Analysis is a major branch of mathematics that has to do with approximating certain mathematical objects, such as numbers or functions, by other objects that are easier to understand or to handle. A simple example of analysis is the calculation of the first few decimal places of π by writing it as the limit of an infinite series. The origins of analysis go back to the 17th century, when people such as Isaac Newton began investigating how to approximate locally – in the neighborhood of a point – the behavior of quantities that vary continuously. This led to an intense study of limits, which form the basis of understanding infinite series, differentiation, and integration.


Modern analysis is subdivided into several areas: real analysis (the study of derivatives and integrals of real-valued functions); functional analysis (the study of spaces of functions); harmonic analysis (the study of Fourier series and their abstractions); complex analysis (the study of functions from the complex plane to the complex plane that are complex differentiable); and non-standard analysis (the study of hyperreal numbers and their functions, which leads to a rigorous treatment of infinitesimals and of infinitely large numbers).