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Contents

- 10 Famous Maths Equations
- Pythagoras’ Theorem
- Logarithms
- The Law of Gravity
- The Theory of Relativity
- Chaos Theory
- Euler’s Identity
- The Fourier Transform
- Maxwell’s Equations
- The Second Law of Thermodynamics
- The Schrödinger Equation
- Prizes And Awards For Maths
- Mathematicians Who Changed The World
- Taking Mathematics Into The Future

Mathematics is always around us**, everywhere we go.**

And this is because Maths is so broad, covering addition and subtraction, dividing, fractions, graphs, rounding, solving equations, algebra, inequalities, variables, statistics, substitution, trig, computation, distributive property, symmetry, whole numbers, prime numbers, probability, value, vectors, shapes, sequence, proportion and more!

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, complex and basic math is truly everywhere.

There is not an object in existence which is not somehow the outcome of **maths in action**, which you will learn during school math and beyond that in further education maths lessons.

Complex equations with many unknowns, radical 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 publisheda book entitled “The 17 Equations That Changed the World” (Ed. Robert Laffont).

Since maths is **a broad application of matter**, rather than a discovery, we cannot credit one person with the invention of maths itself (unless you want to be really deep and say that the creator of our universe is responsible for the birth of maths!). However, we can take a look back at **when maths started** to play a role in the life of humans.

Unsurprisingly, evidence shows that even those living in prehistoric times had **some understanding of maths concepts**, records of which were found on many items, like bones, and wall carvings.

Markings would have shown that they used rational thinking when learning **how to solve simple math problems** like adding things up on a surface area. So what would have driven them to take an interest in mathematics in the first place? The passing of time, for instance, would have intrigued them. Working out how long they had left to hunt for food before sunset, maybe, which they would have looked at astronomy to answer.

The Star Garden states that:

“The Ishango bone is about 20,000 years old and has a series of notches carved into it in three columns. Patterns in these numbers may show that they were made by someone who understood addition, subtraction, multiplication, division, and prime numbers.”

They go on to say that:

“People understood geometry and algebra by about 2000 BCE […] Around this time, both the Babylonians and Ancient Egyptians were aware of the number π (pi) the ratio of a circle’s circumference to its diameter. By about 1500 BCE, the Babylonians were also aware of Pythagoras’ theorem, which shows how the lengths of the sides of right-angled triangles are related.”

The theorem is named after ancient Greek mathematician Pythagoras (though some say the concept predates him) which shows that, although all maths theorems and formulae simply exist and are waiting to be discovered, we can at least praise some individuals for find them or working them out.

In their brief history of Maths, The Star Garden website adds that: “Kepler was also inspired by Pythagoras, and believed that the motion of the planets produces music. He used mathematics to show that the planets orbit the Sun in ellipses and, by 1619, he was able to determine the time it takes each planet to orbit and their relative distances from the Sun.

In 1687, Newton published his law of universal gravitation. This was groundbreaking because it showed, not just that abstract mathematical principles, such as the newly invented calculus, could be applied to what we observe in nature, but that the laws responsible for the movement of the planets are also responsible for the movement of objects on Earth. Newton also believed that the universe could be understood as a mathematical object, and described God as “skilled in mechanics and geometry”.

Newton’s contemporary, Leibniz, discovered another link between mathematics and nature when he first considered the idea of fractals. […] Twentieth-century mathematicians, such as French mathematician Gaston Julia and Polish-French-American mathematician Benoit Mandelbrot, were inspired by Leibniz to create complicated fractals of their own.

By this time, quantum mechanics, and German-Swiss-American physicist Albert Einstein’s theories of special and general relativity, had shown that nature obeys the laws of mathematics, even when this contradicts our common sense understanding of the world.”

So, as we can see, Maths has been ever-present throughout the history of Man, yet a number of** significant breakthroughs** have emerged thanks to skilled mathematicians who have come to find them. We will look at some of the most famous maths equations below.

As a final note on the history of maths, it is important to note that, despite humans not developing with the use of mathematical concepts, maths has always played a part on the planet. Even before the existence of people, maths would have dictated all nature, plus the energy sources and animals that graced Earth, thanks to its necessity in reproduction and life in general.

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. There is, of course, no end to the number of mathematical formulas and expressions that exist (some might say the list is infinite!), but here we focus on some of the better known algebraic equation and prvide some helpful notation.

Don’t forget, Superprof can help you find the perfect math tutor if you become inspired to get math help using a math teacher or online math help!

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

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 geometryto non-Euclidean geometry.

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 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**.

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 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**.

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.

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**.

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.

Even today,

Einstein’s theory of relativityremains 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 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**. A great topic to learn maths.

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.

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

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 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 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.

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 (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.

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

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)?

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 maths problems. They serve us every day in our daily lives, in maths lessons or in more or less direct means.

F**ormulas and equations you may learn about in algebra lessons** are quadratic formula, simultaneous equations, differential equation, multi step, two step, and one step equations, polynomial equation, linear equations, exponential function, as well as systems of equations.

There are a number of integral awards given to people, often nicknamed geniuses, who **excel in various areas of mathematics**, usually offering a solution to a math problem. These are prestigious awards with absolute value and worth, some of which are even thought to be equivalent to winning the Nobel Prize. As such, only a very select few receive these awards highlighting their mathematical excellence.

Below are some of these awards.

*The Fields Medal*

The Fields Medal is **one of the most famous prizes given to mathematicians** who have achieved something amazing during their career working with numbers, equations or more, such as discovering a significant theory or concept.

Officially, winners of this prize are awarded the International Medal for Outstanding Discoveries in Mathematics (you can see why this is more commonly referred to as simply the Fields Medal) and it is only given out once every four years to up to four mathematicians below 40 years of age. This means that most winners of this prestigious award can be classed as young, up and coming mathematicians with much promise for the future.

*Abel Prize*

Yet another reputable nomination, the Abel Prize is presented by the King of Norway to **a mathematician who is outstanding in their field of mathematic study**.

It is named after Niels Henrik Abel who, back in 2001 when it was first created, was a popular Norwegian mathematician.

*Wolf Prize in Mathematics*

The Wolf Foundation of Israel awards six different prizes each year, one of which being the Wolf Prize in Mathematics.

This award has been in existence since 1978 and is seen as a great honour to the nominee. Some well-known names who have won this award include Andrew Wiles, John Milnor and others.

*Chern Medal*

One of the** newer prizes in mathematics** is the Chern Meda, which has been recognising lifetime achievements for maths since 2010.

It is awarded every four years, so only a handful of mathematicians have got their hands on one of these prestigious awards so far. It is given out at the International Congress of Mathematicians and it **includes a monetary prize** of $250,000 (evidently to help fund further research or to enhance tuition in other fields of mathematics).

The first recipient in 2010 was Louis Nirenberg and the 2014 winner was Phillip Griffiths.

So why is there **no Nobel prize for mathematicians**?

In the world of math facts, some would say that the reason for this is because Alfred Nobel’s wife had an affair with a famous mathematician, yet we now know this not to be true as Nobel didn’t even marry during his lifetime. Put simply, his work involving scientific matters meant that he had those subjects closer to his heart than mathematical expression.

If, like us, you are wondering why there has been no mention of any **female mathematicians** across the history of maths, then this section is for you! Talk about equations and inequalities…!

Despite many **women playing a part in mathematical discoveries**, just like their male peers, their ability to solve problems was rarely reported on. Below are just some of the women who have **helped shaped mathematics over time**.

*Hypatia*

Hypatia was the daughter of Greek mathematician Theon and became head of the Platonist School in Alexandria, Egypt, where she taught astronomy and philosophy.

This highly intellectual woman was murdered in 415 AD, labelled a Satanist by religious groups, however, it’s believed that she was **an influential genius** who contributed vastly to her famous father’s published texts during the time she had.

*Sophie Germain*

Marie-Sophie Germain was said to be** obsessed by number theories and calculus**, inspired by reading about Archimedes.

This was the 18th century, and some institutions would not allow women to study under their roof. Therefore, Germain assumed the identity of a male student so she could study at a male-only maths academy in Paris.

Sadly, though some are aware of her brilliant work on Fermat’s Last Theorem, it was never officially recognised and she died known only as a single, profession-less woman.

Lear more abour Sophie Germain with an online maths tutor.

*Caroline Herschel*

Caroline Herschel was the first woman to receive the Royal Astronomical Society’s gold medal in 1828, after she famously discovered seven new comets.

Herschel became the paid assistant of her astronomer brother William and the pair went on to discover the planet Uranus in 1781, before making **numerous significant discoveries** of her own before her death aged 97.

*Ada Lovelace*

With a famous father, poet Lord Byron, it is no wonder that Lovelace was nicknamed an Enchantress! Yet, unlike her dad, she was far more talented when asked to coordinate a number system as opposed to any word problems.

Lovelace, by profession, was a Victorian **computer pioneer** who collaborated with Charles Babbage on the first programmable computers in the mid-19th century. Even now, one can see that her ideas were so far ahead of her time, which could be why she wasn’t recognised for her input to the field of mathematics until the 1950s – more than 100 years after she was carrying out the work.

*Sofia Kovalevskaya*

Last but not least, Kovalevskaya, who was born in Moscow in 1850, made some **very important contributions to mathematical analysis** in Germany.

With her ability being noticed by her uncle in the first instance, she was put through private lessons, however, was later forced into marriage.

She finally defeated the inequality and divide between men and women and became the first woman to gain a professorship in northern Europe.

Take a look below at some more women who have been historically involved in the field of maths, and why.

Name | Mathematical idenity |
---|---|

Emmy Noether | Innovator in higher algebra |

Florence Nightingale | Inventor of pictorial ways to display statistics |

Joan Clarke | Famous code-breaker |

Dame Jocelyn Bell Burnell | Astronomy pioneer |

Radia Perlman | Leader in the field of computer science |

What will be **the next major mathematical innovation**?

What new mathematical revelation will overturn our current conceptions of life as know it? Could you become a renowned mathematician for a quadratic equation, an algebraic function, a trigonometric breakthrough or even for discovering a new concept like the Pythagorean theorem?

Alternatively, maybe you will **bring something totally new to the table** by discovering a new integer we never knew existed, developing an intellectual graphing calculator or by simplifying maths with an equation solver device or a system of equations app.

If any of these are to be true, you must first ace your GCSE math tutorials and more advanced topics like college algebra. You can do this if you place value on your math lessons and textbooks, listen to your teacher’s explanation, do the worksheets to your absolute best and complete your homework assignments even if it means seeking help from a free math tutoring website or app.

Find a tutor for maths revision GCSE or a general maths tutor with Superprof.

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