Geometrically, the digits in a decimal representation can be viewed as being parts of an “address” of the number, with each successive digit locating it more and more accurately. Thus if you have the decimal 1.41421356237, the integer part tells you that the number is between 1 and 2. The first decimal place tells you that the number is between 1.4 and 1.5. The next place says that the number is between 1.41 and 1.42. The first decimal place specifies the number to within an interval of The second decimal place specifies the number to within an interval of length and so on.

If you think of it in this way, you can imagine applying this “address system” to any number, not just finite decimals. For finite decimals the procedure would effectively stop, with all digits beyond a given point being zero. With a number that is not a finite decimal, the process would go on forever, with each successive digit giving the number 10 times more precision. Thus, the finite decimals give you a systematic method for approximating any number to any desired accuracy. In particular, although the reciprocal of an integer will not usually be a finite decimal, you can approximate it by a finite decimal. Thus, is first located between 0 and 1, then between 0.3 and 0.4, then between 0.33 and 0.34, and so on.

But once you have started allowing approximation, there is no need or reason to restrict yourself to rational numbers. All numbers on the number line— even those that are not rational—can be approximated by finite decimals. For example, the number is approximately 1.41421. Expanding the rational number system to include all numbers on the number line brings you to the real number system. Finite decimals give you access to arbitrarily accurate approximate arithmetic for all real numbers. That is one reason for their ubiquitous use in calculators.

   

NOTE: The finite decimals, also called decimal fractions, were first discussed by Stevin, 1585/1959.

The potential of the number line does not stop at providing a simple way to picture all rational numbers geometrically. It also lets you form geometric models for the operations of arithmetic. These models are at the same time more visual and more sophisticated than most interpretations. Consider addition. We have already mentioned that one way to interpret addition of whole numbers is in terms of joining line segments. Now you can refine that interpretation by taking a standard segment of a given (positive) length to be the segment of that length with its left endpoint at the origin. Then the right



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