Mathematicians of yesteryear, with limited access to computing, solved difficult problems by a combination of genius, vivid imaginations and hard work.

In those days, human 'computers' were used to calculate and tabulate the solutions to maths problems ranging from trigs and logs, to astronomical, nautical, banking and insurance information etc. However, those human computing processes were laborious (thus expensive), and error-prone. The invention of mechanical and electrical computing devices in the 19th and 20th centuries by pioneers such as Charles Babbage, Ada Lovelace and Alan Turing allowed such problems to be solved both more cheaply and more reliably. Nowadays, computers are immensely more powerful than they were in either Babbage’s or Turing’s time. We will describe a few mathematical problems that computers can solve, and present some of the key techniques, such as step-by-step iteration, recursion and Monte-Carlo simulation. We touch upon some modern computing tools, such as calculators, spreadsheets, programming languages etc. These tools are the present-day equivalents of Babbage’s and Turing’s machines, and are available to us all (often for free) in order to solve our everyday mathematical problems in work and in life. We end the talk by briefly speculating upon whether computing has anything to say about the nature of our lives and of the universe itself.