Often  called the language of the universe, mathematics is fundamental to our  understanding of the world and, as such, is vitally important in a  modern society such as ours. Everywhere you look it is likely  mathematics has made an impact, from the faucet in your kitchen to the  satellite that beams your television programs to your home. As such,  great mathematicians are undoubtedly going to rise above the rest and  have their name embedded within history. This list documents some such  people. I have rated them based on contributions and how they effected  mathematics at the time, as well as their lasting effect. I also suggest  one looks deeper into the lives of these men, as they are truly  fascinating people and their discoveries are astonishing – too much to  include here.  As always, such lists are highly subjective, and as such  please include your own additions in the comments! 

Greek  Mathematician Pythagoras is considered by some to be one of the first  great mathematicians. Living around 570 to 495 BC, in modern day Greece,  he is known to have founded the Pythagorean cult, who were noted by  Aristotle to be one of the first groups to actively study and advance  mathematics. He is also commonly credited with the Pythagorean Theorem  within trigonometry. However, some sources doubt that is was him who  constructed the proof (Some attribute it to his students, or Baudhayana,  who lived some 300 years earlier in India). Nonetheless, the effect of  such, as with large portions of fundamental mathematics, is commonly  felt today, with the theorem playing a large part in modern measurements  and technological equipment, as well as being the base of a large  portion of other areas and theorems in mathematics. But, unlike most  ancient theories, it played a bearing on the development of geometry, as  well as opening the door to the study of mathematics as a worthwhile  endeavor. Thus, he could be called the founding father of modern  mathematics.
 

The  only currently living mathematician on this list, Andrew Wiles is most  well known for his proof of Fermat’s Last Theorem: That no positive  integers, a, b and c can satisfy the equation a^n+b^n=c^n For n greater  then 2. (If n=2 it is the Pythagoras Formula). Although the  contributions to math are not, perhaps, as grand as other on this list,  he did ‘invent’ large portions of new mathematics for his proof of the  theorem. Besides, his dedication is often admired by most, as he quite  literally shut himself away for 7 years to formulate a solution. When it  was found that the solution contained an error, he returned to solitude  for a further year before the solution was accepted. To put in  perspective how ground breaking and new the math was, it had been said  that you could count the number of mathematicians in the world on one  hand who, at the time, could understand and validate his proof.  Nonetheless, the effects of such are likely to only increase as time  passes (and more and more people can understand it).
 
8Isaac Newton and Wilhelm Leibniz
 
I  have placed these two together as they are both often given the honor  of being the ‘inventor’ of modern infinitesimal calculus, and as such  have both made monolithic contributions to the field. To start, Leibniz  is often given the credit for introducing modern standard notation,  notably the integral sign. He made large contributions to the field of  Topology. Whereas all round genius Isaac Newton has, because of the  grand scientific epic Principia, generally become the primary man hailed  by most to be the actual inventor of calculus. Nonetheless, what can be  said is that both men made considerable vast contributions in their own  manner.
 
Blgollo,  also known as Leonardo Fibonacci, is perhaps one of the middle ages  greatest mathematicians. Living from 1170 to 1250, he is best known for  introducing the infamous Fibonacci Series to the western world. Although  known to Indian mathematicians since approximately 200 BC, it was,  nonetheless, a truly insightful sequence, appearing in biological  systems frequently. In addition, from this Fibonacci also contributed  greatly to the introduction of the Arabic numbering system. Something he  is often forgotten for. 
Haven  spent a large portion of his childhood within North Africa he learned  the Arabic numbering system, and upon realizing it was far simpler and  more efficient then the bulky Roman numerals, decided to travel the Arab  world learning from the leading mathematicians of the day. Upon  returning to Italy in 1202, he published his Liber Abaci, whereupon the  Arabic numbers were introduced and applied to many world situations to  further advocate their use. As a result of his work the system was  gradually adopted and today he is considered a major player in the  development of modern mathematics.
 
Computer  Scientist and Cryptanalyst Alan Turing is regarded my many, if not  most, to be one of the greatest minds of the 20th Century. Having worked  in the Government Code and Cypher School in Britain during the second  world war, he made significant discoveries and created ground breaking  methods of code breaking that would eventually aid in cracking the  German Enigma Encryptions. Undoubtedly affecting the outcome of the war,  or at least the time-scale. 
After  the end of the war he invested his time in computing. Having come up  with idea of a computing style machine before the war, he is considered  one of the first true computer scientists. Furthermore, he wrote a range  of brilliant papers on the subject of computing that are still relevant  today, notably on Artificial Intelligence, on which he developed the  Turing test which is still used to evaluate a computers ‘intelligence’.  Remarkably, he began in 1948 working with D. G. Champernowne, an  undergraduate acquaintance on a computer chess program for a machine not  yet in existence. He would play the ‘part’ of the machine in testing  such programs. 
 

French  Philosopher, Physicist and Mathematician Rene Descartes is best known  for his ‘Cogito Ergo Sum’ philosophy. Despite this, the Frenchman, who  lived 1596 to 1650, made ground breaking contributions to mathematics.  Alongside Newton and Leibniz, Descartes helped provide the foundations  of modern calculus (which Newton and Leibniz later built upon), which in  itself had great bearing on the modern day field. Alongside this, and  perhaps more familiar to the reader, is his development of Cartesian  Geometry, known to most as the standard graph (Square grid lines, x and y  axis, etc.) and its use of algebra to describe the various locations on  such. Before this most geometers used plain paper (or another material  or surface) to preform their art. Previously, such distances had to be  measured literally, or scaled. With the introduction of Cartesian  Geometry this changed dramatically, points could now be expressed as  points on a graph, and as such, graphs could be drawn to any scale, also  these points did not necessarily have to be numbers. The final  contribution to the field was his introduction of superscripts within  algebra to express powers. And thus, like many others in this list,  contributed to the development of modern mathematical notation.
 
Living  around 300BC, he is considered the Father of Geometry and his magnum  opus: Elements, is one the greatest mathematical works in history, with  its being in use in education up until the 20th century. Unfortunately,  very little is known about his life, and what exists was written long  after his presumed death. Nonetheless, Euclid is credited with the  instruction of the rigorous, logical proof for theorems and conjectures.  Such a framework is still used to this day, and thus, arguably, he has  had the greatest influence of all mathematicians on this list. Alongside  his Elements were five other surviving works, thought to have been  written by him, all generally on the topic of Geometry or Number theory.  There are also another five works that have, sadly, been lost  throughout history.
 

Bernhard  Riemann, born to a poor family in 1826, would rise to become one of the  worlds prominent mathematicians in the 19th Century. The list of  contributions to geometry are large, and he has a wide range of theorems  bearing his name. To name just a few: Riemannian Geometry, Riemannian  Surfaces and the Riemann Integral. However, he is perhaps most famous  (or infamous) for his legendarily difficult Riemann Hypothesis; an  extremely complex problem on the matter of the distributions of prime  numbers. Largely ignored for the first 50 years following its  appearance, due to few other mathematicians actually understanding his  work at the time, it has quickly risen to become one of the greatest  open questions in modern science, baffling and confounding even the  greatest mathematicians. Although progress has been made, its has been  incredibly slow. However, a prize of $1 million has been offered from  the Clay Maths Institute for a proof, and one would almost undoubtedly  receive a Fields medal if under 40 (The Nobel prize of mathematics). The  fallout from such a proof is hypothesized to be large: Major encryption  systems are thought to be breakable with such a proof, and all that  rely on them would collapse. As well as this, a proof of the hypothesis  is expected to use ‘new mathematics’. It would seem that, even in death,  Riemann’s work may still pave the way for new contributions to the  field, just as he did in life.
 

Child  prodigy Gauss, the ‘Prince of Mathematics’, made his first major  discovery whilst still a teenager, and wrote the incredible  Disquisitiones Arithmeticae, his magnum opus, by the time he was 21.  Many know Gauss for his outstanding mental ability – quoted to have  added the numbers 1 to 100 within seconds whilst attending primary  school (with the aid of a clever trick). The local Duke, recognizing his  talent, sent him to Collegium Carolinum before he left for Gottingen  (at the time it was the most prestigious mathematical university in the  world, with many of the best attending). After graduating in 1798 (at  the age of 22), he began to make several important contributions in  major areas of mathematics, most notably number theory (especially on  Prime numbers). He went on to prove the fundamental theorem of algebra,  and introduced the Gaussian gravitational constant in physics, as well  as much more – all this before he was 24! Needless to say, he continued  his work up until his death at the age of 77, and had made major  advances in the field which have echoed down through time.
 
If  Gauss is the Prince, Euler is the King. Living from 1707 to 1783, he is  regarded as the greatest mathematician to have ever walked this planet.  It is said that all mathematical formulas are named after the next  person after Euler to discover them. In his day he was ground breaking  and on par with Einstein in genius. His primary (if that’s possible)  contribution to the field is with the introduction of mathematical  notation including the concept of a function (and how it is written as  f(x)), shorthand trigonometric functions, the ‘e’ for the base of the  natural logarithm (The Euler Constant), the Greek letter Sigma for  summation and the letter ‘/i’ for imaginary units, as well as the symbol  pi for the ratio of a circles circumference to its diameter. All of  which play a huge bearing on modern mathematics, from the every day to  the incredibly complex.
As  well as this, he also solved the Seven Bridges of Koenigsberg problem  in graph theory, found the Euler Characteristic for connecting the  number of vertices, edges and faces of an object, and (dis)proved many  well known theories, too many to list. Furthermore, he continued to  develop calculus, topology, number theory, analysis and graph theory as  well as much, much more – and ultimately he paved the way for modern  mathematics and all its revelations. It is probably no coincidence that  industry and technological developments rapidly increased around this  time.