CALCULATION WITH INTEGERS - THE NATURAL NUMBERS - What Is Mathematics? An Elementary Approach to Ideas and Methods, 2nd Edition (1996)

What Is Mathematics? An Elementary Approach to Ideas and Methods, 2nd Edition (1996)

CHAPTER I. THE NATURAL NUMBERS

INTRODUCTION

Number is the basis of modern mathematics. But what is number? What does it mean to say that image, and (–1) (–1) = 1? We learn in school the mechanics of handling fractions and negative numbers, but for a real understanding of the number system we must go back to simpler elements. While the Greeks chose the geometrical concepts of point and line as the basis of their mathematics, it has become the modern guiding principle that all mathematical statements should be reducible ultimately to statements about the natural numbers, 1, 2, 3, · · ·. “God created the natural numbers; everything else is man’s handiwork.” In these words Leopold Kronecker (1823-1891) pointed out the safe ground on which the structure of mathematics can be built.

Created by the human mind to count the objects in various assemblages, numbers have no reference to the individual characteristics of the objects counted. The number six is an abstraction from all actual collections containing six things; it does not depend on any specific qualities of these things or on the symbols used. Only at a rather advanced stage of intellectual development does the abstract character of the idea of number become clear. To children, numbers always remain connected with tangible objects such as fingers or beads, and primitive languages display a concrete number sense by providing different sets of number words for different types of objects.

Fortunately, the mathematician as such need not be concerned with the philosophical nature of the transition from collections of concrete objects to the abstract number concept. We shall therefore accept the natural numbers as given, together with the two fundamental operations, addition and multiplication, by which they may be combined.

§1. CALCULATION WITH INTEGERS

1. Laws of Arithmetic

The mathematical theory of the natural numbers or positive integers is known as arithmetic. It is based on the fact that the addition and multiplication of integers are governed by certain laws. In order to state these laws in full generality we cannot use symbols like 1, 2, 3 which refer to specific integers. The statement

1 + 2 = 2 + 1

is only a particular instance of the general law that the sum of two integers is the same regardless of the order in which they are considered. Hence, when we wish to express the fact that a certain relation between integers is valid irrespective of the values of the particular integers involved, we shall denote integers symbolically by letters a, b, c, · · ·. With this agreement we may state five fundamental laws of arithmetic with which the reader is familiar:

1) a + b = b + a,

2) ab = ba,

3) a + (b + c) = (a + b) + c,

4) a(bc) = (ab)c,

5) a(b + c) = ab + ac.

The first two of these, the commutative laws of addition and multiplication, state that one may interchange the order of the elements involved in addition or multiplication. The third, the associative law of addition, states that addition of three numbers gives the same result whether we add to the first the sum of the second and third, or to the third the sum of the first and second. The fourth is the associative law of multiplication. The last, the distributive law, expresses the fact that to multiply a sum by an integer we may multiply each term of the sum by this integer and then add the products.

These laws of arithmetic are very simple, and may seem obvious. But they might not be applicable to entities other than integers. If a and b are symbols not for integers but for chemical substances, and if “addition” is used in a colloquial sense, it is evident that the commutative law will not always hold. For example, if sulphuric acid is added to water, a dilute solution is obtained, while the addition of water to pure sulphuric acid may result in disaster to the experimenter. Similar illustrations will show that in this type of chemical “arithmetic” the associative and distributive laws of addition may also fail. Thus one can imagine types of arithmetic in which one or more of the laws 1)-5) do not hold. Such systems have actually been studied in modern mathematics.

A concrete model for the abstract concept of integer will indicate the intuitive basis on which the laws 1)-5) rest. Instead of using the usual number symbols 1, 2, 3, etc., let us denote the integer that gives the number of objects in a given collection (say the collection of apples on a particular tree) by a set of dots placed in a rectangular box, one dot for each object. By operating with these boxes we may investigate the laws of the arithmetic of integers. To add two integers a and b, we place the corresponding boxes end to end and remove the partition.

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Fig. 1. Addition.

To multiply a and b, we arrange the dots in the two boxes in rows, and form a new box with a rows and b columns of dots. The rules 1)-5)

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Fig. 2. Multiplication.

will now be seen to correspond to intuitively obvious properties of these operations with boxes.

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Fig. 3. The Distributive Law.

On the basis of the definition of addition of two integers we may define the relation of inequality. Each of the equivalent statements, a < b (read, “a is less than b”) and b > a (read, “b is greater than a”), means that box b may be obtained from box a by the addition of a properly chosen third box c, so that b = a + c. When this is so we write

c = ba,

which defines the operation of subtraction.

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Fig. 4. Subtraction.

Addition and subtraction are said to be inverse operations, since if the addition of the integer d to the integer a is followed by the subtraction of the integer d, the result is the original integer a:

(a + d) – d = a.

It should be noted that the integer ba has been defined only when b > a. The interpretation of the symbol ba as a negative integer when b < a will be discussed later (p. 54 et seq.).

It is often convenient to use one of the notations, ba (read, “b is greater than or equal to a”) or ab (read, “a is less than or equal to b”), to express the denial of the statement, a > b. Thus, 2 ≥ 2, and 3 ≥ 2.

We may slightly extend the domain of positive integers, represented by boxes of dots, by introducing the integer zero, represented by a completely empty box. If we denote the empty box by the usual symbol 0, then, according to our definition of addition and multiplication,

a + 0 = a,

a.0 = 0,

for every integer a. For a + 0 denotes the addition of an empty box to the box a, while a.0 denotes a box with no columns; i.e. an empty box. It is then natural to extend the definition of subtraction by setting

a – a = 0

for every integer a. These are the characteristic arithmetical properties of zero.

Geometrical models like these boxes of dots, such as the ancient abacus, were widely used for numerical calculations until late in the middle ages, when they were slowly displaced by greatly superior symbolic methods based on the decimal system.

2. The Representation of Integers

We must carefully distinguish between an integer and the symbol, 5, V, · · ·, etc., used to represent it. In the decimal system the ten digit symbols, 0, 1, 2, 3, · · ·, 9, are used for zero and the first nine positive integers. A larger integer, such as “three hundred and seventy-two,” can be expressed in the form

300 + 70 + 2 = 3.102 + 7.10 + 2,

and is denoted in the decimal system by the symbol 372. Here the important point is that the meaning of the digit symbols 3, 7, 2 depends on their position in the units, tens, or hundreds place. With this “positional notation” we can represent any integer by using only the ten digit symbols in various combinations. The general rule is to express an integer in the form illustrated by

z = a. 103 + b. 102 + c. 10 + d,

where the digits a, b, c, d are integers from zero to nine. The integer z is then represented by the abbreviated symbol

abcd.

We note in passing that the coefficients d, c, b, a are the remainders left after successive divisions of z by 10. Thus

image

The particular expression given above for z can only represent integers less than ten thousand, since larger integers will require five or more digit symbols. If z is an integer between ten thousand and one hundred thousand, we can express it in the form

z = a. 104 + b. 103 + c. 102 + d. 10 + e,

and represent it by the symbol abcde. A similar statement holds for integers between one hundred thousand and one million, etc. It is very useful to have a way of indicating the result in perfect generality by a single formula. We may do this if we denote the different coefficients, e, d, c, · · ·, by the single letter a with different “subscripts,” a0, a1, a2, a3, · · ·, and indicate the fact that the powers of ten may be as large as necessary by denoting the highest power, not by 103 or 104 as in the examples above, but by 10n, where n is understood to stand for an arbitrary integer. Then the general method for representing an integer z in the decimal system is to express z in the form

(1)   z = an · 10n + an-1 · 10n-1 + · · · + a1 · 10 + a0,

and to represent it by the symbol

anan-1an-2 · · · a1a0.

As in the special case above, we see that the digits a0, a1, a2, · · ·, an are simply the successive remainders when z is divided repeatedly by 10.

In the decimal system the number ten is singled out to serve as a base. The layman may not realize that the selection of ten is not essential, and that any integer greater than one would serve the same purpose. For example, a septimal system (base 7) could be used. In such a system, an integer would be expressed as

(2)   bn · 7n + bn-1 · 7n-1 + · · · + b1 · 7 + b0,

where the b′s are digits from zero to six, and denoted by the symbol

bnbn-1 · · · b1b0.

Thus “one hundred and nine” would be denoted in the septimal system by the symbol 214, meaning

2. 72 + 1. 7 + 4.

As an exercise the reader may prove that the general rule for passing from the base ten to any other base B is to perform successive divisions of the number z by B; the remainders will be the digits of the number in the system with base B. For example:

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It is natural to ask whether any particular choice of base would be most desirable. We shall see that too small a base has disadvantages, while a large base requires the learning of many digit symbols, and an extended multiplication table. The choice of twelve as base has been advocated, since twelve is exactly divisible by two, three, four, and six, and, as a result, work involving division and fractions would often be simplified. To write any integer in terms of the base twelve (duodecimal system), we require two new digit symbols for ten and eleven. Let us write α for ten andβ for eleven. Then in the duodecimal system “twelve” would be written 10, “twenty-two” would be 1α, “twenty-three” would be 1β, and “one hundred thirty-one” would be αβ.

The invention of positional notation, attributed to the Sumerians or Babylonians and developed by the Hindus, was of enormous significance for civilization. Early systems of numeration were based on a purely additive principle. In the Roman symbolism, for example, one wrote

CXVIII = one hundred + ten + five + one + one + one.

The Egyptian, Hebrew, and Greek systems of numeration were on the same level. One disadvantage of any purely additive notation is that more and more new symbols are needed as numbers get larger. (Of course, early scientists were not troubled by our modern astronomical or atomic magnitudes.) But the chief fault of ancient systems, such as the Roman, was that computation with numbers was so difficult that only the specialist could handle any but the simplest problems. It is quite different with the Hindu positional system now in use. This was introduced into medieval Europe by the merchants of Italy, who learned it from the Moslems. The positional system has the agreeable property that all numbers, however large or small, can be represented by the use of a small set of different digit symbols (in the decimal system, these are the “Arabic numerals” 0, 1, 2, · · ·, 9). Along with this goes the more important advantage of ease of computation. The rules of reckoning with numbers represented in positional notation can be stated in the form of addition and multiplication tables for the digits that can be memorized once and for all. The ancient art of computation, once confined to a few adepts, is now taught in elementary school. There are not many instances where scientific progress has so deeply affected and facilitated everyday life.

3. Computation in Systems Other than the Decimal

The use of ten as a base goes back to the dawn of civilization, and it undoubtedly due to the fact that we have ten fingers on which to count. But the number words of many languages show remnants of the use of other bases, notably twelve and twenty. In English and German the words for 11 and 12 are not constructed on the decimal principle of combining 10 with the digits, as are the “teens,” but are linguistically independent of the words for 10. In French the words “vingt” and “quatrevingt” for 20 and 80 suggest that for some purposes a system with base 20 might have been used. In Danish the word for 70, “halvfirsindstyve,” means half-way (from three times) to four times twenty. The Babylonian astronomers had a system of notation that was partly sexagesimal (base 60), and this is believed to account for the customary division of the hour and the angular degree into 60 minutes.

In a system other than the decimal the rules of arithmetic are the same, but one must use different tables for the addition and multiplication of digits. Accustomed to the decimal system and tied to it by the number words of our language, we might at first find this a little confusing. Let us try an example of multiplication in the septimal system. Before proceeding, it is advisable to write down the tables we shall have to use:

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Let us now multiply 265 by 24, where these number symbols are written in the septimal system. (In the decimal system this would be equivalent to multiplying 145 by 18.) The rules of multiplication are the same as in the decimal system. We begin by multiplying 5 by 4, which is 26, as the multiplication table shows.

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We write down 6 in the units place, “carrying” the 2 to the next place. Then we find 4·6 = 33, and 33 + 2 = 35. We write down 5, and proceed in this way until everything has been multiplied out. Adding 1,456 + 5,630, we get 6 + 0 = 6 in the units place, 5 + 3 = 11 in the sevens place. Again we write down 1 and keep 1 for the forty-nines place, where we have 1 + 6 + 4 = 14. The final result is 265·24 = 10,416.

To check this result we may multiply the same numbers in the decimal system. 10,416 (septimal system) may be written in the decimal system by finding the powers of 7 up to the fourth: 72 = 49, 73 = 343, 74 = 2,401. Hence 10,416 = 2,401 + 4.49 + 7 + 6, this evaluation being in the decimal system. Adding these numbers we find that 10,416 in the septimal system is equal to 2,610 in the decimal system. Now we multiply 145 by 18 in the decimal system; the result is 2,610, so the calculations check.

Exercises: 1) Set up the addition and multiplication tables in the duodecimal system and work some examples of the same sort.

2) Express “thirty” and “one hundred and thirty-three” in the systems with the bases 5, 7, 11, 12.

3) What do the symbols 11111 and 21212 mean in these systems?

4) Form the addition and multiplication tables for the bases 5, 11, 13.

From a theoretical point of view, the positional system with the base 2 is singled out as the one with the smallest possible base. The only digits in this dyadic system are 0 and 1; every other number z is represented by a row of these symbols. The addition and multiplication tables consist merely of the rules 1 + 1 = 10 and 1·1 = 1. But the disadvantage of this system is obvious: long expressions are needed to represent small numbers. Thus seventy-nine, which may be expressed as 1·26 + 0·25 + 0·24 + 1·23 + 1·22 + 1·2 + 1, is written in the dyadic system as 1,001,111.

As an illustration of the simplicity of multiplication in the dyadic system, we shall multiply seven and five, which are respectively 111 and 101. Remembering that 1 + 1 = 10 in this system, we have

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which is thirty-five, as it should be.

Gottfried Wilhelm Leibniz (1646-1716), one of the greatest intellects of his time, was fond of the dyadic system. To quote Laplace: “Leibniz saw in his binary arithmetic the image of creation. He imagined that Unity represented God, and zero the void; that the Supreme Being drew all beings from the void, just as unity and zero express all numbers in his system of numeration.”

Exercise: Consider the question of representing integers with the base a. In order to name the integers in this system we need words for the digits 0, 1, · · ·, a –1 and for the various powers of a: a, a2, a3, · · ·. How many different number words are needed to name all numbers from zero to one thousand, for a = 2, 3, 4, 5, · · ·, 15? Which base requires the fewest? (Examples: If a = 10, we need ten words for the digits, plus words for 10, 100, and 1000, making a total of 13. For a = 20, we need twenty words for the digits, plus words for 20 and 400, making a total of 22. If a = 100, we need 100 plus 1.)