Numerical analysis in its purest sense has been around for a very long time. As early as 1700 B.C., the Egyptians and the Greeks were forming algorithms to find values for things such as the square root of two. They did not understand or even accept the concept of irrational numbers, but needed the repetitive algorithms to get a workable value for irrational square roots. One of the the students of Aristotle actually proved that the square root of two was not rational. He was given a set of chains and asked to swim in the Adriatic Sea. The Dark Ages interrupted math's progress, and it was only when Isaac Newton and Gottfried Leibnitz invented calculus that numerical analysis was needed to get real-world answers theoretical ones offered by calculus.
You might consider that numerical analysis actually came into existence in the late 1940s, after the end of World War II. This was the dawn of the computer age. Algorithms were no longer painstakingly handled by humans, but rather written by humans and executed by machines. Herman Goldstine and John von Neumann published a paper on matrices in the November 1947 Bulletin of the AMS that dealt numerically using a computer, rather than math theoretically with this problem. Since then computers have flourished and so has theoretical numerical analysis.
Until the 1950s, linear algebra dealt with matrices of a very small size (no larger than 10x10). They were tedious and extremely slow to solve. With the advent of computers, theoretical numerical analysts came up with better and better algorithms that stimulated linear algebra into an important role in modern math, science and business. In 2010, a handheld calculator can handle a 150x150 matrix and come up with extremely accurate estimates.
One of the most common tools in engineering and economics is the differential equation. In theory, a differential equation can be proved to have a solution. Yet the mechanics of finding that solution is nearly impossible. With numerical techniques, the actual solution is not found, but an approximation of a value at a specific point can help the engineer, scientist or economist handle a real world application.
A theoretical numerical analyst strives to find new ways of dealing with problems generated on a continuous domain. In 2010 he will basically be trying to find a way of translating one of these problems into a concise algorithm, which in turn can be fed to a computer.