The Greek mathematician Pythagoras
The Greeks were much more advanced than other ancient societies at math. They were the first to apply deductive logic to geometry. This was also when basic geometry originated. A Greek mathematician, Euclid, created his Elements, which contain the fundamental axioms of geometry. The Greeks also coined the term "mathematics." Pythagoras is credited with first discovering the Pythagorean Theorem, although there is no evidence that he even tried to prove this.
The Chinese did not have a huge impact on mathematics. They did, however, create a positional decimal notation (sort of like the system we use today) and the first decimal multiplication table. They also invented the first abacus, which they called the "Suan Pan." They believed that the magic square was very important, both spiritually and religiously. They also developed the Chinese Remainder Theorem (shocker), which helps people find solutions to systems of modular congruences.
Leibniz (ignore the t in the picture) developed modern calculus (a torture to students everywhere), including most of the notation we use. He developed this independently around the same time as Isaac Newton. He had the ideas of a derivative, antiderivative, and even the infinitesimal. He first figured out how to find the area under a curve (y = f(x)) on November 11, 1675. He used an elongated S for summation and a d for differential, which we still use today.
Euler was a very important modern mathematician. He created notation that we still use today. This includes function notation f(x), the sigma to mean summation (Σ), and the i to represent the imaginary unit (ⅈ). He also started the first work on Graph Theory, a field which is very important to computer science and mathematics today. This includes his formula V - E + F = 2, used to describe polyhedrons.
George Cantor is a very important name in set theory. He first showed that there are more real numbers than rational numbers, which means that there is more than one type of infinity. We call the infinity of the natural numbers (and equivalently the integers and rationals) the countable infinity (aleph-null). We call any other infinities uncountable. He devoted some of his life to attempting to prove the Continuum Hypothesis. This, informally, says that there is no set that is bigger than the set of the naturals but less than the set of the reals.
Kurt Godel (properly pronounced, his last name rhymes with noodle), developed his two Incompleteness Theorems. The First says that, if you create a consistent logical system, there is at least one thing in the system that you cannot prove or disprove. The Second says that any system cannot prove the consistency of itself, either directly or indirectly.
John Nash was a game theorist. Game theory is about the interactions ("games") of rational "agents," things that will always make the most logical choice. He developed the concept of "Nash equilibrium." This means that, assuming the other agents' strategies are constant, that any agent could not benefit by changing their strategy. For example, any game in which you can benefit yourself by hurting a partner will have a Nash equilibrium when both people choose to benefit themselves, because, regardless of the other's strategy, both people will choose to benefit themselves.
THIS CONCLUDES A BRIEF HISTORY OF MATHEMATICS. THANK YOU!
Bibliography: All facts, excepting some of the Chinese section, were originally found on Wikipedia and verified independently. Some facts in the Chinese section were found on: http://www.storyofmathematics.com/chinese.html