2.2 Decimals and Real Numbers

We have a nice way to represent numbers including fractions, and that is as decimal expansions. Suppose we consider numbers like , (which is the same as ), , and so on.

We write them as .1, .2, .3, and so on. The decimal point is a code that tells us that the digit just beyond it is to be divided by ten.

We can extend this to integers divided by one hundred, by adding a second digit after the decimal point. Thus .24 means . And we can keep right on going and describe integers divided by a thousand or by a million and so on, by longer and longer strings of integers after the decimal point.

However we do not get all rational numbers this way if we stop. We will only get rational numbers whose denominators are powers of ten. A number like 1/3 will become .33333...., where the threes go on forever. (This is often written as .3*, the star indicating that what precedes it is to be repeated endlessly)

To get all rational numbers using this decimal notation you must therefore be willing to go on forever. If you do so, you get even more than the rational numbers. The set of all sequences of digits starting with a decimal point give you all the rational numbers and even more. What you get are called the real numbers between 0 and 1. The rational numbers turn out to be those that repeat endlessly, like .33333...., or .1000...., or .14141414...., (aka .(14)*).

Now neither you or I or any computer are really going to go on forever writing a number so there is a sense of unreality about this notion of real numbers, but so what? In your imagination you can visualize a stream of numbers going on forever. That will represent a real number.

If you stop a real number after a finite number of digits, you get a rational number, and as a result, the rules of addition, subtraction, uglification and derision that work for rational numbers can be used to do the same things for real numbers as well. Fortunately, the digits that are far to the right of the decimal point in a number have little effect on computations when there are non-zero digits much closer to the decimal point.

Since we cannot in real life go on forever to describe a non-rational real number, to do so we have to describe it some other way. Here is an example of such a description.
We define the number that has decimal expansion .1101001000100001....; between each consecutive pair of 1's there is a number of 0's that is one more than between the previous consecutive pair of 1's. This number is not rational; it does not repeat itself.

We do not have to, but just for the fun of it, we will go one step further and extend our numbers once more.