Real Numbers - a logical development

If we don't look too closely at the historical facts, the development of number systems can be thought of as a logical development. We begin with the natural numbers and gradually extend the number system though positive integers, integers and rationals to real numbers. At each stage some defect in the previous number system is remedied, resulting in a better system, subsuming the previous one and having a wider range of applications.

1. Natural Numbers count. These are the numbers we use for counting, starting from one, going on indefinitely as we count upwards. As well as counting we can do addition with these numbers, but as soon as we consider the opposite of addition, subtraction, we find that it doesn't always work. Multiplication is also always possible, but division only sometimes.
2. Adding Zero makes a difference. Adding zero into the natural numbers, giving the non-negative integers, may seem today such a small step, but it was a significant development in its time.
3. Integers make subtraction work. The next step is to make subtraction work properly, and to do this negative numbers are required. This is really just extending the notation to provide a notation for the solution to a subtraction where the number subtracted is larger than the original, and we call the resulting number system the integers. Now addition, subtraction and multiplication work all the time, but division doesn't.
4. Rationals divide. A number system is closed under an operation if the operation can be applied to any two numbers and the result is still a number. The integers are not closed under division. Devising a number system closed under division is hard, but for most practical purposes close under division by non-zero numbers is close enough. To make the closure we just consider ratios as numbers and call them the rational numbers.
5. Real numbers complete the line. From the days of ancient Greece it has been known that rational numbers do not suffice to give an account of magnitudes in geometry. This is known because the ratio between certain geometric magnitudes (for example the ratio of the side of a square to its diagonal) are not the same as any ratio between whole numbers, and are called incommensurable. A number system which remedied this defect was not devised until additional pressures arose from the development of the calculus. In this context it was clear that some calculations could get arbitrarily precise in their results and yet there be no rational number which could possibly be the result. The result of fixing this problem, the real number system, may be thought of, once again, as agreeing to add into the system solutions to this new class of problems. A real number may be thought of as the limit of a convergent series of rationals.
Other Ways of Looking at Numbers
This simple story provides neither a good historical account of how numbers actually developed, nor a very good logical picture of how number systems can practically be developed for today's mathematical purposes. But its not a bad place to start.

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