Lecture 1 - 1 Ultimate Aim Numbers were first used to...

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1. Ultimate Aim Numbers were first used to count things. As such the first numbers that people used were whole positive numbers. One thing that is nat- ural to do is to add two numbers, since that represents counting the total number of objects belonging to two groups. Becoming quite a bit more sophisticated, one can then ask: “Given that there are three objects in one pile and five objects in both piles, then how many objects are there in the second pile?”. Abstracting this question even more, and letting x denote the number of objects in the second pile, we get an equation x + 3 = 5 . As soon as one reaches this level of generality, there are obvious problems. What is the solution to the equation x + 3 = 3? It took people a very long time to realise that the right answer is x = 0, in other words that we should consider zero as a number. From there it is not such a big leap to consider equations of the form x + 4 = 3 , and realise the need for negative numbers. Becoming even more so- phisticated, we might consider an equation of the form 3 x = 9 . Once again, as soon as you write this down, some trouble maker is going to ask how to solve 3 x = 2 . The solution, as everyone knows, is to allow fractions (or better we have gone from whole numbers, to the natural numbers, from there to the integers and finally to the rationals). It seems at this stage, that we are finished, since any equation, of the form ax + b = c, where a , b and c are rational numbers, has a unique rational solution (or none exists for obvious reasons). Indeed one can solve this equation,
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