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till they finally terminate in unity, for the operation exhibited above may be continued in the same manner, while the remainder is greater than 1, since P is a prime number. Now when the remainder becomes unity, we have the product A× 1, which must be divisible by P; therefore A also must be divisible by P.

Hence, if the prime number P, which we have supposed not to divide B, will not divide A, it will not divide the product of these numbers.

(This demonstration is taken principally from the Théorie des nombres of M. Legendre.)

b

98. Now when the fraction is irreducible, there is no prime

a

number, which will divide, at the same time, b and a; but, from the preceding demonstration, it is evident, that every prime number, which will not divide a, will not divide a Xa, or a2, every prime number, which will not divide b, will not divide b× b, or b2; the numbers a2 and 62 are, therefore, in this case, prime b a' being irreducible, as well as as the fraction itself, cannot become an entire number (B).

b2
a2

to each other; and consequently the square of the fraction

99. From this last proposition it follows, that entire numbers, except only such as are perfect squares, admit of no assignable root, either among whole numbers or fractions. Yet it is evident, that there must be a quantity, which, multiplied by itself, will produce any number whatever, 2276, for instance, and that, in the present case, this quantity lies between 47 and 48; for 47 × 47 gives a product less than this number, and 48 × 48 gives one greater. Dividing then the difference between 47 and 48 by means of fractions, we may obtain numbers that, multiplied by themselves, will give products greater than the square of 47, but less than that of 48, and which will approach nearer and nearer to the number 2276.

The extraction of the square root, therefore, applied to numbers, which are not perfect squares, makes us acquainted with a new species of numbers, in the same manner, as division gives rise to fractions; but there is this difference between fractions and the roots of numbers, which are not perfect squares; that the former, which are always composed of a certain number of parts of unity, have with unity a common measure, or a relation

which may be expressed by whole numbers, which the latter have not.

If we conceive unity to be divided into five parts, for example, we express the quotient arising from the division of 9 by 5, or , by nine of these parts; then, being contained five times in unity, and nine times in, is the common measure of unity and the fraction, and the relation these quantities have to each other is that of the entire numbers 5 and 9.

Since whole numbers, as well as fractions, have a common measure with unity, we say that these quantities are commensurable with unity, or simply that they are commensurable; and since their relations or ratios, with respect to unity, are expressed by entire numbers, we designate both whole numbers and fractions, by the common name of rational numbers.

On the contrary, the square root of a number, which is not a perfect square, is incommensurable or irrational, because, as it cannot be represented by any fraction, into whatever number of parts we suppose unity to be divided, no one of these parts will be sufficiently small to measure exactly, at the same time, both this root and unity.

In order to denote, in general, that a root is to be extracted, whether it can be exactly obtained or not, we employ the character, which is called a radical sign;

16 is equivalent to 4,

2 is incommensurable or irrational.

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100. Although we cannot obtain, either among whole numbers or fractions, the exact expression for 2, yet we may approximate it, to any degree we please, by converting this number into a fraction, the denominator of which is a perfect square. The root of the greatest square contained in the numerator will then be that of the proposed number expressed in parts, the value of which will be denoted by the root of the denominator.

If we convert, for example, the number 2 into twenty-fifths, we have. As the root of 50 is 7, so far as it can be expressed in whole numbers, and the root of 25 exactly 5, we obtain 7, or 1 for the root of 2, to within one fifth.

101. This process, founded upon what was laid down in article 96, that the square of a fraction is expressed by the square of the numerator divided by the square of the denominator, may evidently be applied to any kind of fraction whatever, and more

readily to decimals than to others. It is manifest, indeed, from the nature of multiplication, that the square of a number expressed by tenths will be hundredths, and that the square of a number expressed by hundredths will be ten thousandths, and so on; and consequently, that the number of decimal figures in the square is always double that of the decimal figures in the root. The truth of this remark is further evident from the rule observed in the multiplication of decimal numbers, which requires that a product should contain as many decimal figures, as there are in both the factors. In any assumed case, therefore, the proposed number, considered as the product of its root multiplied by itself, must have twice as many decimal figures as its root.

From what has been said, it is clear, that in order to obtain the square root of 227, for example, to within one hundredth, it is necessary to reduce this number to ten thousandths, that is, to annex to it four cyphers, which gives 2270000 ten thousandths. The root of this may be extracted in the same manner, as that of an equal number of units; but to show that the result is hundredths, we separate the two last figures on the right by a comma. We thus find that the root of 227 is 15,06, accurate to hundredths. The operation may be seen below;

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If there are decimals already in the proposed number, they should be made even. To extract, for example, the root of 51,7, we place one cypher after this number, which makes it hundredths; we then extract the root of 51,70. If we proposed to have one decimal more, we should place two additional cyphers after this number, which would give 51,7000; we should then obtain 7,19 for the root.

If it were required to find the square root of the numbers 2 and 3 to seven places of decimals, we should annex fourteen cyphers to these numbers; the result would be

√2 = 1,4142136, √3 = 1,7320508.

102. When we have found more than half the number of figures, of which we wish the root to consist, we may obtain the rest simply by division. Let us take, for example, 32976; the square root of this number is 181, and the remainder, 215. If

we divide this remainder 215, by 362, double of 181, and extend the quotient to two decimal places, we obtain 0,59, which must be added to 181; the result will be 181,59 for the root of 32976, which is accurate to within one hundredth.

In order to prove that this method is correct, let us designate the proposed number by N, the root of the greatest square contained in this number by a, and that which it is necessary to add to this root to make it the exact root of the proposed number by b; we have then

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From this result it is evident, that the first member may be

b2

taken for the value of b, so long as the quantity is less than

2 a

a unit of the lowest place found in b. But as the square of a number cannot contain more than twice as many figures as the number itself, it follows, that if the number of figures in a ex

b2

ceeds double those in b, the quantity will then be a fraction.

2 a

In the preceding example, a = 181 units, or 18100 hundredths, and consequently contains one figure more than the square of 59 hundredths; the fraction then becomes, in this case,

(59)

=

b2

2 a

3481 and is less than a unit of the second part

2 X 18100 36200'

59, or than a hundredth of a unit of the first.

103. This leads to a method of approximating the square root of a number by means of vulgar fractions. It is founded on the circumstance, that a, being the root of the greatest square contained in N, b is necessarily a fraction, and

ler than b, may be neglected.

b2

2 a

being much smal

If it were required, for example, to extract the square root of 2; as the greatest square contained in this number is 1, if we subtract this, we have a remainder, 1. Dividing this remainder by double of the root, we obtain ; taking this quotient for the value of the quantity b, we have, for the first approximation to

2.

the root, 1+, or . Raising this root to its square, we find, which, subtracted from 2 or, gives for a remainder

this case the formula

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Substituting

12

for b, we have for the second approximation

1; ; taking the square of 1, we find, a quantity, which still exceeds 2 or 2. Substituting for a, we obtain

288
144
1

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12 X 34

2a

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This operation may be easily continued to any extent we please. I shall give, in the Supplement to this treatise, other formulas more convenient for extracting roots in general.

104. In order to approximate the square root of a fraction, the method, which first presents itself, is, to extract, by approximation, the square root of the numerator and that of the denominafor; but with a little attention it will be seen, that we may avoid one of these operations by making the denominator a perfect square. This is done by multiplying the two terms of the proposed fraction by the denominator. If it were required, for example, to extract the square root of, we might change this fraction into

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by multiplying its two terms by the denominator, 7. Taking the root of the greatest square contained in the numerator of this fraction, we have 4 for the root of, accurate to within 4.

If a greater degree of exactness were required, the fraction # must be changed by approximation or otherwise into another, the denominator of which is the square of a greater number than 7. We shall have, for example, the root sought within, if we convert into 225ths, since 225 is the square of 15; thus the fraction becomes 75 of one 225th, or, within the root of

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