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ON ARITHMETIC, AND IN PARTICULAR FRACTIONS AND LOGARITHMS.
ARITHMETIC is divided into two parts. The first is based on the decimal system of notation and on the manner of arranging numeral characters to express numbers. This first part comprises the four common operations of addition, subtraction, multiplication, and division,—operations which, as you know, would be different if a different system were adopted, but, which it would not be difficult to transform from one system to another, if a change of systems were desirable.
Systems of numeration
The second part is independent of the system of numeration. It is based on the consideration of quantities and on the general properties of numbers. The theory of fractions, the theory of powers and of roots, the theory of arithmetical and geometrical progressions, and, lastly, the theory of logarithms, fall under this head. I purpose to advance, here, some remarks on the different branches of this part of arithmetic.
It may be regarded as universal arithmetic, having an intimate affinity to algebra. For, if instead of particularising the quantities considered, if instead of assigning them numerically, we treat them in quite a general way, designating them by letters, we have algebra.
You know what a fraction is. The notion of a fraction is slightly more composite than that of whole numbers. In whole numbers we consider simply a quantity repeated. To reach the notion of a fraction it is necessary to consider the quantity divided into a certain number of parts. Fractions represent in general ratios, and serve to express one quantity by means of another. In general, nothing measurable can be measured except by fractions expressing the result of the measurement, unless the measure be contained an exact number of times in the thing to be measured.
You also know how a fraction can be reduced to its lowest terms. When the numerator and the denominator are both divisible by the same number, their greatest common divisor can be found by a very ingenious method which we owe to Euclid. This method is exceedingly simple and lucid, but it may be rendered even more palpable to the eye by the following consideration. Suppose, for example, that you have a given length, and that you wish to measure it. The unit of measure is given, and you wish to know how many times it is contained in the length. You first lay off your measure as many times as you can on the given length, and that gives you a certain whole number of measures. If there is no remainder your operation is finished. But if there be a remainder, that remainder is still to be evaluated. If the measure is divided into equal parts, for example, into ten, twelve, or more equal parts, the natural procedure is to use one of these parts as a new measure and to see how many times it is contained in the remainder. You will then have for the value of your remainder, a fraction of which the numerator is the number of parts contained in the remainder and the denominator the total number of parts into which the given measure is divided.
Greatest common divisor.
I will suppose, now, that your measure is not so divided but that you still wish to determine the ratio of the proposed length to the length which you have adopted as your measure. The following is the procedure which most naturally suggests itself.
If you have a remainder, since that is less than the measure, naturally you will seek to find how many times your remainder is contained in this measure. Let us say two times, and that a remainder is still left. Lay this remainder on the preceding remainder. Since it is necessarily smaller, it will still be contained a certain number of times in the preceding remainder, say three times, and there will be another remainder or there will not; and so on. In these different remainders you will have what is called a continued fraction. For example, you have found that the measure is contained three times in the proposed length. You have, to start with, the number three. Then you have found that your first remainder is contained twice in your measure. You will have the fraction one divided by two. But this last denominator is not complete, for it was supposed there was still a remainder. That remainder will give another and similar fraction, which is to be added to the last denominator, and which by our supposition is one divided by three. And so with the rest. You will then have the fraction
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
as the expression of your ratio between the proposed length and the adopted measure.
Fractions of this form are called continued fractions, and can be reduced to ordinary fractions by the common rules. Thus, if we stop at the first fraction, i. e., if we consider only the first remainder and neglect the second, we shall have 3 + ½, which is equal to 7/2. Considering only the first and the second remainders, we stop at the second fraction, and shall have 3 + 1/2 + 1/3. Now 2 + 1/3 = 7/3. We shall have therefore 3 + 3/7, which is equal to [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. And so on with the rest. If we arrive in the course of the operation at a remainder which is contained exactly in the preceding remainder, the operation is terminated, and we shall have in the continued fraction a common fraction that is the exact value of the length to be measured, in terms of the length which served as our measure. If the operation is not thus terminated, it can be continued to infinity, and we shall have only fractions which approach more and more nearly to the true value.
Terminating continued fractions.
If we now compare this procedure with that employed for finding the greatest common divisor of two numbers, we shall see that it is virtually the same thing; the difference being that in finding the greatest common divisor we devote our attention solely to the different remainders, of which the last is the divisor sought, whereas by employing the successive quotients, as we have done above, we obtain fractions which constantly approach nearer and nearer to the [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. fraction formed by the two numbers given, and of which the last is that fraction itself reduced to its lowest terms.
As the theory of continued fractions is little known, but is yet of great utility in the solution of important numerical questions, I shall enter here somewhat more fully into the formation and properties of these fractions. And, first, let us suppose that the quotients found, whether by the mechanical operation, or by the method for finding the greatest common divisor, are, as above, 3, 2, 3, 5, 7, 3. The following is a rule by which we can write down at once the convergent fractions which result from these quotients, without developing the continued fraction.
The first quotient, supposed divided by unity, will give the first fraction, which will be too small, namely, 3/1. Then, multiplying the numerator and denominator of this fraction by the second quotient and adding unity to the numerator, we shall have the second fraction, 7/2, which will be too large. Multiplying in like manner the numerator and denominator of this fraction by the third quotient, and adding to the numerator the numerator of the preceding fraction, and to the denominator the denominator of the preceding fraction, we shall have the third fraction, which will be too small. Thus, the third quotient being 3, we have for our numerator (7 × 3 = 21) + 3 = 24, and for our denominator (2 × 3 = 6) + 1 = 7. The third convergent, therefore, is [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. We proceed in the same manner for the fourth convergent. The fourth quotient being 5, we say 24 times 5 is 120, and this plus 7, the numerator of the fraction preceding, is 127; similarly, 7 times 5 is 35, and this plus 2 is 37. The new fraction, therefore, is [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. And so with the rest.
In this manner, by employing the six quotients 3, 2, 3, 5, 7, 3 we obtain the six fractions
3/1, 7/2, 24/7, 127/37, 913/266, 2866/835,
of which the last, supposing the operation to be completed at the sixth quotient 3, will be the required value of the length measured, or the fraction itself reduced to its lowest terms.
The fractions which precede the last are alternately smaller and larger than the last, and have the advantage of approaching more and more nearly to its value in such wise that no other fraction can approach it more nearly except its denominator be larger than the product of the denominator of the fraction in question and the denominator of the fraction following. For example, the fraction [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is less than the true value which is that of the fraction [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], but it approaches to it more nearly than any other fraction does whose denominator is not greater than the product of 7 by 37, that is, 259. Thus, any fraction expressed in large numbers may be reduced to a series of fractions expressed in smaller numbers and which approach as near to it as possible in value.
The demonstration of the foregoing properties is deduced from the nature of continued fractions, and from the fact that if we seek the difference between one of the convergent fractions and that next adjacent to it we shall obtain a fraction of which the numerator is always unity and the denominator the product of the two denominators; a consequence which follows a priori from the very law of formation of these fractions. Thus the difference between 7/2 and 3/1 is 1/2, in excess; between [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and 7/2, 1/14, in defect; between [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], in excess; and so on. The result being, that by employing this series of differences we can express in another and very simple manner the fractions with which we are here concerned, by means of a second series of fractions of which the numerators are all unity and the denominators successively the products of every two adjacent denominators. Instead of the fractions written above, we have thus the series:
3/1 + 1/[1 × 2] - 1/[2 × 7] + 1/[7 × 37] - 1/[37 × 266] + 1/[266 × 835].
A second method of expression.
The first term, as we see, is the first fraction, the first and second together give the second fraction 7/2, the first, the second, and the third give the third fraction [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], and so on with the rest; the result being that the series entire is equivalent to the last fraction.
There is still another way, less known but in some respects more simple, of treating the same question—which leads directly to a series similar to the preceding. Reverting to the previous example, after having found that the measure goes three times into the length to be measured and that after the first remainder has been applied to the measure there is left a new remainder, instead of comparing this second remainder with the preceding, as we did above, we may compare it with the measure itself. Thus, supposing it goes into the latter seven times with a remainder, we again compare this last remainder with the measure, and so on. until we arrive, if possible, at a remainder which is an aliquot part of the measure,—which will terminate the operation. In the contrary event, if the measure and the length to be measured are incommensurable, the process may be continued to infinity. We shall have then, as the expression of the length measured, the series
3 + 1/2 - 1/[2 × 7] + ····
A third method of expression.
It is clear that this method is also applicable to ordinary fractions. We constantly retain the denominator of the fraction as the dividend, and take the different remainders successively as divisors. Thus, the fraction [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] gives the quotients 3, 2, 7, 18, 19, 46, 119, 417 835; from which we obtain the series
3 + 1/2 - 1/[2 × 7] + 1/[2 × 7 × 18] – 1[/2 × 7 × 18 × 19] + ····;
and as these partial fractions rapidly diminish, we shall have, by combining them successively, the simple fractions,
7/2, 48/[2 × 7], 865/[2 × 7 × 18], ····,
which will constantly approach nearer and nearer to the true value sought, and the error will be less than the first of the partial fractions neglected.
Origin of continued fractions.
Our remarks on the foregoing methods of evaluating fractions should not be construed as signifying that the employment of decimal fractions is not nearly always preferable for expressing the values of fractions to whatever degree of exactness we wish. But cases occur where it is necessary that these values should be expressed by as few figures as possible. For example, if it were required to construct a planetarium, since the ratios of the revolutions of the planets to one another are expressed by very large numbers, it would be necessary, in order not to multiply unduly the number of the teeth on the wheels, to avail ourselves of smaller numbers, but at the same time so to select them that their ratios should approach as nearly as possible to the actual ratios. It was, in fact, this very question that prompted Huygens, in his search for its solution, to resort to continued fractions and that so gave birth to the theory of these fractions. Afterwards, in the elaboration of this theory, it was found adapted to the solution of other important questions, and this is the reason, since it is not found in elementary works, that I have deemed it necessary to go somewhat into detail in expounding its principles.
We will now pass to the theory of powers, proportions, and progressions.
As you already know, a number multiplied by itself gives its square, and multiplied again by itself gives its cube, and so on. In geometry we do not go beyond the cube, because no body can have more than three dimensions. But in algebra and arithmetic we may go as far as we please. And here the theory of the extraction of roots takes its origin. For, although every number can be raised to its square and to its cube and so forth, it is not true reciprocally that every number is an exact square or an exact cube. The number 2, for example, is not a square; for the square of 1 is 1, and the square of 2 is four; and there being no other whole numbers between these two, it is impossible to find a whole number which multiplied by itself will give 2. It cannot be found in fractions, for if you take a fraction reduced to its lowest terms, the square of that fraction will again be a fraction reduced to its lowest terms, and consequently cannot be equal to the whole number 2. But though we cannot obtain the square root of 2 exactly, we can yet approach to it as nearly as we please, particularly by decimal fractions. By following the common rules for the extraction of square roots, cube roots, and so forth, the process may be extended to infinity, and the true values of the roots may be approximated to any degree of exactitude we wish.
Involution and evolution.
But I shall not enter into details here. The theory of powers has given rise to that of progressions, before entering on which a word is necessary on proportions.
Every fraction expresses a ratio. Having two equal fractions, therefore, we have two equal ratios; and the numbers constituting the fractions or the ratios form what is called a proportion. Thus the equality of the ratios 2 to 4 and 3 to 6 gives the proportion 2: 4:: 3: 6, because 4 is the double of 2 as 6 is the double of 3. Many of the rules of arithmetic depend on the theory of proportions. First, it is the foundation of the famous rule of three, which is so extensively used. You know that when the first three terms of a proportion are given, to obtain the fourth you have only to multiply the last two together and divide the product by the first. Various special rules have also been conceived and have found a place in the books on arithmetic; but they are all reducible to the rule of three and may be neglected if we once thoroughly grasp the conditions of the problem. There are direct, inverse, simple, and compound rules of three, rules of partnership, of mixtures, and so forth. In all cases it is only necessary to consider carefully the conditions of the problem and to arrange the terms of the proportion correspondingly.
I shall not enter into further details here. There is, however, another theory which is useful on numerous occasions,—namely, the theory of progressions. When you have several numbers that bear the same proportion to one another, and which follow one another in such a manner that the second is to the first as the third is to the second, as the fourth is to the third, and so forth, these numbers form a progression. I shall begin with an observation.
The books of arithmetic and algebra ordinarily distinguish between two kinds of progression, arithmetical and geometrical, corresponding to the proportions called arithmetical and geometrical. But the appellation proportion appears to me extremely inappropriate as applied to arithmetical proportion. And as it is one of the objects of the École Normale to rectify the language of science, the present slight digression will not be considered irrelevant.
I take it, then, that the idea of proportion is already well established by usage and that it corresponds solely to what is called geometrical proportion. When we speak of the proportion of the parts of a man's body, of the proportion of the parts of an edifice, etc.; when we say that a plan should be reduced proportionately in size, etc.; in fact, when we say generally that one thing is proportional to another, we understand by proportion equality of ratios only, as in geometrical proportion, and never equality of differences as in arithmetical proportion. Therefore, instead of saying that the numbers, 3, 5, 7, 9, are in arithmetical proportion, because the difference between 5 and 3 is the same as that between 9 and 7, I deem it desirable that some other term should be employed, so as to avoid all ambiguity. We might, for instance, call such numbers equi-different, reserving the name of proportionals for numbers that are in geometrical proportion, as 2, 4, 6, 8, etc.
Excerpted from Lectures on Elementary Mathematics by Joseph Louis Lagrange, Thomas J. McCormack. Copyright © 2008 Dover Publications, Inc.. Excerpted by permission of Dover Publications, Inc..
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Posted March 25, 2012