# data[].attributes.attachment_link.readable_text
Newton discovered the relationship between the motion of the Moon and the motion of a body falling freely on Earth. By his dynamical and gravitational theories, he explained Kepler's laws and established the modern quantitative science of gravitation. Newton assumed the existence of an attractive force between all massive bodies, one that does not require bodily contact and that acts at a distance. By invoking his law of inertia (bodies not acted upon by a force move at constant speed in a straight line), Newton concluded that a force exerted by Earth on the Moon is needed to keep it in a circular motion about Earth rather than moving in a straight line. He realized that this force could be, at long range, the same as the force with which Earth pulls objects on its surface downward. When Newton discovered that the acceleration of the Moon is 1/3,600 smaller than the acceleration at the surface of Earth, he related the number 3,600 to the square of the radius of Earth. He calculated that the circular orbital motion of radius R and period T requires a constant inward acceleration A equal to the product of 4π ² and the ratio of the radius to the square of the time:

The Moon's orbit has a radius of about 384,000 km (239,000 miles; approximately 60 Earth radii), and its period is 27.3 days (its synodic period, or period measured in terms of lunar phases, is about 29.5 days). Newton found the Moon's inward acceleration in its orbit to be 0.0027 metre per second per second, the same as (1/60) ² of the acceleration of a falling object at the surface of Earth.

In Newton's theory every least particle of matter attracts every other particle gravitationally, and on that basis he showed that the attraction of a finite body with spherical symmetry is the same as that of the whole mass at the centre of the body. More generally, the attraction of any body at a sufficiently great distance is equal to that of the whole mass at the centre of mass. He could thus relate the two accelerations, that of the Moon and that of a body falling freely on Earth, to a common interaction, a gravitational force between bodies that diminishes as the inverse square of the distance between them. Thus, if the distance between the bodies is doubled, the force on them is reduced to a fourth of the original.

Newton saw that the gravitational force between bodies must depend on the masses of the bodies. Since a body of mass M experiencing a force F accelerates at a rate F/ M, a force of gravity proportional to M would be consistent with Galileo's observation that all bodies accelerate under gravity toward Earth at the same rate, a fact that Newton also tested experimentally. In Newton's equation F ₁₂ is the magnitude of the gravitational force acting between masses M ₁ and M ₂ separated by distance r ₁₂. The force equals the product of these masses and of G, a universal constant, divided by the square of the distance.

The constant G is a quantity with the physical dimensions (length) ³/(mass)(time) ²; its numerical value depends on the physical units of length, mass, and time used. ( G is discussed more fully in subsequent sections.)

The force acts in the direction of the line joining the two bodies and so is represented naturally as a vector, F. If r is the vector separation of the bodies, then In this expression the factor r/ r ³ acts in the direction of r and is numerically equal to 1/ r ².

The attractive force of a number of bodies of masses M ₁ on a body of mass M is where Σ ₁ means that the forces because of all the attracting bodies must be added together vectorially. This is Newton's gravitational law essentially in its original form. A simpler expression, equation (5), gives the surface acceleration on Earth. Setting a mass equal to Earth's mass M E and the distance equal to Earth's radius r E, the downward acceleration of a body at the surface g is equal to the product of the universal gravitational constant and the mass of Earth divided by the square of the radius:

The weight W of a body can be measured by the equal and opposite force necessary to prevent the downward acceleration; that is M g. The same body placed on the surface of the Moon has the same mass, but, as the Moon has a mass of about 1/81 times that of Earth and a radius of just 0.27 that of Earth, the body on the lunar surface has a weight of only 1/6 its Earth weight, as the Apollo program astronauts demonstrated. Passengers and instruments in orbiting satellites are in free fall. They experience weightless conditions even though their masses remain the same as on Earth.

Equations ( 1) and ( 2) can be used to derive Kepler's third law for the case of circular planetary orbits. By using the expression for the acceleration A in equation (1) for the force of gravity for the planet G M P M S/ R ² divided by the planet's mass M P, the following equation, in which M S is the mass of the Sun, is obtained:

Kepler's very important second law depends only on the fact that the force between two bodies is along the line joining them.

Newton was thus able to show that all three of Kepler's observationally derived laws follow mathematically from the assumption of his own laws of motion and gravity. In all observations of the motion of a celestial body, only the product of G and the mass can be found. Newton first estimated the magnitude of G by assuming Earth's average mass density to be about 5.5 times that of water (somewhat greater than Earth's surface rock density) and by calculating Earth's mass from this. Then, taking M E and r E as Earth's mass and radius, respectively, the value of G was which numerically comes close to the accepted value of 6.6743 × 10 −11 m ³ s −2 kg −1, first directly measured by Henry Cavendish.

Comparing equation ( 5) for Earth's surface acceleration g with the R ³/ T ² ratio for the planets, a formula for the ratio of the Sun's mass M S to Earth's mass M E was obtained in terms of known quantities, R E being the radius of Earth's orbit:

The motions of the moons of Jupiter (discovered by Galileo) around Jupiter obey Kepler's laws just as the planets do around the Sun. Thus, Newton calculated that Jupiter, with a radius 11 times larger than Earth's, was 318 times more massive than Earth but only 1/4 as dense.

# data[].attributes.attachment_link.title
Gravity - Newton's Law, Universal Force, Mass Attraction

# data[].attributes.attachment_link.description
Gravity - Newton's Law, Universal Force, Mass Attraction: Newton discovered the relationship between the motion of the Moon and the motion of a body falling freely on Earth. By his dynamical and gravitational theories, he explained Kepler's laws and established the modern quantitative science of gravitation.

# data[].attributes.attachment_link.readable_text
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 published a book entitled "The 17 Equations That Changed the World" (Ed. Robert Laffont).

How old are Maths Formulas?

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.

10 Famous Maths Equations

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 an online math tutor!

Pythagoras' Theorem

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

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.

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

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.

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.

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

There are 3 types of logarithm:

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.

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.

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.

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 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. 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.

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 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 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.

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.

The Maxwell-Gauss equation

The Maxwell-Thomson equation

The Maxwell-Faraday equation

The Maxwell-Ampere Equation

There are 4 forms of Maxwell's equations:

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.

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.

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)?

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.

Prizes And Awards For Maths

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.
The Fields Medal
Below are some of these awards.

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.

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

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

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.

Mathematicians Who Changed The World

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

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.
Caroline Herschel
Lear more abour Sophie Germain with an online math tutor.

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

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.
Sofia Kovalevskaya
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.

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.

Other Influential Female Mathematicians

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

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

Taking Mathematics Into The Future

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.

# data[].attributes.attachment_link.title
10 Maths Equations And Formulas that Changed the World

# data[].attributes.attachment_link.description
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!

# data[].attributes.attachment_link.readable_text
The Fibonacci sequence is one of the most famous formulas in mathematics.

Each number in the sequence is the sum of the two numbers that precede it. So, the sequence goes: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, and so on. The mathematical equation describing it is Xn+2= Xn+1 + Xn

A mainstay of high-school and undergraduate classes, it's been called "nature's secret code," and "nature's universal rule." It is said to govern the dimensions of everything from the Great Pyramid at Giza, to the iconic seashell that likely graced the cover of your school math textbook.

And odds are, almost everything you know about it is wrong.

Scattered history

So then, what's the real story behind this famous sequence?

Many sources claim it was first discovered or "invented" by Leonardo Fibonacci. The Italian mathematician, who was born around A.D. 1170, was originally known as Leonardo of Pisa, said Keith Devlin, a mathematician at Stanford University. Only in the 19th century did historians come up with the nickname Fibonacci (roughly meaning, "son of the Bonacci clan"), to distinguish the mathematician from another famous Leonardo of Pisa, Devlin said. [ Large Numbers that Define the Universe]

But Leonardo of Pisa did not actually discover the sequence, said Devlin, who is also the author of "Finding Fibonacci: The Quest to Rediscover the Forgotten Mathematical Genius Who Changed the World," (Princeton University Press, 2017). Ancient Sanskrit texts that used the Hindu-Arabic numeral system first mention it, and those predate Leonardo of Pisa by centuries.

"It's been around forever," Devlin told Live Science.

However, in 1202 Leonardo of Pisa published the massive tome "Liber Abaci," a mathematics "cookbook for how to do calculations," Devlin said. Written for tradesmen, "Liber Abaci" laid out Hindu-Arabic arithmetic useful for tracking profits, losses, remaining loan balances and so on, Devlin said.

In one place in the book, Leonardo of Pisa introduces the sequence with a problem involving rabbits. The problem goes as follows: Start with a male and a female rabbit. After a month, they mature and produce a litter with another male and female rabbit. A month later, those rabbits reproduce and out comes - you guessed it - another male and female, who also can mate after a month. (Ignore the wildly improbable biology here.) After a year, how many rabbits would you have? The answer, it turns out, is 144 - and the formula used to get to that answer is what's now known as the Fibonacci sequence. [ The 11 Most Beautiful Mathematical Equations]

"Liber Abaci" first introduced the sequence to the Western world. But after a few scant paragraphs on breeding rabbits, Leonardo of Pisa never mentioned the sequence again. In fact, it was mostly forgotten until the 19th century, when mathematicians worked out more about the sequence's mathematical properties. In 1877, French mathematician Édouard Lucas officially named the rabbit problem "the Fibonacci sequence," Devlin said.

(Image credit: Shutterstock)

Imaginary meaning

But what exactly is the significance of the Fibonacci sequence? Other than being a neat teaching tool, it shows up in a few places in nature. However, it's not some secret code that governs the architecture of the universe, Devlin said.

It's true that the Fibonacci sequence is tightly connected to what's now known as the golden ratio (which is not even a true ratio because it's an irrational number). Simply put, the ratio of the numbers in the sequence, as the sequence goes to infinity, approaches the golden ratio, which is 1.6180339887498948482... From there, mathematicians can calculate what's called the golden spiral, or a logarithmic spiral whose growth factor equals the golden ratio. [ The 9 Most Massive Numbers in Existence]

The golden ratio does seem to capture some types of plant growth, Devlin said. For instance, the spiral arrangement of leaves or petals on some plants follows the golden ratio. Pinecones exhibit a golden spiral, as do the seeds in a sunflower, according to "Phyllotaxis: A Systemic Study in Plant Morphogenesis" (Cambridge University Press, 1994). But there are just as many plants that do not follow this rule.

"It's not 'God's only rule' for growing things, let's put it that way," Devlin said.

And perhaps the most famous example of all, the seashell known as the nautilus, does not in fact grow new cells according to the Fibonacci sequence, he said.

When people start to draw connections to the human body, art and architecture, links to the Fibonacci sequence go from tenuous to downright fictional.

"It would take a large book to document all the misinformation about the golden ratio, much of which is simply the repetition of the same errors by different authors," George Markowsky, a mathematician who was then at the University of Maine, wrote in a 1992 paper in the College Mathematics Journal.

Much of this misinformation can be attributed to an 1855 book by the German psychologist Adolf Zeising. Zeising claimed the proportions of the human body were based on the golden ratio. The golden ratio sprouted "golden rectangles," "golden triangles" and all sorts of theories about where these iconic dimensions crop up. Since then, people have said the golden ratio can be found in the dimensions of the Pyramid at Giza, the Parthenon, Leonardo da Vinci's "Vitruvian Man" and a bevy of Renaissance buildings. Overarching claims about the ratio being "uniquely pleasing" to the human eye have been stated uncritically, Devlin said.

All these claims, when they're tested, are measurably false, Devlin said.

"We're good pattern recognizers. We can see a pattern regardless of whether it's there or not," Devlin said. "It's all just wishful thinking."

# data[].attributes.attachment_link.description
The Fibonacci sequence is one of the most famous formulas in mathematics. Each number in the sequence is the sum of the two numbers that precede it. So, the sequence goes: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, and so on.

# data[].attributes.attachment_link.readable_text
Right Triangles - Pythagorean Theorem

The Pythagorean theorem was first known in ancient Babylon and Egypt (beginning about 1900 B.C.). The relationship was shown on a 4000 year old Babylonian tablet now known as Plimpton 322. However, the relationship was not widely publicized until Pythagoras stated it explicitly.
Pythagorean

Pythagoras lived during the 6th century B.C. on the island of Samos in the Aegean Sea. He also lived in Egypt, Babylon and southern Italy. Pythagoras was a teacher and a philosopher.

Pythagoras found out that for a right angle triangle (with one of the angles being 90 o), the square of the hypotenuse is equal to the sum of the squares of the other two sides: a²+b²=c².

In other words, the green square's area (with area c ²) equals the sum of two others. It is exactly the yellow square's area (a ²) plus the blue square's (b ²).

Now take a look at the tile pattern.

[The proof of Pythagorean Theorem is in the following figure.]

Count the triangles within the squares. The number of green triangles equals the total of those that are yellow and blue!

Do You Know ...

Archeologists use the Pythagorean in field excavations. When they start a dig, they place a rectangular grid over the site surface. To lay out an accurate grid system, archeologists use the Theorem, X 2+Y ²=Z ². After deciding how long the baselines (X- and Y- axis) should be, the proper length of the diagonal is calculated using the Pythagorean theorem to make sure the quadrant is a rectangle and not a lopsided parallelogram. Corner stakes will then be placed to mark the accurate location of the site.

An overlaying rectangular grid uses a mathematical idea called Cartesian coordinate system.

# data[].attributes.attachment_link.description
Right Triangles - Pythagorean Theorem The Pythagorean theorem was first known in ancient Babylon and Egypt (beginning about 1900 B.C.). The relationship was shown on a 4000 year old Babylonian tablet now known as Plimpton 322. However, the relationship was not widely publicized until Pythagoras stated it explicitly.

# data[].attributes.attachment_link.readable_text
Pythagoras

Over 2000 years ago there was an amazing discovery about triangles:
When a triangle has a right angle (90°) ...... and squares are made on each of the three sides, ...... then the biggest square has the exact same area as the other two squares put together!

It is called "Pythagoras' Theorem" and can be written in one short equation:

Note:

Definition

The longest side of the triangle is called the "hypotenuse", so the formal definition is:

In a right angled triangle:
the square of the hypotenuse is equal to
the sum of the squares of the other two sides.

Sure ... ?

Let's see if it really works using an example.

Let's check if the areas are the same:

3 ² + 4 ² = 5 ²

Calculating this becomes:

9 + 16 = 25
It works ... like Magic!

Why Is This Useful?

If we know the lengths of two sides of a right angled triangle, we can find the length of the third side. (But remember it only works on right angled triangles!)

How Do I Use it?

Write it down as an equation:

a ² + b ² = c ²

Then we use algebra to find any missing value, as in these examples:

Example: Solve this triangle

Read Builder's Mathematics to see practical uses for this.

Also read about Squares and Square Roots to find out why √169 = 13

Example: Solve this triangle.

Example: What is the diagonal distance across a square of size 1?

It works the other way around, too: when the three sides of a triangle make , then the triangle is right angled.

Example: Does this triangle have a Right Angle?

They are equal, so ...

Yes, it does have a Right Angle!

Example: Does an 8, 15, 16 triangle have a Right Angle?

So, NO, it does not have a Right Angle

Example: Does this triangle have a Right Angle?

Yes, it does!

So this is a right-angled triangle

And You Can Prove The Theorem Yourself !

Get paper pen and scissors, then using the following animation as a guide:

Draw a right angled triangle on the paper, leaving plenty of space.

Draw a square along the hypotenuse (the longest side)

Draw the same sized square on the other side of the hypotenuse

Draw lines as shown on the animation, like this:

Cut out the shapes

Arrange them so that you can prove that the big square has the same area as the two squares on the other sides

Another, Amazingly Simple, Proof

Here is one of the oldest proofs that the square on the long side has the same area as the other squares.

Watch the animation, and pay attention when the triangles start sliding around.

You may want to watch the animation a few times to understand what is happening.

The purple triangle is the important one.

becomes

We also have a proof by adding up the areas.

Historical Note: while we call it Pythagoras' Theorem, it was also known by Indian, Greek, Chinese and Babylonian mathematicians well before he lived.

511,512,617,618, 1145, 1146, 1147, 2359, 2360, 2361

# data[].attributes.attachment_link.description
Pythagoras Over 2000 years ago there was an amazing discovery about triangles: When a triangle has a right angle (90°) ... ... and squares are made on each of the three sides, ... ... then the biggest square has the exact same area as the other two squares put together!