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Calculus is one of the most important breakthroughs in modern mathematics, answering questions that had puzzled mathematicians, scientists, and philosophers for more than two thousand years. In particular, calculus gave a clear and precise definition of [infinity, both in the case of the infinitely large and the infinitely small. It provided an answer to Zeno's paradoxes and gave the first clear definition of what Aristotle called "the quality of motion".

Calculus was not discovered all at once. In the ancient world, Eudoxus and Archimedes proposed mathematical ideas that can now be seen as similar to calculus. In Twelfth Century India, Bhaskara conceived of differential calculus and two centuries later, Madhava and the Kerala school studied infinite series, convergence, differentiation and other concepts of calculus. In the 17th century, Kowa Seki in Japan elaborated some of the fundamental principles of integral calculus. At roughly the same time, in Europe, Wallis and Barrow proposed ideas that would now be considered integrals, derivatives, and the fundamental theorem of calculus, which was first proved by James Gregory. But it is Isaac Newton and Gottfried Leibniz who are credited with bringing all these ideas together, and they are usually credited with the independent and nearly simultaneous creation of calculus. Even so, it was generations after Newton and Leibniz that Cauchy and other mathematicians finally put calculus on a rigorous basis, with the definition of the limit.

From a mathematical standpoint, calculus gives the definitions and properties of three linear operators, the limit, the derivative, and the integral. All of these depend on the definition of the limit. Roughly speaking, the limit allows us to control an otherwise uncontrollable output, the derivative is the slope of a graph, and the integral is the area under a curve. In scientific applications, the derivative is often used to find a changing velocity given a changing position, and...