An important theme that runs through most of mathematics, from the geometry of the early Greeks to modern signal processing, is analysis and synthesis, or, less formally, breaking up and putting back together. The Greeks estimated the area of a circle by breaking it up into sectors that approximated triangles. The Riemann approach to integration involves breaking up the area under a curve into pieces that approximate rectangles or other simple shapes. Viewed differently, the Riemann approach is first to approximate the function to be integrated by a step function and then to integrate the step function.

Along with geometry, Euclid includes a good deal of number theory, in which we find analysis and synthesis. His theorem that every positive integer is divisible by a prime is analysis; division does the breaking up and the simple pieces are the primes. The fundamental theorem of arithmetic, which asserts that every positive integer can be written in an essentially unique way as the product of powers of primes, is synthesis, with the putting back together done by multiplication.