The numbers in any number system

A positional number system uses r different kinds of marks called digits, as we already know. By using just these digits we can represent zero or any positive integer. By using one additional kind of mark called the "minus sign", we can represent any negative integer as well. By using still another additional kind of mark called the "point", we can represent any real number either exactly or approximately, as discussed below.

Consider first the irrationals. No irrational can be exactly represented in any number system, nor will the digits of the number approximating it form indefinitely repeating cycles however far they are extended toward the right. For instance, no irrational is either an exact decimal or a repeating decimal. No matter how far we extend digits toward the right, there will be no indefinitely repeating cycle of digits. In general it can be proved that in any system an irrational number is a non-terminating and non-repeating positional number, and conversely.

Now consider the rationals. Any integer can be exactly represented in any system, as we have already seen. Likewise in any given number system, some of the nonintegers which are rational can be represented exactly (that is, they terminate, as one-fourth is exactly 0.25 in the decimal system). The rest of the nonintegers which are rational cannot be represented exactly in the given system (as one-seventh cannot be represented exactly in the decimal system), but each of them can be exactly represented in some other system (as one-seventh is exactly 0.2 in the system with radix 14). It will be obvious that any rational number can be exactly represented in some system. Also it can be proved that if a rational number cannot be exactly represented in a given system, then its digits when carried far enough to the right begin to repeat indefinitely in some cycle.

In concluding these general remarks, let us note the theorem that any real number whatsoever can be represented as closely as we please. More exactly, let m be any real number and e be any preassigned positive number as small as desired. Then, in any given number system, there is some number n such that m-n›e.