22391824. Whatever this root may be, we may suppose it capable of being resolved into tens and units, as in the preceding examples. As the square of the tens has no figure inferior to hundreds, the two last figures 24 cannot make a part of it; we may therefore separate them, and the question will be reduced to this, to find the greatest square contained in the part 223918, which remains on the left. This part consisting of more than two figures, we may conclude, that the number, which expresses the tens in the root sought, will have more than one figure; it may therefore be resolved, like the others, into tens and units. As the square of the tens does not enter into the two last figures 18 of the number 223918, it must be sought in the figures 2239, which remain on the left; and since 2239 still consists of more than two figures, the square, which is contained in it must have a root, which consists of at least two; the number which expresses the tens sought will therefore have more than one figure; it is then, lastly, in 22 that we must seek the square of that, which represents the units of the highest place in the root required. By this process, which may be extended to any length we please, the proposed number may be divided into portions of two figures from right to left; it must be understood however, that the last figure on the left may consist of only one figure.

16

63,9
60 9

87

943

9462

301,8

2829

1892,4

1892 4

Having divided the proposed number into portions as below, we proceed with the three first portions, as 22,39,18,24 4732 in the preceding article; and when we have found the three first figures 473 of the root, to the remainder 189 we bring down the fourth portion 24, and consider the number 18924, as containing double the product of the 473 tens already found by the units sought, plus the square of these units. We separate the last figure 4; divide those, which remain on the left, by 946, double of 473, and then make trial of the quotient 2, as in the preceding examples.

0000 0

Here the operation, in the present case, terminates; but it is very obvious, that if we had one portion more, the four figures already found 4732, would express the tens of a root, the units of which would remain to be sought; we should proceed therefore

to divide the remainder now found, together with the first figure of the following portion, by double of these tens, and so on for each of the portions to be successively brought down.

0 00 0

94. If, after having brought down a portion, the remainder, joined to the first figure of this portion, does not contain double of the figures already found, a cypher must be placed in the root; for the root, in this case, will have no units of this rank; the following portion is then to be brought down, and the operation to be continued as before. The example subjoined vill illustrate this case. The quantities to be subtracted are 49,42,09 | 708 not put down, but the subtractions are supposed 04,20, 9| 1403 to be performed mentally, as in division. 95. Every number, it will be perceived, is not a perfect square. If we look at the table given, page 100, we shall see that between the squares of each of the nine primitive numbers, there are intervals comprehending many numbers, which have no assignable root; 45, for instance, is not a square, since it falls between 36 and 49. It very often happens, therefore, that the number, the root of which is sought, does not admit of one; but if we attempt to find it, we obtain for the result the root of the greatest square, which the number contains. If we seek, for example, the root of 2276, we obtain 47, and have a remainder 67, which shows, that the greatest square, contained in 2276, is that of 47 or 2209.

As a doubt may sometimes arise, after having obtained the root of a number, which is not a perfect square, whether the root found be that of the greatest square contained in the number, I shall give a rule, by which this may be readily determined. As the square of a + b is

a2+2ab+b2,

if we make b= 1, the

square

of

a+1 will be

a2+2a+1,

a quantity which differs from a3, the square of a, by double of a, plus unity. Therefore if the root found can be augmented by unity, or more than unity, its square, subtracted from the proposed number, will leave a remainder at least equal to twice this root plus unity. Whenever this is not the case, the root obtained will be, in fact, that of the greatest square contained in the number proposed.

96. Since a fraction is multiplied by another fraction, when their numerators are multiplied together, and their denominators

together, it is evident that the product of a fraction multiplied by itself, or the square of a fraction is equal to the square of its numerator, divided by the square of its denominator. Hence it follows, that to extract the square root of a fraction, we extract the square root of its numerator and that of its denominator. Thus the root of 25 is, because 5 is the square root of 25, and 8 that of 64.

It is very important to remark, that not only are the squares of fractions, properly so called, always fractions, but every fractional number, which is irreducible, (Arith. 59) will, when multiplied by itself, give a fractional result, which is also irreducible.

97. This proposition depends upon the following; Every prime number P, which will divide the product A B of two numbers A and B, will necessarily divide one of these numbers.

Let us suppose, that it will not divide B, and that B is the greater; if we designate the entire part of the quotient by q, and the remainder by B', we have

B=qP+B',

multiplying by A, we obtain

AB=qAP+AB,

and dividing the two members of this equation by P, we have

from which it appears, that if A B be divisible by P, the product A B' will be divisible by the same number. Now B', being the remainder after the division of B by P, must be less than P; therefore B' cannot be divided by P; if we divide P by B' we have a quotient q' and a remainder B"; if further we divide P by B", we have a quotient q'' and a remainder B", and so on, since P is a prime number.

We have therefore the following series of equations;

P=qB'+B", P=q"B" + B", &c.

multiplying each of these by A, we obtain

From these results it is evident, that if A B' be divisible by P,

the products A B", A B", &c. will also be divisible by it. But the remainders B', B", B", &c. are becoming less and less continually,

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 x 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 × a, or a2, every prime number, which will not divide b, will not divide b × b, or b2; the numbers a2 and b2 are therefore, in this case, prime to

b2 square of the fraction, be

a2

a

each other; and consequently the ing irreducible, as well as the fraction itself, cannot become an entire number (B).

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 rela

tion 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 2, 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.

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 }, or 13 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