Lesson THE KINDS of real numbers

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All numbers are products of our imagination. Some numbers require less imagination than others.

As we move on from arithmetics and begin to study algebra, you keep encountering more and more kinds of numbers. They fill the "gaps", making more and more operations possible. For instance, zero and negative numbers were invented so that greater numbers could be subtracted from smaller numbers.

Then fractions were invented to allow division of any integer number by another integer. A decimal was used to write fractions, and some were represented by infinitely long decimals.

Real numbers are another step in this direction. They address several problems that were unsolvable within the world of signed fractions.

One is the problem of finding a number whose square is equal to 2 (or generally, being able to compute a square root of any positive fraction). There is no fraction whose square is equal to two.

The second problem is dealing with decimals that have an endless sequence of digits after the decimal point.

To resolve these problems, we now declare that any decimal numbers, even the ones that have an endless sequence of digits, are valid numbers and call such numbers "real numbers".

For an in depth discussion of real numbers, go to <A HREF=Real-number.wikipedia>this article</A>.

Some real numbers can be computed with computer programs, to any precision. These numbers are called "computable" and include such numbers as pi, {{{sqrt( 2 )}}}, and so on. As it turns out, the majority of real numbers cannot be computed with any computer programs! We can only speak about their existence hypothetically, and we cannot have anyw ay of writing them down.