Rational Mathematics

by G. P. Jelliss

The Study of Numbers

By numbers we mean those words or symbols used to describe the sizes of sets. Over the years, mathematicians have allowed the meaning of the term ‘number’ to drift, to the extent that now almost any system of concepts that have properties similar to the original counting numbers are considered a type of number. In this study, following our morphological principle, numbers are defined as those concepts that describe the structure of the simplest systems, namely sets. A set is a concept defined in terms of other concepts, known as its members. Two sets have the same number of members, or are of the same size, if they can be placed into one-to-one correspondence.

The study of numbers should logically be termed numerology. It is unfortunate that this name has been misapplied to the use of numbers in fortune telling, for which the proper term should be numeromancy, or to the metaphysical interpretation of numbers, better termed arithmosophy, so that mathematicians, wishing to disassociate themselves from these practices, tend to use clumsier phrases such as theory of numbers or higher arithmetic for this branch of study. We propose to reinstate this term to its rightful domain. These comments should not be taken to imply however that numbers have no romance or mysticism properly associated with them.

The process of finding the number of elements in a set is called enumeration, and can take various forms, which may be simple or complicated. An enumeration problem may not just be a matter of calculating the total number of things of a given type, but may also involve listing them or actually constructing them in a systematic manner that ensures that all are included, none omitted and none counted twice. Such a process or algorithm may in some special cases provide a recurrence for calculating the successive totals, and in even more special cases may lead to an explicit formula for the total, but many cases are not of this helpful type. The proof that a given number is the answer to a question may be a matter of simple logic, may involve complex reasoning, or may require extensive historical or scientific research.

In solving enumeration problems the same formulae often appear in different contexts, indicating an underlying similarity in the structure of the problems. Often we do not recognise two problems as being the same, simply because we express them in different terms. In the following sections we bring together basic results of this type, and define various useful notations.

Numbers: Their Names and Symbols

The earliest representation of numbers was in physical form as sets of pebbles or sticks or fingers, each of these counters representing one element in the set being evaluated. The method of comparison in which the elements of a set are matched one by one with counters is counting. The word calculate, meaning to manipulate numbers, is derived from the Latin calculus meaning a small stone used for counting.

From the representation of numbers by sets of counters, it is a small step to the written tally method, whereby the set is represented by a group of repeated identical signs within a designated space, the tally marks representing the pebbles or sticks used for counting. Thus, for example, using round brackets, (, and ), for spacing and a vertical stroke, |, as the tally mark, we get the following symbols for the numbers in small sets: (), (|), (||), (|||), (||||), (|||||), and so on.

If the grouping of the tally marks is shown simply by spacing them, this results in there being no clear symbol for the number of elements in a null set. As a consequence of this the size of a null set, termed zero (or nought), and denoted above by (), was often omitted from the list of numbers.

Sometimes, like zero, the number one, (or unity) denoted above by (|), was also not counted as a number, on the grounds that a ‘number of things’ ought to be plural.

There is a natural temptation to arrange the tally marks in geometrical patterns. This is most familiar these days from the spots on dominos and playing cards. The zero on dominoes is represented by a blank square.

As the sizes of the tally symbols increase they become more difficult to distinguish from one another. A common way of making them easier to read is to cross them through in groups. Another way, that adopted by the ancient Romans is to introduce a new sign for a particular group of unit signs. They used I as the unit symbol and V for the group IIIII which gives the system I, II, III, IIII, V, VI, VII, VIII, VIIII, VV, and so on. This is a simplified version of Roman numerals. The Romans and their successors introduced further signs: X for VV, L for XXXXX, C for LL, D for CCCCC, M for DD, and adopted other conventions, such as that a smaller symbol placed before a larger one indicates removal of elements from the set instead of addition of further elements to it. Thus the fully developed Roman system runs: I, II, III, IV, V, VI, VII, VIII, IX, X, XI, XII, XIII, XIV, XV, XVI, XVII, XVIII, XIX, XX, XXI, XXII, and so on. This scheme is still employed occasionally for special purposes, such as on clock-faces or for dates, especially on memorials.

Another simple way of representing numbers is to have a standard sequence of arbitrarily chosen verbal names or visual signs, such as the letters of the alphabet, and to take these as representing the successive numbers. This type of system was used, for example, by the ancient Phoenicians, Jews and Greeks, and we still use it for labelling lists, (a), (b), (c) ... or A, B, C, ... though not for doing calculations.

Individual non-systematic names for the first few numbers in English are zero, one, two, three, four, five, six, seven, eight, nine, ten, eleven, twelve. We could, if we so wished, continue to invent individual names for as far as we wish to count, but it is more economical, and practical for performing calculations, to name the higher numbers in a systematic manner, such as that derived from the positional numeration described below.

We have also acquired the separate specialised sequence of numerals (to which we give the same names as the numbers they represent): 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, derived originally from usage in India, and transmitted via Arab culture to Europe around the tenth century. Various symbols have been proposed to extend the series to ten and eleven, such as the inverted 2 and 3 (resembling script T and E) proposed by Isaac Pitman in the 19th century. The use of the hexadecimal system in computing is beginning to popularise the straight-forward, if unimaginative, alphabetical sequence a, b, c, d, e, f, for the digits following nine.

When the numbers are used for labelling successive items in a sequence slightly different ordinal (as opposed to cardinal) forms of the number names tend to be used. We call the item labelled 1 the first, 2 the second, 3 the third, 4 the fourth, 5 the fifth, 6 the sixth, 7 the seventh, 8 the eighth, 9 the ninth, and so on, and generally n the nth. When counting things, people often say the names of the numbers out loud, beginning with one, not zero, this results in the last number stated giving the number of objects in the set. This is why lists usually begin with 1, whereas the list of numbers begins with 0. If 0 is used as a label it usually marks the datum from which counting starts. For instance in our calendar there is no year zero, but there is a zero point of time marking the start of the first year. One sometimes encounters a book with a zeroth chapter, which contains all data needed before trying to understand the rest of the book. It was formerly the custom to list the digits in the sequence 1, 2, 3, 4, 5, 6, 7, 8, 9, 0, with zero occurring last. The last year in a decade is the 10th, the last year in a century is the 100th, and the last year in a millennium the 1000th; though many people recently treated the year 2000 as the start of a new millenium instead of the end of one. Perhaps, as a compromise, zero should be seen as marking a transition, rather than an end or beginning.

Positional Numeration

When counting large numbers with pebbles we are liable to run out of counters, but we can reach higher numbers by agreeing that when we have a heap of a certain size, n, we will remove it and represent it by a single pebble alongside, to the left, starting a second heap, and, when we have filled this second heap with n pebbles we empty its space and start a third heap. This is the origin of the method of positional numeration with base n.

Translating the heaps of pebbles into symbols 0, 1, ..., m (where m = n−1) we arrive at the system 0, 1, ..., m, 10, 11, ..., 1m, ... 100, 101, ..., 10m, and so on. The system with base n uses only n symbols, representing the numbers 0 to n−1, and these symbols are called digits.

The principle of positional numeration was known to the ancient Babylonians, though their system was somewhat hybrid in that the base they used was sixty and to denote the numbers up to sixty they employed a tally system based on grouping in tens. The base ten has come to be accepted as standard, though for no particularly good reasons. It seems to have been adopted purely because humans are endowed with ten bony appendages at the ends of their arms which are convenient for counting. Reformers have tried to get the system changed to another base, the most popular being 2, 6, 8, twelve or sixteen. Here we conform to custom and use the base ten (denary) system for our numbers. The symbols for our numbers thus follow the sequence: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, and so on.

Instead of counting in tens any number greater than one can be used as the base for a system of positional numeration. If we are writing a lot of numbers in the same base then it is clear what the base is from the largest digit that appears in the numbers, since every digit is statistically likely to appear in a sufficiently large sample. However, if we are using two different bases it is advisable to state the base in use each time. A notation for this is to put the base as a subscript at the end of the numerical expression, expressed in some agreed standard system; we will use base ten. Thus, for example, the number of playing cards in a standard pack, expressed in various different bases is: 1101002 = 12213 = 1246 = 52 = 4412 = 3616 in bases two, three, six, ten, twelve and sixteen.

The names given to the numbers after twelve start to get more systematic: thirteen, fourteen, fifteen, sixteen, seventeen, eighteen, nineteen where the suffix ‘teen’ means add ten; then we get twenty, twenty-one, ... twenty-nine, thirty, thirty-one ... thirty-nine, forty, ... fifty, ... sixty, ... seventy, ... eighty, ..., ninety, ... but instead of ‘tenty’ we have one hundred, one hundred and one and so on (in American usage the 'and' is often left out).