He is regarded as the founder of modern algebraic notation, and his work In artem analyticem isagoge introduced the use of letters to represent both known and unknown quantities, as well as the rules for manipulating them.
He is credited with the invention of analytic geometry, which combines algebra and geometry, and his work La Géométrie introduced the Cartesian coordinate system and the notion of functions.
He is one of the pioneers of calculus, which is a branch of algebra that deals with limits, derivatives, and integrals, and his work Principia Mathematica contains many applications of algebra to physics and astronomy.
He is one of the founders of abstract algebra, and his work Réflexions sur la résolution algébrique deséquations introduced the concepts of groups, fields, and permutations, as well as the Lagrange's theorem and the Lagrange resolvent.
He is considered the prince of mathematics, and his work Disquisitiones Arithmeticae is a landmark in algebraic number theory, while his contributions to modular arithmetic, polynomial equations, and quadratic forms are also significant.