No list of mathematical achievements would be complete without the inclusion of the most seminal and influential mathematical work to come out of Greek antiquity.

Written around 300BC, Euclid's work built the foundation for modern mathematics by introducing a set of axioms and proceeding to demonstrate by mathematical rigor a collection of theorems that naturally followed.

Covering subjects ranging from algebra to plane geometry (also now known as Euclidean Geometry), Elements remained a cornerstone of mathematical teaching for over 2,000 years following its creation.

http://www.policymic.com/articles/29778/pi-day-5-greatest-mathematical-discoveries-in-history

Yes, if we see Applied Mathematics, but if we focus on Theoretical Mathematics, then I think that Gödel's incompleteness theorems are the 'discovery' that put, for a first time in history of Maths, a limit in what we can create with them:

http://en.wikipedia.org/wiki/G%C3%B6del's_incompleteness_theorems

Euler's formula : e^(i*x) = cos(x) + i*sin(x)

(http://en.wikipedia.org/wiki/Euler%27s_formula) - 1722

[Euler's identity : e^(i*pi) + 1 = 0]

(http://en.wikipedia.org/wiki/Euler%27s_identity) - 1740

I think that the more general theorem of Weierstrass is more important then the Fourier Series.

Actually, Fourier Series (FS) mentioned in this question is the entry point into the world of Fourier analysis (FA) and its wide-ranging extensions and generalizations such as Fourier Transform (FT), Discrete Fourier Transform (DFT) leading to the Fast Fourier Transform (FFT), wavelets, etc.

An entire industry is devoted to further developing the theory and enlarging the scope of applications of Fourier–inspired methods. New directions in FA continue to be discovered and exploited in a broad range of physical, mathematical, engineering, chemical, biological, financial, and other systems.

Instead of answer my dear @Mahmoud! Read Laplace's words on ".... expressing all numbers by means of ten symbols, each symbol receiving a value of position as well as an absolute value..."!

Set theory – G.Cantor (http://en.wikipedia.org/wiki/Set_theory)

1926 - «No one shall expel us from the Paradise that Cantor has created» - David Hilbert (http://en.wikiquote.org/wiki/David_Hilbert)

1910 - “Problem solving, long mystery enshrouding mathematical infinity, is probably the greatest achievement that should be proud of our age" - Bertrand Russell (http://en.wikipedia.org/wiki/Bertrand_Russell).

Leopold Kronecker, considered Cantor a scientific charlatan (http://en.wikipedia.org/wiki/Leopold_Kronecker).

How many emotions regarding the Set theory !

Dear Demetris, Sylantyev, Ljubomir and Abedallah,

I meant the most widely applicable mathematical discoveries. With due respect your examples are not widely applicable.

Let me elaborate further. An important application of DFT is in signal and image processing. For example, telecommunication, automotive industry, computer industry, IT and modern digital media such as CD’s, DVD’s and MP3’s, and medical devices such as MRI, CT, ultrasound's and FTIR, are all based on discrete data, not continuous functions. Digital world is benefited from FA and FFT most.

@Mahmoud, do not forget another field of use of Discrete Fourier Transform for Control Design, Robust Control Design....

Dear @Ljubomir You are right. And there are many more! The really cool thing about fourier series is that first, almost any kind of a wave can be approximated.

In fact, Fourier Analysis has many scientific and engineering applications – in physics, partial differential equations, number theory, combinatorics, approximation theory, signal processing, image processing, control theory, probability theory, statistics, option pricing, cryptography, numerical analysis, acoustics, oceanography, sonar, optics, diffraction, geometry, protein structure analysis and many many other areas.

Dears @Ljubomir and @Mahmoud, As a mathematician, I always give more weight for the theory. The theorem of Weierstrass enables the formal study and development of Fourier, Taylor, Hermite, Newton, and Bernstein series. The Fourier series is the trigonometric version of the Weierstrass theorem.

Dear Abedallah

What concrete theorem of Weierstrass you mean?

In (http://en.wikipedia.org/wiki/Weierstrass_theorem) there exists 8 theorem of Weierstrass.

Dear @Sylantev, You are right: several theorems are named after Karl Weierstrass. The one I mean is the Weierstrass approximation theorem which states that every continuous function defined on a closed interval [a, b] can be uniformly approximated as closely as desired by a polynomial function; this theorem has many proofs, the most practical one is the Bernstein proof.

http://en.wikipedia.org/wiki/Weierstrass_approximation_theorem

I agree with @Kamal in that the invention of 0 is one of the strong candidates for being the most important mathematical discovery.

Yes, the number zero!

No list of mathematical achievements would be complete without the inclusion of the most seminal and influential mathematical work to come out of Greek antiquity.

Written around 300BC, Euclid's work built the foundation for modern mathematics by introducing a set of axioms and proceeding to demonstrate by mathematical rigor a collection of theorems that naturally followed.

Covering subjects ranging from algebra to plane geometry (also now known as Euclidean Geometry), Elements remained a cornerstone of mathematical teaching for over 2,000 years following its creation.

http://www.policymic.com/articles/29778/pi-day-5-greatest-mathematical-discoveries-in-history

I still believe the rapid advancement of Hi-Tech and digital world owe to the Fourier discovery.

It is a tedious task to think of writing a list of the most important mathematical discovery; it can be easier to take an era or a time period and study the mathematical discoveries at that time and thereafter making a list for voting.

Dear Kamal. You are right. However my question (what is the single most important discovery in all of mathematics?) was specific not general. ... the single most important ...

Yes dear Kamal. As a rule of tumb all in RG are open minded to learn and gain experience. Nevertheless, I expected more participation and feedback on this and similar questions from RG members. But it seems Pareto Principle at work in RG.

Paul Glendinning describes in his book 'Maths in minutes' (Quercus Publishing Plc 2012) 200 key concepts in Mathematics. He starts with the concept 'numbers'. Zero is introduced as concept '4', the theorem of Kurt Gödel is introduced as concept '32', and Fourier series are described as concept '112'. Do you know concepts in mathematics where the concept of 'numbers' is assumed not to exist?

Thank you dear Marcel for introducing the book 'Maths in minutes' by Glendinning. . As you pointed out 'numbers' are everything in mathematics. In fact, everyone use and manipulate numbers all the time. Some of these 200 key concepts have changed our life more. Fourier Series is one such concept even it is '112' step further from 'numbers' in Glendinning's list.

Dear Mahmoud,

I understand that there is significant evolution in mathematical thinking, but without the invention of the concept 'number' the subsequent evolutions in mathematics would not have been possible. Numbers exist without Fourier Series, not the opposite. Therefore I think that the invention of what people call 'Number' is the most important mathematical discovery.

Just to stimulate discussion:

Perhaps the importance of the concepts are scale dependent? Can Fourier Series resolve all problems related to the understanding of the functioning and organisation of the world? For instance, if we accept that each spatiotemporal moment is unique in physical expression, can mathematically defined Fourier Series truly exist if all physical details of each spatiotemporal point are taken into account?

Dear Marcel Thank you for your comments.

Honestly i could not understand your question about 'scale' at first and search the web! Also, when I looked at your profile in RG I realized you are ecologist and scale plays the central role in determining the outcome of observations. Acts are played out on various scales of space and time (Hutchinson (1965) called it the “ecological theatre” ).

According to Levin (1992) "The problem of pattern and scale is the central problem in ecology, unifying population biology and ecosystems science, and marrying basic and applied ecology. Applied challenges ... require the interfacing of phenomena that occur on very different scales of space, time, and ecological organization. Furthermore, there is no single natural scale at which ecological phenomena should be studied; systems generally show characteristic variability on a range of spatial, temporal, and organizational scales."

Surprisingly, Fourier theory is relevant here. when you want to deal with space-time phenomenon Wavelet, STFA (Short-time Fourier analysis) (both have roots in Fourier Analysis) can be used!

The spatial distributions of krill populations of the Southern Ocean have been shown to be patchy on almost every scale of description (Levin, 1992). This has fundamental importance both for the dynamics of krill, and for their predator species. Various studies have characterized the Fourier spectrum of variability of krill, and shown that variance decreases with scale. Yet, no single mechanism explains pattern on all scales.

Dear Mahmoud,

what I wanted to say is that Maths always gives a simplified vision of the world's physical/chemical/biological complexity. There will always be at least a (tiny) difference between the theoretical Maths presentation of a phenomenon and the true physical presentation of the same phenomenon.

Example that I already exposed before:

The simplest mathematical procedure is the addition of two phenomena of which we assume they share characteristics. 1 + 1 = 2. If we apply this to natural phenomena a scientist might claim: one tree (t1) plus another tree (t2) = 2 trees (t1 + t2). This is often done in statistical analyses that define classes of phenomena that share characteristics. The physical reality is that the two trees will always differ in physics in at least one scale of analysis or perception. The physics that is shared among the trees is human-perception defined accepting imprecisions in the definition of what a tree physically truly is (e.g. each tree is unique in physical expression even for characteristics of which we assume perceived physical traits are shared).

The question then is: Is this difference between theory and practice scientifically or philosophically speaking important or not?

Dear Marcel. Your remarks sound philosophical.

Albert Einstein said: “We can't solve problems by using the same kind of thinking we used when we created them.”

Dear Mahmoud,

Great citation!

There are a lot of important results in mathematics. The importance of a results is judge

on the power of the result to unify or make unnecessary previous results and theories. Since the question is asked for just one, I would single out one event in the history of mathematics: The introduction of Category Theory. This change the whole fielf of mathematics and unify all previous mathematics. As an example I can note that Category Theory is behind of what we usually call "real mathematics": Algebraic Topology and Algebraic Geometry.

It is Zero by Arya Bhatta. Thanks

Physicist Lord Kelvin remarked in 1867: “Fourier’s theorem is not only one of the most beautiful results of modern analysis, but it may be said to furnish an indispensable instrument in the treatment of nearly every recondite question in modern physics.”

https://www.researchgate.net/post/What_is_the_most_beautiful_theorem_in_mathematics

Maybe, the most important discovery in mathematics was the introduction by Eilemberg and MacLane of the Category Theory, and then the work of Grothendieck on Algebraic Geometry.

Dear Colleagues,

Good Day,

Please, see this ....

Richard Feynman (May 11, 1918 – February 15, 1988) was an American theoretical physicist known for his work in the path integral formulation of quantum mechanics, the theory of quantum electrodynamics, and the physics of the superfluidity of supercooled liquid helium, as well as in particle physics for which he proposed the parton model. For his contributions to the development of quantum electrodynamics, Feynman, jointly with Julian Schwinger and Sin-Itiro Tomonaga, received the Nobel Prize in Physics in 1965. He developed a widely used pictorial representation scheme for the mathematical expressions governing the behavior of subatomic particles, which later became known as Feynman diagrams. During his lifetime, Feynman became one of the best-known scientists in the world. In a 1999 poll of 130 leading physicists worldwide by the British journal Physics World he was ranked as one of the ten greatest physicists of all time) said that Euler's formula:

e^i*x=cos(x)+i*sin(x)

was the most useful formula in all of mathematics. He called it a "gem".

Here are some basic examples of Euler's formula in use:

One of my favorites is this: What is i^i ?

Using Euler's formula

i=cos(π/2)+i*sin(π/2)= e^i*(π/2)

then

i^i=(e^i*(π/2))^i= e^−π/2 .

Would you have thought that iiii is a real number? If anyone can make an intuitive argument for why i^i is a real number I'd like to understand why.

Dear Colleagues,

Good D.

ay,

"Pi Day: 5 Greatest Mathematical Discoveries in History,  By Ben Beder March 14, 2013.

list of the five greatest mathematical discoveries in history:

Please, see the link for more detail...

https://mic.com/articles/29778/pi-day-5-greatest-mathematical-discoveries-in-history#.7T1ixyrGT

Dear Hazim Thank you for your comments. These 5 discoveries you mentioned are indeed very influential.

Differential and Integral Calculus is the most important mathematical discovery. Its make the way to various disciplines and technologies development. It is really difficult to imagine the modern world without Calculus.

Analytical Geometry

"Analytic geometry was independently invented by René Descartes and Pierre de Fermat,[6][7] although Descartes is sometimes given sole credit.[8][9] Cartesian geometry, the alternative term used for analytic geometry, is named after Descartes.

Descartes made significant progress with the methods in an essay titled La Geometrie (Geometry), one of the three accompanying essays (appendices) published in 1637 together with hisDiscourse on the Method for Rightly Directing One's Reason and Searching for Truth in the Sciences, commonly referred to as Discourse on Method. This work, written in his native Frenchtongue, and its philosophical principles, provided a foundation for calculus in Europe. Initially the work was not well received, due, in part, to the many gaps in arguments and complicated equations. Only after the translation into Latin and the addition of commentary by van Schooten in 1649 (and further work thereafter) did Descartes's masterpiece receive due recognition.[10]

Pierre de Fermat also pioneered the development of analytic geometry. Although not published in his lifetime, a manuscript form of Ad locos planos et solidos isagoge (Introduction to Plane and Solid Loci) was circulating in Paris in 1637, just prior to the publication of Descartes' Discourse.[11][12][13] Clearly written and well received, the Introduction also laid the groundwork for analytical geometry. The key difference between Fermat's and Descartes' treatments is a matter of viewpoint: Fermat always started with an algebraic equation and then described the geometric curve which satisfied it, whereas Descartes started with geometric curves and produced their equations as one of several properties of the curves.[10] As a consequence of this approach, Descartes had to deal with more complicated equations and he had to develop the methods to work with polynomial equations of higher degree. It was Leonard Euler who first applied the coordinate method in a systematic study of space curves and surfaces." (see link)

https://en.wikipedia.org/wiki/Analytic_geometry

"Life is good for only two things, discovering mathematics and teaching mathematics." ~Siméon Poisson

"Mathematics is the Queen of Science, and Arithmetic the Queen of Mathematics." ~C. F. Gauss

I agree with the view that sees zero as the main mathematical discovery. However, it should not be ruled Greek discovery of pi, the imaginary number, the number of Euler, the calculus, the concept of joint and infinite Cantor, Gödel's proof of incompleteness, among many other discoveries that have marked the history of mathematics.

Dear Ricaedo is right. W. B. Smith once said "Calculus is the most powerful weapon of thought yet devised by the wit of man."

Good response @ all 

The Calculus of variation 

I like famous formula of EULER and discovery of non-Euclidean space by Carl Friedrich Gauss

Leonhard Euler (1707- 1783): most prolific mathematician of all time, publishing close to 900 books. When he went blind in his late 50s his productivity in many areas increased.

His famous formula eiπ + 1 = 0, where e is the mathematical constant sometimes known as Euler's number and i is the square root of minus one, is widely considered the most beautiful in mathematics. He later took an interest in Latin squares – grids where each row and column contains each member of a set of numbers or objects once. Without this work, we might not have had sudoku.

Carl Friedrich Gauss (1777-1855): His revolutionary discovery of non-Euclidean space (that it is mathematically consistent that parallel lines may diverge) was found in his notes after his death. During his analysis of astronomical data, he realised that measurement error produced a bell curve – and that shape is now known as a Gaussian distribution.

https://www.theguardian.com/culture/2010/apr/11/the-10-best-mathematicians

Augustus De Morgan: The moving power of mathematical invention is not reasoning, but imagination.

P.S.: He formulated De Morgan's laws and was the first to introduce the term, and make rigorous the idea of mathematical induction. De Morgan crater on the Moon is named after him.

https://en.wikiquote.org/wiki/Augustus_De_Morgan

As recorded by George Dantzig’s father Tobias Dantzig, the 19th century mathematician Pierre-Simon Laplace explained:

It is India that gave us the ingenious method of expressing all numbers by means of ten symbols, each symbol receiving a value of position as well as an absolute value; a profound and important idea which appears so simple to us now that we ignore its true merit. But its very simplicity and the great ease which it has lent to all computations put our arithmetic in the first rank of useful inventions; and we shall appreciate the grandeur of this achievement the more when we remember that it escaped the genius of Archimedes and Apollonius, two of the greatest men produced by antiquity. [Dantzig2007, pg. 19]

http://experimentalmath.info/blog/2010/02/the-greatest-mathematical-discovery/

Applied mathematics is oriented towards problem solving. There are fine examples of discoveries in this area.

https://www.nsf.gov/discoveries/disc_summ.jsp?cntn_id=117540&org=NSF

https://www.nsf.gov/discoveries/disc_summ.jsp?cntn_id=136523&org=NSF

Dear @Artur thank you for your answer. With regard to your nature like discoveries, I just quote from Paul Dirac: God used beautiful mathematics in creating the world.

The Discovery of ZERO

Quite aside from its practical application, the discovery of ZERO represents an extremely important step in humankind's intellectual evolution. Its adoption by Indian mathematicians in the 5th century AD allowed the first use of negative numbers and decimal fractions.

The concept of zero seems to have originated around 520 AD with the Indian Aryabhata who used a symbol he called “kha” as a place holder. Brahmagupta, another Indian mathematician who lived in the 5th century, is credited for developing the Hindu-Arabic number system which included zero as an actual number. Other mathematicians like al-Khwarizmi and Leonardo Fibonacci expanded the use of zero.

How important is zero? It is the number around which the negative numbers to its left stretch into infinity and the positive numbers to the right do likewise. 

Zero allowed consideration of maths as a concept as opposed to simply a counting system for tangible or quasi-tangible problems. 

I guess this discussion was made possible by the advent of the Information Theory which was developed by Claude Shannon and many others. 

Tolstoy: "Some mathematician, I believe, has said that true pleasure lies not in the discovery of truth, but in the search for it."

Benjamin Franklin: "No employment can be managed without arithmetic, no mechanical invention without geometry."

5 brilliant mathematicians and their impact on the modern world by SHEA GUNTHER

Math allowed us to tease out the secrets of DNA, create and transmit electricity over hundreds of miles to power our homes and offices, and gave rise to computers and all that they do for the world. Without math, we'd still be living in caves getting eaten by cave tigers.

http://www.mnn.com/green-tech/research-innovations/blogs/5-brilliant-mathematicians-and-their-impact-on-the-modern

There are mathematicians whose creativity, insight and taste have the power of driving anyone into a world of beautiful ideas, which can inspire the desire, even the need for doing mathematics, or can make one to confront some kind of problems, dedicate his life to a branch of math, or choose an specific research topic.

Leonhard Euler: He made important discoveries in pretty much every mathematical field there was at his time.

• He discovered graph theory.
• He is responsible for much of the current mathematical notation we use today, including Σ, i, e, f(x), π, and sin/cos.
• EVERYTHING is named after him
• His combined works fill 80 (!) volumes.
• And last but not least, Euler’s identity eiπ + 1 = 0

http://math.stackexchange.com/questions/815/which-mathematicians-have-influenced-you-the-most

MANJUL BHARGAVA awarded Fields Medal in mathematics

"Manjul Bhargava, the Brandon Fradd, Class of 1983, Professor of Mathematics, was awarded the 2014 Fields Medal, one of the most prestigious awards in mathematics, in recognition of his work in the geometry of numbers. The International Mathematical Union (IMU) presents the medal every four years to researchers under the age of 40 based on the influence of their existing work and on their “promise of future achievement.”

The honor, often referred to as the “Nobel Prize of mathematics,” was awarded to four researchers at the 2014 IMU International Congress of Mathematicians held in Seoul, South Korea. The prize committee commended Bhargava “for developing powerful new methods in the geometry of numbers, which he applied to count rings of small rank and to bound the average rank of elliptic curves.”..."

https://discovery.princeton.edu/tag/mathematics-2/

https://discovery.princeton.edu/2015/11/21/manjul-bhargava-awarded-fields-medal-in-mathematics/

If we talk about field medalists, we should also mention Dr. Maryam Mirzakhani, the first woman ever to win a Fields Medal (some call it Noble prize of Maths)

Research work of Dr. Mirzakhani

• Discovered a formula expressing the volume of a moduli space with a given genus as a polynomial in the number of boundary components.
•  Her thesis showed how to compute the Weil-Petersson volumes of moduli spaces of bordered Riemann surfaces.
•  Her research interests include Teichmüller theory, hyperbolic geometry, ergodic theory, and symplectic geometry

http://www.theguardian.com/science/2014/aug/13/interview-maryam-mirzakhani-fields-medal-winner-mathematician

http://en.wikipedia.org/wiki/Maryam_Mirzakhani

http://www.stanforddaily.com/2014/08/13/stanford-professor-maryam-mirzakhani-wins-fields-medal-first-woman-to-do-so/

https://www.researchgate.net/post/What_is_your_opinion_about_awarding_the_2014_Fields_Medal_to_an_Iranian_woman_for_the_first_time_in_the_history?

Thales' theorem

Thales was the first to inscribe in a circle a right-angle triangle. In geometry, Thales' theorem states that if A, B and C are points on a circle where the line AC is a diameter of the circle, then the angle ∠ABC is a right angle.

https://en.wikipedia.org/wiki/Thales%27_theorem

The Greatest Mathematical Discovery of All Time

Albert Einstein said that compound interest is "the greatest mathematical discovery of all time."

It's surprisingly easy to calculate the future value of your investments, and it's something even a novice investor can do for himself or herself. To calculate the future value (FV) of your investment, you can use this formula: 

FV = PV x (1 + i)n  

PV = present value 

i = annual interest rate

n = number of years the investment will compound

http://www.investinganswers.com/education/time-value-money/greatest-mathematical-discovery-all-time-1449

Srinivasa Ramanujan was one of India's greatest mathematical geniuses. He made substantial contributions to the analytical theory of numbers and worked on elliptic functions, continued fractions, and infinity .He displayed a natural ability in mathematics at an early age. His talent was recognized by G. N. Hardy, who arranged for him to be a student at Cambridge University. In the following few years, before an early death at age 32, Ramanujan produced an exceptional output in mathematical analysis, number theory, infinite series and continued fractions.

Ramanujan’s incredible genius was brought to the attention of the world of mathematics and science through his work at Cambridge. During his five-year stay at Cambridge, he published 21 research papers containing theorems on the following topics:

• Definite integral
• Modular equations and functions
• Riemann’s zeta function
• Infinite series
• Summation of series
• Analytic number theory
• Asymptotic formulae
• Partitions and combinatorial analysis

http://www.newworldencyclopedia.org/entry/Srinivasa_Ramanujan

https://en.m.wikipedia.org/wiki/Srinivasa_Ramanujan

Laplace optimization and inverse of a square matrix

Who invented math?

The oldest clay tablets with mathematics date back over 4,000 years ago in Mesopotamia. The oldest written texts on mathematics are Egyptian papyruses. Since these are some of the oldest societies on Earth, it makes sense that they would have been the first to discover the basics of mathematics.

More advanced mathematics can be traced to ancient Greece over 2,500 years ago. Ancient mathematician Pythagoras had questions about the sides of a right triangle. His questioning, research, and testing led to a basic understanding of triangles we still study today, known as the Pythagorean Theorem.

Most experts agree that it was around this time (2,500 years ago) in ancient Greece that mathematics first became an organized science. Since that time, mathematical discoveries have spurred other mathematicians and scientists to build upon the work of others, constantly expanding our understanding of mathematics and its relation to the world around us.

http://wonderopolis.org/wonder/who-invented-math

Dear Professor Omid,

I'm under average when it comes to mathematics, but from the video and articles I'm attaching here, I believe Albert Einstein's general theory of relativity is one of the most important discoveries. Here are excerpts from both articles: (the first one is with a video)

# 1

"A team of scientists announced on Thursday that they had heard and recorded the sound of two black holes colliding a billion light-years away, a fleeting chirp that fulfilled the last prediction of Einstein's general theory of relativity.

That faint rising tone, physicists say, is the first direct evidence of gravitational waves, the ripples in the fabric of space-time that Einstein predicted a century ago...."

 http://www.nytimes.com/2016/02/12/science/ligo-gravitational-waves-black-holes-einstein.html

# 2

"The idea of gravitational waves emerged from the general theory of relativity, Albert Einstein’s fundamental exposition of gravity, unveiled almost exactly 100 years before GW150914’s discovery. Mass, Einstein realised, deforms the space and time around itself. Gravity is the effect of this, the behaviour of objects dutifully moving along the curves of mass-warped spacetime. It is a simple idea, but the equations that give it mathematical heft are damnably hard to solve. Only by making certain approximations can solutions be found. And one such approximation led Einstein to an odd prediction: any accelerating mass should make ripples in spacetime."

http://www.economist.com/news/science-and-technology/21692851-gravitational-waves-at-LIGO-century-after-Albert-Einstein-predicted-them?fsrc=scn/fb/te/pe/ed/mergerandacquisition

Sincerely,

Cameen

Dear Colleagues,

Good Day,

"LIST OF IMPORTANT MATHEMATICIANS

This is a chronological list of some of the most important mathematicians in history and their major achievements, as well as some very early achievements in mathematics for which individual contributions can not be acknowledged.

Where the mathematicians have individual pages in this website, these pages are linked; otherwise more information can usually be obtained from the general page relating to the particular period in history, or from the list of sources used. A more detailed and comprehensive mathematical chronology can be found at

http://www-groups.dcs.st-and.ac.uk/~history/Chronology/full.html.

Please, check the link below for more information...

http://www.storyofmathematics.com/mathematicians.html

Dear Colleagues,

Good Day,

One of the most important mathematical discovery by:

William Playfair, inventor of charts

William Playfair, a Scottish engineer, was the founder of graphical statistics. Besides that signature accomplishment, he was at various times in his life a banker, an accountant, a journalist, an economist, and one of the men to storm the Bastille. 

It's difficult to overstate his importance. He was the inventor of the line graph, bar chart, and the pie chart.  He also pioneered the use of timelines. You're probably familiar with his work.

Is mathematics discovered or invented? I posit that humans invent the mathematical concepts—numbers, shapes, sets, lines, and so on—by abstracting them from the world around them. They then go on to discover the complex connections among the concepts that they had invented; these are the so-called theorems of mathematics. Whar you suggest?

http://www.pbs.org/wgbh/nova/blogs/physics/2015/04/great-math-mystery/

Maxwell Equations

All the phenomena observed in classical electricity and magnetism can be explained by means of just four mathematical equations. Moreover, physicist James Clerk Maxwell (after whom those four equations of electromagnetism are named) showed in 1864 that the equations predicted that varying electric or magnetic fields should generate certain propagating waves. These waves—the familiar electromagnetic waves (which include light, radio waves, x-rays, etc.)—were eventually detected by the German physicist Heinrich Hertz in a series of experiments conducted in the late 1880s.

https://en.wikipedia.org/wiki/Maxwell's_equations

Dear Colleagues,

Good Day,

One of the most important mathematical discovery by:

Alan Turing, World War II code-breaker

Alan Turing is a British mathematician who is hailed as the father of computer science. 

Turing is especially unique on his efforts during the Second World War. Working at the famous Bletchley Park, Turing is credited as one of most important people in devising the techniques for breaking the German Enigma cipher.

He developed the method by which the Bombe – a massive electromechanical machine built by the Allies – could crack the Enigma on an industrial scale, allowing them to read nearly all German communication. In that regard, he is one of the founders of modern cryptanalysis, and by all rights played one of the most crucial parts in winning the Battle of the Atlantic for the Allies.

A New Discovery About Prime Numbers

Recently Yitang “Tom” Zhang stunned the world of pure mathematics when he announced that he had proven the “bounded gaps” conjecture about the distribution of prime numbers—a crucial milestone on the way to the even more elusive twin primes conjecture, and a major achievement in itself.

Full story at the attached link:

http://www.slate.com/articles/health_and_science/do_the_math/2013/05/yitang_zhang_twin_primes_conjecture_a_huge_discovery_about_prime_numbers.html

Error function

In mathematics, the error function (also called the Gauss error function) is a special function (non-elementary) of sigmoid shape that occurs in probability, statistics, and partial differential equations describing diffusion. It is defined as:

https://en.wikipedia.org/wiki/Error_function

Seven equations that rule your world

• The wave equation,
• Maxwell’s four equations,
• Fourier transform
• Schrödinger’s equation

https://www.newscientist.com/article/mg21328516-600-seven-equations-that-rule-your-world/

Zero could have been invented for easier calculations and abstract mathematics. There is a saying by an eminent mathematician that "Today's abstract math is tomorrow's applied math".

http://yaleglobal.yale.edu/about/zero.jsp

The greatest equations ever

These 20 equations are listed below in order of the number of people who proposed them. The first two received about 20 mentions each out of a total of about 120; the rest received between two and 10 each. Equations are given, where appropriate, in their most common form.

http://www.fisica.net/blog/2009/02/greatest-equations-ever.html