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Number systems are odd
M ATHEMATICS is often presented as if it is independent of us. It isn’t. It is a free construction of the human mind, created for some specific use.
It may possess a structure beyond what is factored in — just as a poem might have facets that the poet never thought of. One might study that structure for its own sake.
Some areas of mathematics are so simple that no alternative can be imagined and these would, presumably, be developed by all intelligent species. Among the first systems to be developed in mathematics were the number systems. The numbers that are used for counting represent the simplest such system.
It appears that even animals have a sense of such a system. Animals also seem to have a sense of distance. From this, however, we cannot conclude that arithmetic and geometry exist in themselves, independently of humans. An important feature of counting numbers is that they give the same result, no matter in which order they are added. This structure was not inserted for the purpose of counting but is a property that these numbers have.
One can construct number systems, as needed in some applications, that do not have this property. However, simple number systems, like counting numbers, cannot have much extra structures. More complicated number systems, however, can and do.
There can be situations where simple counting will not do. If a room uses 10 tiles across the floor, counting is fine. But what if 10 are not enough and 11 do not fit? One might find that 21 would fit two rooms. We now have the fractional number 10.5.
These numbers did not exist before the need for them arose. When this happened, the counting numbers were extended to fit general needs. For some applications even these numbers may not be general enough.
Think of two roads that are both a mile long and lie at right angles to each other. The straight path across from the start of one road to the end of the other is more than 1 but less than 2 miles. In fact, it is less than 1.5 miles.
It can be shown that there is no ratio of counting numbers that gives the exact distance. To deal with applications to such general measuring problems, one has to go beyond ratios and define new numbers. Though the two types of numbers seem similar, when their structures are studied it is found that they are quite different.
Numbers are used to deal with general arithmetical problems using symbols for given quantities and also those that have to be determined. For the simplest problems, if the given numbers are counting numbers those that need to be determined may be ratios, in general. However, if the given numbers are ratios so are those that are to be determined.
For slightly more complicated problems, the quantities to be determined appear in the problem, multiplied by themselves. In this case, the ratio numbers are not adequate to solve the problem and we have to proceed further.
To some extent the extended numbers that dealt with the right-angled triangle may be adequate. However, more generally one needs a number such that when it is multiplied by itself the result is the negative of 1. Since the product of two positive numbers is positive and of two negative numbers is also positive, the previously defined numbers do not contain such a number.
Therefore, one constructs a new number that serves the purpose. Such numbers will solve not only these problems, but also all algebraically formulated problems! It is this type of unexpected generality that leads to the (false) impression that numbers exist in themselves.
The fact remains that one had to go on constructing new number systems as the previous ones were proven inadequate, even though numbers were usable beyond the original purpose of their construction. Mathematicians construct other, still more complicated number systems, when a certain number system does not serve the purpose.
Let’s look now at the technical names given to various number systems. Counting numbers are called “Natural Numbers,” presumably because they arise so naturally. Fractions or ratios are called “Rational Numbers” for obvious reasons. Extensions for more general measurements are called “Real Numbers,” as against further generalization to numbers that give negative numbers when multiplied by themselves. The latter numbers are called “Imaginary Numbers,” a most unfortunate misnomer that leads to their being thought of as less real than the other number systems. And, compositions of real and imaginary numbers are called “Complex Numbers,” another unfortunate choice of name that leads to an anecdote.
When a friend of mine and I were invited abroad to attend an advanced course on Complex Analysis, the vice-chancellor of our university denied us the permission to travel, saying: “Pakistan does not need complex analysis, it needs simple analysis.”
The name of the event lent to the confusion, though it was supposed to refer to “a complex of Real and Imaginary Numbers.”
The writer, a distinguished national professor, is director of the Centre for Advanced Mathematics and Physics, National University of Sciences and Technology, Rawalpindi. His emails addresses are
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