A Number for your Thoughts

Facts and Speculations About Numbers from Euclid to the Latest Computers

M. E. Lines

• Publisher: CRC Press
• Year: 2020
• eBook ISBN: 9781000154245
• Edition: 1
• Topic: Mathematics
• Subtopic: Abstract Algebra
• Index: Mathematics

Why do we count the way we do? What is a prime number or a friendly, perfect, or weird one? How many are there and who has found the largest yet known? What is the Baffling Law of Benford and can you really believe it? Do most numbers you meet in every day life really begin with a 1, 2, or 3? What is so special about 6174? Can cubes, as well as squares, be magic? What secrets lie hidden in decimals? How do we count the infinite, and is one infinity really larger than another? These and many other fascinating questions about the familiar 1, 2, and 3 are collected in this adventure into the world of numbers. Both entertaining and informative, A Number for Your Thoughts: Facts and Speculations about Numbers from Euclid to the Latest Computers contains a collection of the most interesting facts and speculations about numbers from the time of Euclid to the most recent computer research. Requiring little or no prior knowledge of mathematics, the book takes the reader from the origins of counting to number problems that have baffled the world's greatest experts for centuries, and from the simplest notions of elementary number properties all the way to counting the infinite.

COUNTING

1. COUNTING

From prehistoric times man has had the need to count. The stone-age hunter or hunting scout would doubtless have found it of great use to be able to give his hunting colleagues some indication of the number of animals he had located, in addition to their kind and approximate location. Although terms such as one, few, and many, may well have sufficed for a while, a more precise counting scheme would be needed eventually, perhaps for bartering, and some concept of number does seem to be possessed by even the most primitive tribes today. Counting, of course, can be performed without the verbal possession of number words. This can be achieved, for example, by placing the objects to be counted in a one-to-one correspondence with fingers, toes, or ‘counting stones’, but words for the most commonly occurring numbers (usually the smallest) are obviously convenient, and seem to have developed at quite an early stage in all forms of human society.

In order to proceed to large numbers in the counting process it soon becomes clear that some grouping arrangement is highly desirable. Thus, the number twenty three is much more conveniently recorded by two ‘marking-stones’ designating tens, and three perhaps smaller ones designating units, than by twenty three separate ‘unit’ stones. The grouping number is, in modern usage, referred to as the base of the counting system. It need not be equal to ten, of course, and systems based on five, twenty, and even sixty, have occurred in other cultures. Indeed, remnants of such systems are still with us today in the measurement of time (hours, minutes, and seconds - with base sixty) and in the words dozen (base twelve) and score (base twenty). It is even possible to evolve multi-base counting systems - the Mayans used one - and such systems abound among the English units of measurement which answer the question of ‘how much?’ rather than ‘how many?’. Some readers may still recall the tribulations of working out the old English money system of twelve pence in one shilling and twenty shillings in one pound before it was mercifully decimalized in the early 1970’s. Others may be more familiar with at least most of the units of weight such as tons, hundred-weights, stones, pounds, and ounces, but possibly without having a precise recollection of how many of these happen to be in one of those.

Doing arithmetic in these multi-base counting systems can be quite tricky, although familiarity helps to a surprising degree, and when the English currency was finally decimalized, many of the older generation found the new simplified system quite confusing and continued to convert everything back to the old multi-base pounds, shillings, and pence before deciding on the advisability of a particular purchase. Fundamentally, however, the single-base system is the simplest, and such a system with base ten is used almost universally for counting today. For this reason the present book can very largely be restricted to this system alone. It is referred to as the decimal system of counting, and the choice of the number ten presumably arose from counting on the fingers, with the word ‘digit’ for any numeral between 0 and 9 seemingly attesting to this fact.

Numbers are an abstract concept and have no physical form. I cannot therefore write down the number 5. But you just did, I hear you say. Well, not really - I wrote down a particular mark (called a numeral) to represent it. Had I been a Roman I should have written V. There is fortunately nothing absolute about any one representation, and the fundamental properties of numbers are not at all dependent on the notation used. The fact that 5 is not exactly divisible by 2 is still true if I think of it as V divided by II. It is also just as true if I use a different base or grouping terminology. This means that if these properties can be demonstrated in our own familiar base-ten system, then we do not have to worry further about verifying them in other counting systems.

Counting in groups of ten, with the symbols 0,1,2,3,4,5,6,7,8,9, is so ingrained in us that the fact that it is quite an arbitrary choice comes almost as a surprise. That one can count quite happily in systems with other bases, right down to ‘base-two’, which uses only the two symbols 0 and 1, is something of an alien concept to most of us. After all, counting in groups of ten (or to the base ten as we should more formally say) has been a common procedure in many civilizations since the early Egyptians at least. On the other hand, the precise system which we use today, and so take for granted, is of much more recent origin. It contains within it one of the most important inventions ever made, a property which all earlier counting systems, even those using the base ten, did not have. Once again familiarity breeds contempt, and you may well be wondering what attribute of our simple counting system, which somehow seems so natural, could possibly deserve such an accolade. Wouldn’t anyone, you may feel, who chose to count in tens, proceed roughly as we do with possibly different symbols for the numerals? The answer is almost certainly not.

The most natural way to count in groups of ten is first to choose symbols for the first nine digits, and then to choose other symbols to represent ten, twenty, etc., up to ninety, and still others for one hundred, two hundred, and so on. Roman numerals, with which we are all acquainted tc some degree, are just such an example. They are based in groups of ten as follows:

Units: I, II, III, IV, V, VI, VII, VIII, IX

Tens: X, XX, XXX, XL, L, LX, LXX, LXXX, XC

Hundreds: C, CC, CCC, CD, D, DC, DCC, DCCC, CM

Thousands: M, MM, MMM, MMMM, MMMMM, and so on.

Thus the number 8888 becomes MMMMMMMMDCCCLXXXVIII in Roman numerals, and a routine shopping list for our everyday Roman Centurian might look something like this:

III pairs sandals	III x VII	equals	XXI	den.
IV tunics	IV x IX	equals	XXXVI	den.
I ceremonial toga	I x XL	equals	XL	den.
II plumed helmets	II x XVIII	equals	XXXVI	den.
I sword (regular)	I x XXVIII	equals	XXVIII	den.
I shield	I x XXXIX	equals	XXXIX	den.
	TOTAL	equals	CC	den.

where den. stands for denarius, which was a Roman unit of currency.

Now I do not known whether the cost of living during any period of the Roman empire was such as to make these values realistic, but it is quite possible since rampant inflation was as much a part of Roman lives as it is of our own. The point which we are trying to make does not depend on this of course; it is that checking this shopping list does seem to be a bit tricky without converting it to more familiar numbers. Similarly, multiplying that Roman equivalent of 8888 set out above by say IX or XI seems to be even more difficult. Are these difficulties due only to our lack of familiarity with the system, or is it more than that? Well, our lack of familiarity is certainly no help, but there is indeed a fundamental difficulty with the Roman system over and above this. We should probably sense it first in this way; there are no units columns and no tens columns. Now it is true that methods can be devised for putting the Roman addition and multiplication into some kind of column format (although there is no evidence that the Romans actually did this) which, when combined with special rules for transfering from some columns to others, enable these tasks to be carried out for relatively small numbers. For large numbers, however, the situation is so bad that to represent a million, let alone multiply it by anything, a Roman would have to fill several pages of this book entirely with M’s. It is true that more and more letters could be introduced to represent larger and larger groups of ten, but then the system itself rapidly gets out of hand in any case.

The arithmetic associated with these kinds of problems is eased a little bit if we have just one symbol for each numeral between 1 and 9, and then use a separate set of symbols to designate whether we are dealing with tens, hundreds, thousands, etc. For example, if we designate thousands by M, hundreds by C, and tens by X, to follow the Roman precedent, then the number 8888 referred to above would now appear as 8M8C8X8, or its equivalent with the numeral 8 replaced by whatever symbol might have been chosen to represent eight units. In fact, just such grouping systems as these were used at one time by the Chinese and the Japanese (with suitably oriental characters for the numerals and the group symbols of course). But with this system there are still significant problems remaining concerning the writing ...

Table of contents

• Cover
• Half Title
• Title Page
• Copyright Page
• Table of Contents
• 1. Counting.
• 2. The Search for Prime Numbers.
• 3. The World Record Holders.
• 4. The Distribution of Primes.
• 5. Prime Races, Emirps, and More.
• 6. The Baffling Law of Benford.
• 7. What is so Special about 6174?
• 8. Number Patterns and Symmetries.
• 9. Numbers Perfect, Friendly, and Weird.
• 10. How do These Series End?
• 11. Fermat’s Legendary Last Theorem.
• 12. Shapely Numbers and Mr. Waring.
• 13. Magic Squares and Cubes.
• 14. How can Anything so Simple be so Difficult?
• 15. Nearly All Numbers are Insane.
• 16. Cyclic Numbers and their Secret.
• 17. Pi, a Transcendental Number.
• 18. Most Numbers are Normal, but it’s Tough to Find One.
• 19. A Different Way of Counting; Geometric Numbers.
• 20. Two Dimensional Numbers.
• 21. Counting the Infinite.
• 22. Update (September 1985).
• Appendices
• Bibliography
• Index