Even though not much is known from his life, he was a philosopher who was part of the Pythagorean Brotherhood. He is credited with the discovery of irrational numbers, which is coincidentally the reason he was killed. (https://www.google.com/search?q=Hippasus&tbm=isch&source=iu&ictx=1&fir=h2vH3_u1lJoumM%253A%252CCbW5IYWNcJxEhM%252C%252Fm%252F03q0l5&vet=1&usg=AI4_-kQphnakykSmwkWC66KmMl4ks3_3VQ&sa=X&ved=2ahUKEwi9pKHBktLoAhVRiOAKHXVqCv4Q_B0wFnoECAoQAw#imgrc=h2vH3_u1lJoumM:).

He is considered the most prominent mathematician of the Greco-Roman time period. He is most known for his development of geometry and "The Elements." This is a 13 book compilation of mathematical definitions, concepts and proofs. Image:(https://medium.com/@sunfaceman/math-and-magic-euclid-defines-space-ea987f61709c).

What he seems to be most known for in the world of mathematics, is his proof that "the surface area of any sphere of radius r is four times that of its greatest circle (in modern notation, S = 4πr^{2)} and that the volume of a sphere is two-thirds that of the cylinder in which it is inscribed (leading immediately to the formula for the volume, V = 4/3πr^{3)."} (https://www.britannica.com/biography/Archimedes). (https://www.britannica.com/biography/Archimedes#/media/1/32808/57320).

His contributions to trigonometry are especially important. For instance, Ptolemy’s table of the lengths of chords in a circle is the earliest surviving table of a trigonometric function. (britannica.com/biography/Ptolemy)

Often called the "father of algebra" because he greatly contributed to number theory, mathematical notation and his Arithmetica. This is the major work of Diophantus and the most prominent work on algebra in Greek mathematics. It is a collection of problems giving numerical solutions of both determinate and indeterminate equations. (https://en.wikipedia.org/wiki/Diophantus). (https://www.google.com/url?sa=i&url=https%3A%2F%2Fwww.maa.org%2Fpress%2Fperiodicals%2Fconvergence%2Fmathematical-treasure-bachets-arithmetic-of-diophantus&psig=AOvVaw3wVkKeq8Xx08xlalVvZWhG&ust=1586203675989000&source=images&cd=vfe&ved=0CAIQjRxqFwoTCOD3vqmL0ugCFQAAAAAdAAAAABAD).

He was a german mathematician who made significant contributions to elliptic functions, dynamics, differential equations, determinants, and number theory. "Jacobi formulated a theory of elliptic functions based on four theta functions. The quotients of the theta functions yield the three Jacobian elliptic functions: sn z, cn z, and dn z. His results in elliptic functions were published in Fundamenta Nova Theoriae Functionum Ellipticarum (1829; “New Foundations of the Theory of Elliptic Functions”). In 1832 he demonstrated that, just as elliptic functions can be obtained by inverting elliptic integrals, so too can hyperelliptic functions be obtained by inverting hyperelliptic integrals. This success led him to the formation of the theory of Abelian functions, which are complex functions of several variables." (https://www.britannica.com/biography/Carl-Jacobi). (https://www.google.com/search?q=jacobi&tbm=isch&source=iu&ictx=1&fir=ThLB9iX51V9-QM%253A%252C40oB-sNNmuC6HM%252C%252Fm%252F01dw80&vet=1&usg=AI4_-kSJfeaRCdDzprOZDVlSmEJyIhJzcQ&sa=X&ved=2ahUKEwiq6tugk9LoAhUOm-AKHa3yD00Q_B0wCnoECAsQAw#imgrc=ThLB9iX51V9-QM:).

His biggest contribution to mathematics is in the area of abstract algebra called Galois Theory. "Galois Theory uncovers a relationship between the structure of groups and the structure of fields. It then uses this relationship to describe how the roots of a polynomial relate to one another." (https://www.math3ma.com/blog/what-is-galois-theory-anyway).

Kronecker was primarily an arithmetician and. algebraist. Heis biggest contribution to mathematics was in the field of elliptical functions, the theory of algebraic equations, and the theory of algebraic numbers. For example, Kronecker's theory of algebraic magnitudes (1882). Thi theory revolves around algebraic numbers, "which are real numbers for which there exists a polynomial equation with integer coefficients such that the given real number is a solution. Algebraic numbers include all of the natural numbers, all rational numbers, some irrational numbers, and complex numbers of the form pi + q, where p and q are rational, and i is the square root of −1." (https://www.britannica.com/science/algebraic-number). Furthermore, "he was the first to doubt the significance of nonconstructive existence proofs (proofs that show something must exist, often by using a proof through contradiction, but that give no method of producing them)." (https://www.britannica.com/biography/Leopold-Kronecker). v(https://www.google.com/url?sa=i&url=https%3A%2F%2Fen.wikipedia.org%2Fwiki%2FLeopold_Kronecker&psig=AOvVaw3rIwdpu8LAeqKR4vvg1tbU&ust=1586205300510000&source=images&cd=vfe&ved=0CAIQjRxqFwoTCPCYhq-R0ugCFQAAAAAdAAAAABAD).

Hardy was an English pure mathematician whose work revolved around number theory. For example, he contributed fundamentally to the Riemann zeta function and the distribution of primes. However, when asked what his greatest contribution to math was, he said it was discovering Ramanujan. Image: (https://www.pinterest.ca/pin/284993482650246754/).

He was an English mathematician, computer scientist, logician, cryptanalyst, philosopher, and theoretical biologist. Turing was highly influential in the development of theoretical computer science, providing a formalization of the concepts of algorithm and computation with the Turing machine, which can be considered a model of a general-purpose computer. Turing is widely considered to be the father of theoretical computer science and artificial intelligence. Furthermore, he played a key role in breaking the German Enigma machine during the Second World War. (https://en.wikipedia.org/wiki/Alan_Turing). (https://www.google.com/imgres?imgurl=https%3A%2F%2Fupload.wikimedia.org%2Fwikipedia%2Fcommons%2Fa%2Fa1%2FAlan_Turing_Aged_16.jpg&imgrefurl=https%3A%2F%2Fen.wikipedia.org%2Fwiki%2FAlan_Turing&tbnid=Uz9peieEHvMCMM&vet=12ahUKEwiUyPrcvNboAhURPN8KHVQNAmAQMygAegUIARCWAg..i&docid=Xqrl7aDPKekzYM&w=675&h=919&itg=1&q=alan%20turing&ved=2ahUKEwiUyPrcvNboAhURPN8KHVQNAmAQMygAegUIARCWAg).