Mathematicians develop abstract structures to study complex realities. For example, it is often convenient to group like items together, as when the apples in the market are placed separately from the oranges and the pears, and mathematicians developed the notion of ''set'' to capture this. It sometimes happens that members of one set can be paired up with members of another set, and we use the concept of ''number'' to describe this tendency. In our daily lives we encounter numerous examples of related quantities, such as the relationship between temperature and the time of day, or between the distance we have travelled and the time during which we have been travelling. The notion of ''function'' is an abstract way of describing such relationships. Sometimes it is necessary to organize and manipulate large collections of data, and ''matrices'' are often useful for that purpose.<br><br>The pure mathematician takes the attitude that if abstractions like sets, numbers, functions and matrices are routinely useful for studying so many aspects of our daily lives, then they are also worth studying for their own sake. History records many instances of the usefulness of this approach. When Isaac Newton sought to understand the trajectories of projectiles, he found success not by studying projectiles, but by studying continuous functions. When Einstein was working out his theory of relativity, non-Euclidean geometry, a branch of mathematics developed for reasons having nothing to do with physics, proved to be indispensable. These are just two of many possible examples.