PQR THEORY

Now that we have the integers we can add, multiply or subtract any two of them and our result will always be another integer. But when we try to divide one by another, we often end up with a fraction rather than an integer. To accommodate these, we need to supplement the integers and include all possible fractions that can be made by dividing one integer by another. (Note that division by zero is forbidden.) This gives us the Rational numbers, so called because each one is the ratio of two integers.

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The Rational numbers include all possible fractions like ½, -2¾, or 355/113. Note that the Integers are included in the Rational numbers (e.g. 6/3=2), and that there are many ways of obtaining each Rational number. For example, 1½ can be written as 3/2, or 6/4, or -300/-200, but these three divisions all result in the same Rational number.

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We can regard the Rational numbers as filling in the gaps between the integers on the line drawn above, just like the markings on a ruler would do if they could show every fraction of an inch. Now, at last, we can add, subtract, divide or multiply any two Rational numbers (other than dividing by zero) and the answer will still always be a Rational number.

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But the Rational numbers are still not enough to answer all problems in mathematics. For instance, the ancient Greeks wondered what number, when multiplied by itself, would give the answer 2. (We call this number √2, the square root of 2.) They could never solve this problem exactly using the Rational numbers, although they could get very close. For example, 140/99 turns out to be a little bit too small, and 99/70 is a little bit too big. If you take the average of these two Rational numbers, you get 19601/13860 which is very close, but still about one part in a billion too big. In fact, as the ancient Greeks showed, there is no Rational number whose square is 2. You can find lots of them, as close to √2 as you want, but they will never get you there exactly.

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Nor can you do this using decimals, since decimals (which express numbers in tenths, hundredths, thousandths and so on) can by definition only represent Rational numbers. (This applies to repeating decimals as well as to those that terminate.) Using decimals, our modern computers calculate √2 to be 1.414213562373095048801688724209..., but this is still only an approximation, even though a highly accurate one. The dots to the right of the decimal mean that a computer program could, in principle, go on producing more and more digits indefinitely. Of course in practice, the program would have to stop sooner or later. But the digits don’t. Some computer in Japan with nothing better to do has calculated the first two hundred billion digits of √2 (which can’t leave its hard drive with much room for useful information).

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So it seems that the Rational numbers do not completely fill the gaps in the number line illustrated above. Although √2 appears to belong at a specific point on this line, we cannot actually find it among the Rational numbers. You can approach it from above or below, getting as close as you like, but you will never quite get there using only Rational numbers. There are many other mathematical quantities (such as π, the ratio of a circle’s circumference to its diameter, and e, the base of natural logarithms) which also “slip between the cracks” of the Rational numbers in this way. Mathematicians overcome this difficulty by referring to these infinitesimal gaps in the Rational number line as “Irrational numbers,” thus neatly defining their problem out of existence. (Whether right or wrong, this is probably better than the pragmatic approach used by the ancient Greeks who, according to legend, got rid of this problem by throwing its discoverer off a ship.)

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When we supplement the Rational numbers with the Irrational numbers, the combined set is referred to as the Real numbers. These can be regarded as spread out along the number line illustrated above, which is now continuous in the sense that it has no gaps, not even infinitesimally small ones: everywhere you look, you will find a Real number.

The concept of this continuous real-number line is fundamental to mathematics. Yet I believe that in constructing it, mathematicians have made an assumption which has no validity in our physical Universe and that as a result, Real numbers have no particular physical significance.

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I will examine this assumption and its consequences further in Part II of this article.

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Note: In this article I have capitalized certain adjectives (like Real and Natural) to emphasize that these are names for the various types of numbers, rather than actually describing these numbers. Mathematicians study many other types of numbers, such as Complex and Algebraic numbers, which I have not mentioned here because it is not necessary to consider them now, and this article is quite long enough already.