They underpin our understanding of the relationship between electricity and magnetism, and are among the essential, fundamental laws of modern physics.

They underpin all electric, optical and radio technologies, including WiFi. Maxwell’s equations form the foundation of classical electromagnetism (Source: commons.wikimedia.org)

There are 4 forms of Maxwell’s equations:

  • The Maxwell-Gauss equation
  • The Maxwell-Thomson equation
  • The Maxwell-Faraday equation
  • The Maxwell-Ampere Equation

The Second Law of Thermodynamics

The second law of thermodynamics (also known as the Carnot principle after its discoverer, in 1824) proves irrefutably that physical phenomena are irreversible, especially when thermal changes occur.

The principles of thermodynamics are the principal laws governing thermodynamics.

This principle has been modified and reformulated on several occasions, and gained widespread popularity in 1873 thanks to Ludwig Boltzmann and Max Planck.

The second law of thermodynamics states that the total entropy of an isolated system can only increase over time. Sadi Carnot, the first to formulate the second law of thermodynamics (Source: MacTutor History of Mathematics archive)

While the first law of thermal dynamics specifies that energy can be exchanged between physical systems as heat and work. The second law introduces another quantity, known as entropy.

It is a principle of change and evolution since it determines in which direction potential energy transformations are possible.

Therefore, some chemical transformations are possible while others will never be. You can state with certainty, for example, that if you put an ice cube in a cup of hot coffee, the ice cube will melt, while the coffee will never freeze.

The Schrödinger Equation

The Schrödinger equation, conceived by the Austrian physicist Erwin Schrödinger in 1925, is a fundamental equation in quantum mechanics.

A mathematical equation that describes the changes over time of a physical system downloaded Time-dependent Schrödinger equation (Source: wikipedia.org)

As Einstein’s theory of general relativity helped explain the universe on a large scale, this equation sheds light on the behaviour of atoms and subatomic particles.

The Schrödinger equation explains the changes over time of a particle. It describes the states of a particle, from which it is possible to describe any state.

This equation poses a real philosophical question: Is matter made up of the presence of possible physical states (solids, liquids, gases)?

Schrodinger's discoveries formed the basis of wave mechanics Erwin Schrödinger, the Nobel Prize-winning Austrian physicist (Source: commons.wikimedia.org)

The application of this equation can be found in modern technology including nuclear energy, solid-state computers and lasers.

As we can see, throughout human history and especially since the 18th century, mathematical equations have transformed our understanding of the world in which we live and our ability to solve problems. They serve us every day in our daily lives, in maths lessons or in more or less direct means.

What will be the next major mathematical innovation? What new mathematical revelation will overturn our current conceptions of life as know it?

Superprof remains on standby, and will be the first to let you know!


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