As long as we are content simply with counting, numbers are practically just identification marks, like “Oakdene” or “Mon Abri,” attached to objects in a row. But we can do things with sets of objects, merge them together, or break them up into parts. This is the origin of arithmetic.
Two sets of objects are said to be added when we think of the set of objects belonging to either one or the other of the given sets. This new set is called their sum. For example, the set of my limbs is the sum of the set of my arms and the set of my legs. The numbers of these sets of objects undergo a corresponding addition, which we denote by the sign + (plus). I have two arms and two legs, and four limbs. This (and all other similar additions) is represented by the formula 2 + 2 = 4. Here the sign =, read as “equals,” means that the process of calculation indicated on the left-hand side gives the same result as the process of calculation indicated on the right-hand side, or that the result is the number written on the right-hand side.
It is a common experience that it does not make any difference in what order we add sets of objects. This is reflected in formulae such as 2 + 3 = 3 + 2. That such a rule always holds is called the commutative law of addition.
Another important rule is called the associative law; this is indicated by the formula
(2 + 3) + 1 = 2 + (3 + 1).
The bracket notation means that whatever process is indicated inside the brackets must be done first, and that then the brackets may be removed. Thus the lefthand side means that we are to add 3 to 2, and then 1 to the result. The right-hand side means that we are to add 1 to 3, and then the result is to be added to 2. Both the processes lead to the number 6, and that the results are the same is just what the whole formula means.
Rules such as the commutative law are laws which we command our numbers to obey, so that they shall represent certain processes which are usually carried out with physical objects. We could command them to obey different laws, for example that 2 + 3 should not be equal to 3 + 2, but to something different. They would do this provided that the proposed laws were not inconsistent with each other, but then of course the whole system would mean something quite different from what it ordinarily does.
Next suppose, for example, that I have three sets each containing four members, and that I then count all the members straight through as one set. There are twelve in all. This is multiplication. It corresponds to the multiplication of numbers expressed by the formula 3 × 4 = 12. The number obtained by multiplying two other numbers is called their product. There are commutative and associative laws for multiplication, as examples of which we may give
2 × 3 = 3 × 2,
and (2 × 3) × 4 = 2 × (3 × 4).
There is also a “distributive law” relating addition and multiplication, exemplified by
3 × (2 + 1) = (3 × 2) + (3 × 1).
The symbol − is the inverse of +; 3 − 2 means a number which, when it has 2 added to it, gives 3; or subtraction can be defined in terms of the “logical subtraction” of sets. Similarly ÷, the symbol of division, is the inverse of ×; 4 ÷ 2 means a number which, when multiplied by 2, gives 4.
A negative number is just the same thing as an ordinary number, except that it carries round with it the sign −. Thus −1, −2, −3 denote negative numbers; − 2 means, count 2 and then, if there are ordinary numbers about too, subtract instead of adding.
This seems a simple enough idea to us, but it took a long time to get itself clear in people’s minds. This is probably because an ordinary number, say 2, at once calls up a vision in our minds of two objects; and − 2 seems to mean that two objects are doing something even less than not existing, a difficult situation to visualize. It is however quite unnecessary to perform this feat. It is perfectly easy to count and to give a rule of operation as well, and this is all that a negative number does.
One can of course think of negative numbers as representing liabilities, if ordinary numbers represent assets.
To distinguish them from negative numbers, the ordinary numbers are called positive numbers. It is usual to set out the whole array of positive and negative numbers, together with 0, as a row extending endlessly in both directions
… −3, −2, −1, 0, 1, 2, 3, …
The use of negative numbers enables us to attach a meaning to formulae such as 2 − 3, which otherwise we should have to avoid. It can now just mean − 1.
To multiply a negative number by a positive number, we first multiply without thought of sign, and then preserve the minus sign; thus 2 × (−3) = −6. To agree with the commutative law, we must also say that (− 3) × 2 = −6. But this implies that multiplication by a negative number is equivalent to multiplication, together with reversal of sign.
If this is to be true also when one negative number is multiplied by another, it must mean for example that − 3 multiplied by −2 is 6; that is, in multiplication, two minuses make a plus.
If I start pairing off my fingers against a shilling’s worth of pennies, I find that there are some of the latter left over. In such a case we say that the number of fingers is less than the number of pennies. The symbol < is used to mean “less than”; thus 10 < 12. The opposite to “less than” is “greater than”; and this is denoted by >; thus 12 > 10. We use ≤ to mean “less than or equal to.” Similarly ≥ means “greater than or equal to.”
These symbols are easy to remember, since the bigger number is always at the bigger end.
We bring negative numbers and 0 into this scheme by thinking of them in the order written down above. Thus we say that −3 is less than −2, and we write −3 < −2; similarly − 1 < 0, − 1 < 1, and so on.
Nature has provided us with ten objects, our fingers, with which to compare other sets of objects. No doubt this is the origin of the use of ten as a basic number in counting. This number was so used by most of the ancient races whose records have come down to us, the Egyptians, the Babylonians, the Greeks and the Romans.
The system of numbers which we all use today is as follows. The first nine numbers in order are denoted by the symbols 1, 2, 3, 4, 5, 6, 7, 8 and 9. Nought or zero is denoted by 0. These are known as the Arabic numerals. After 9 we do not have any new symbols, but the next number, ten, is denoted by 10, the symbol 1 being one place to the left. The next number is 11, then 12, and all numbers up to 99 can be formed in this way. After this the next number is 100, the symbol 1 being moved two places to the left to denote ten multiplied by ten. Similarly 1000 means ten multiplied by ten multiplied by ten, and so on. This gives us a simple way of writing down numbers indefinitely.
This system has the great advantage that it requires only ten symbols to express all numbers. When it has been learnt in the first few cases, it can easily be used to any extent. It is known as the scale of ten, or the decimal system.
The old notations for numbers seem very cumbrous compared with this. The ancient Greeks usually used letters for numbers. Thus for example A, B and T meant 1, 2 and 3; I, K and ∧ meant 10, 20 and 30, and P, ∑ and T meant 100, 200 and 300. Thus 321 would be TKA. This is quite a convenient system for fairly small numbers, but it has to be elaborated very much to express large numbers, and it gives no idea how to go on indefinitely. A special system was invented by Archimedes in order to show Gelon, King of Syracuse, that it was not beyond the power of language to express the number of grains of sand with which the universe, on a reasonable estimate, could be filled. A very interesting account of these ancient systems of numbers is given by T. L. Heath, A Manual of Greek Mathematics (Oxford, 1931).
The Roman numerals, with I as the first number, V for 5, × for 10, C for 100, D for 500, and M for 1000, are well known, and are still sometimes used.
In these old methods of counting, though ten was a specially important number, no use was made of position to denote multiplication by ten, in the sense in which we use it now. The only exception to this seems to have been a notation used by the ancient Babylonians. They had a system of signs in which ∨ meant 1 and < meant 10; thus 
meant 26, and 
meant 40. But 
meant 26 × 60 + 40, the displacement of the symbol for 26 to the left meaning that it was to be multiplied by 60. This is equivalent to counting in the scale of 60. It does not seem to be known why they attached such special importance to the number 60.
A relic of this system has come down to us in our method of measuring angles, and of counting time. This has come via the Greeks, who copied the Babylonian system of astronomy. We divide a complete turn into 360 parts, called degrees; each degree is divided into 60 parts, called minutes; and each minute into 60 parts called seconds (i.e., second-sixtieths). A similar system is of course used for time.
Ten is a very convenient number for the scale, somehow not too large or too small. It has two factors, two and five, which makes it easy to see when any number is divisible by these numbers. But other numbers, such as twelve, would do equally well. We English have a sort of traditional affection for the number twelve (twelve inches make a foot, twelve pence make a shilling). In this country the scale of ten is used for counting but not for measuring, so that problems about money, weights and so on involve endless tiresome arithmetic. In other countries this has been avoided by the adoption of the metric system of measurement, and similar systems for weights and other things.
Any number greater than one could be used as the basic number of the scale. In a way, the simplest choice would be the number two. In the scale of two, the only symbols which we should have to use would be 0 and 1. These would have the same meanings as before. The number two is then denoted by 10, the displacement of 1 one place to the left meaning, in this scale, that it is multiplied by two. Three is 11 (i.e. two and one). Four is 100 (1 displaced twice to the left is multiplied by two twice). Similarly the remaining numbers up to twelve are 101, 110, 111, 1000, 1001, 1010, 1011, 1100. The disadvantage of this system is that it takes so many figures to represent moderately large numbers.
When we multiply two numbers together, we obtain another number; for example, 2 × 3 = 6. It is then natural to ask about the opposite process. If we are given a number, can it be obtained by multiplying two other numbers? And if so, what are they? It is soon found that there are various possible answers to this question. The number 6 is the product of 2 and 3; the number 12 is the product of 2 and 6, and it is also the product of 3 and 4. On the other hand, the number 5 is not the product of any two smaller numbers. It is only the product of 1 × 5, which is not at all interesting.
If a number can be expressed as the product of two other smaller numbers, these are called factors of it; thus 2 and 3 are factors of 6, and 3 and 4 of 12. We can find out whether a number is a factor of another number by dividing and seeing whether there is a remainder left. Thus on dividing 7 into 45, we take 7’s away from 45 as often as possible, that is 6 times, and then 3 is left. This amounts to expressing 45 as (6 × 7) + 3. Clearly 7 is not a factor of 45; but it is a factor of 42, since nothing is left after the division. In fact we find that 42 = 6 × 7.
Sometimes one of the factors of a number has factors of its own; thus 12 = 3 × 4, and 4 = 2 × 2. In this case we can express the original number as a product of more than two factors; thus 12 = 3 × 2 × 2. There may even be several factors; thus 60 = 2 × 2 × 3 × 5.
One thing which emerges from all this is the special position occupied by those numbers which have no factors, such as 2, 3, 5 and 7. These are called prime numbers. It is easy to write down a great many such numbers. The next few are 11, 13, 17, 19, 23 and 29.
Now in some of the above cases in which we have expressed a number in factors, these factors are prime numbers. In such a case we say that the number is expressed as a product of prime factors. For example 6 = 2 × 3 and 21 = 3 × 7 are such expressions. In other cases, such as 42 = 6 × 7, one or more of the factors is not prime (here 6 is not prime). But now we can write 6 = 2 × 3,andso42 = 2 × 3 × 7. We have thus expressed 42 as the product of three prime factors.
These examples suggest that any number whatever can be expressed as a product of prime factors. The reader should try a few examples, and will find that it always works. Another example is the above factorization of the number 60. But there is more in it than this. In all simple cases such as we have considered, there is only one way of expressing any given number as a product of prime factors, apart from trivial variations like changing the order of the factors. For example 60 is equal to 6 × 10, and also to 4 × IS; writing 6 = 2 × 3, 10 = 2 × 5, we obtain 60 = 2 × 3 × 2 × 5, and writing 4 = 2 × 2 and 15 = 3 × 5, we obtain 60 = 2 × 2 × 3 × 5. So the result is the same (apart from the order in which the factors are multiplied) in whichever way we do it.
These calculations suggest strongly the following result: a number can be expressed as a product of prime factors in one way only. This is true, and not so very difficult to prove, but it is not quite obvious. We are so used to thinking of a number as the product of its prime factors that we are apt to regard the result as quite obvious. In trying to prove it, the main difficulty is to avoid assuming the truth of something equivalent to what is to be proved. For example, is it obvious that the product of two primes cannot be equal to the product of two different primes? However, we must leave the matter to books which deal with this subject.
The factorization of fairly small numbers is quite easy. If a very large number is written down at random, or expressed by means of some formula, it is often very hard to find out whether it has any factors or not. In a sense there is no difficulty about it, because one would merely have to divide by all smaller numbers to see whether any of them left no remainder. It is just that this would take so long to do, that (if the number were very large) one could not finish it in a lifetime. The invention of electronic computers has made it possible to factorize large numbers far more quickly. Even so, the numbers which we can factorize remain limited.
There are some simple rules for factors. If the figure representing units is even, 2 is a factor. If all the separate figures added together give a number divisible by 3, then the original number is divisible by 3. There are a few more rules of this kind, but they do not take us very far. The general problem of finding the factors of very large numbers is unsolved.
The prime numbers can be regarded as the raw material out of which all other numbers are made up. All numbers can be expressed as products of primes. Let us then think how to find the primes, and how many of them there are.
A simple method of finding the prime numbers, used since ancient times, is called the sieve of Eratosthenes. Suppose for example that we want to find all primes less than 100. Write out all numbers from 2 to 100 in a row. Now 2 is a prime, but no other even number is, since they are all divisible by 2. So cross out all the even numbers (4, 6,8, …) after 2.
The first number not crossed out is 3, which is then a prime. Now cross out all subsequent multiples of 3 (6, 9, …). Some numbers, such as 6, will have been crossed out twice, but this does not matter.
Proceeding in this way, we come to the primes 5 and 7, and cross out multiples of them. All other numbers up to 10, and a great many others, are crossed out. Now look at the numbers not crossed out. They are all the primes up to 100.
Why are they primes? Because if a number less than 100 has two factors, they obviously cannot both be greater than, or equal to, 10. So one at least must be less than 10. But we have crossed out all numbers with prime factors less than 10, and so all numbers with any factors less than 10. Hence the numbers left are primes.
This “sieve” method, which can of course be used up to any limit, is a good wholesale method of finding prime numbers. But it does not tell us much about how many of them there are.
It was proved by Euclid, or some mathematician of his period, that the sequence of primes is endless; or that, as we say, there are an infinity of primes. In other words, however far we go along the sequence of the numbers, there are always more prime numbers beyond. The proof of this is very ingenious but it is really quite simple, once you have thought of it (or Euclid has thought of it for you).
Let us take a fair-sized prime number, for example 97 (this is the greatest prime revealed by the sieve for primes less than 100). We want to prove the existence of at least one still larger prime. To do this, consider the number
(2 × 3 × 5 × 7 × … × 97) + 1,
which is the product of all the primes up to 97, plus 1. Now this number itself may be a prime. I do not know whether it is or not, but, if it is, the fact that it is proves at once what is required. It is an example of a prime greater than 97. Next suppose on the other hand that the number written above is not a prime. Then it has prime factors. But none of these prime factors can be the same as any of the primes 2, 3, … up to 97; for on dividing by any such number we obviously get the remainder 1 (this of course was why we added on the 1). Hence any of the prime factors must be greater than 97, and this again proves the existence of primes greater than 97.
Of course we have merely taken 97 as an example. The argument does not depend in any way on this particular prime number, and could be applied equally well to any other. We must therefore conclude that Euclid’s theorem is true.
In modern times mathematicians have spent a great deal of time and energy in investigating the distribution of prime numbers. Such problems are usually very difficult, and a great part of their interest lies in the ingenious methods which are required to solve them. I will mention only one such problem. A glance down a table of prime numbers shows that they often go in pairs, separated only by a single even number. For example, 11,13; 17,19; 59, 61; are such prime-pairs. It is an obvious suggestion that such pairs, like the primes themselves, form an endless succession. But no method is known of showing whether this is true or not, and it remains a completely unsolved problem.
By the square of a number we mean the product of the number by itself; and we denote it by affixing a little 2 to the top right-hand corner of the number. Thus 1 × 1 = l2 = 1, 2 × 2 = 22 = 4, 3 × 3 = 32 = 9, are the squares of 1, 2 and 3. By the cube, we mean a number multiplied by itself, and then by itself again; for this we use a little 3; thus: 1 × 1 × 1 = l3 = 1, 2 × 2 × 2 = 23 = 8, 3 × 3 × 3 = 33 = 27 are the cubes of 1, 2 and 3.
The square of a number is also called the second power of the number, and the cube is called the third power. Similarly the product of four equal factors is called the fourth power, and so on. The notation for fourth and fifth powers is quite similar to that for squares and cubes; for example we write
2 × 2 × 2 × 2 = 24 = 16,
3 × 3 × 3 × 3 × 3 = 35 = 243
and so forth.
The little number in the top right-hand corner, which says how many factors are to be multiplied, is called an index.
The introduction of indices gives us an opportunity of illustrating the difficulty of determining the factors of a given number. Fermat, a famous French mathematician of the 17th century, conjectured that all the numbers 22 + 1, 24 + 1, 28 + 1, 216 + 1, 232 + 1, and so on, are prime numbers. Here each index is twice the one before, and the numbers increase very rapidly as we proceed. The first three are equal to 5, 17, and 257, and these are all prime numbers; and the next one is also a prime number. Fermat’s conjecture obviously rested on these facts, but it was not a very good idea, because the next two such numbers are not primes. Euler, a great German mathematician of the 18th century, proved that
232 + 1 = 641 × 6700417,
and in the 19th century it was proved also that
264 + 1 = 274177 × 67280421310721.
The labour involved in doing arithmetic on this scale can be imagined. In recent times still more of Fermat’s numbers have been factorized.
Very large numbers can be expressed by writing a number with a moderately large index. According to Archimedes, the number of grains of sand which the universe could contain would not exceed 1063; in the decimal notation this is 1 followed by sixty-three 0’s. The idea of calculating the number of particles in the universe has been revived recently. According to Eddington, this number is 2 × 136 × 2256. Written out in full, this would be a number of eighty figures.
In this chapter we have been able to illustrate the general course which mathematical writings take. They begin with certain axioms or primary assumptions, which are supposed to be agreed upon between the writer and the reader. Such for example are the laws governing addition and multiplication, and the axiom of infinity. These are, so to speak, the rules of the game.
The object of mathematics is to prove theorems, that is, particularly striking and important results which follow from the axioms. For example, the statement that “a number can be factorized in one way only,” and “the set of prime numbers is endless” are important theorems.
These theorems are derived from the axioms by means of proofs; a proof is a chain of reasoning, each link of which should be obviously valid according to the axioms, but the final result of which may be far from obvious. Many of the proofs in mathematics are very long and intricate. Others, though not long, are very ingeniously constructed. Some theorems are capable of being proved in several different ways.
The proof that the set of prime numbers is endless is a good example. It is not long, but it involves an ingenious idea which catches the attention of anyone capable of being interested in mathematics.
In many cases, mathematicians first guess theorems, and afterwards supply the proofs. Such a guess is called a hypothesis. Naturally it requires much experience of mathematics to be able to put forward reasonable hypotheses. The power to make hypotheses which are both interesting and reasonable is a sign of mathematical originality. It leads to many advances in mathematics. As examples of hypotheses, we may take the statements “the numbers 22 + 1, 24 + 1, … are all primes,” and “the number of prime-pairs is infinite.” The former, due to Fermat, turned out to be false. The latter has never been either proved or disproved.