CHAPTER XVII. ON THE EXTRACTION OF SQUARE ROOTS IN SYMBOLICAL ALGE- √-1. different duce the 645. It will follow, from the Rule of Signs, (Art. 569.) There are always two that, in Symbolical Algebra, there are always two roots, differ- roots, with ing from each other in their sign only, which correspond to the signs, same square: thus a' may equally arise from the product a xa which proand ax - a: (a + b) may equally arise from the product same (a + b) × (a + b), and − (a+b) × −(a+b): (a−b)' may equally square. arise from the product (a - b) × (a-b) and (ba) × (b− a ),* and similarly for all other squares. It follows, therefore, that in passing from the square to the square root, we shall always find two roots, which only differ from each other in their sign; but it is the positive square root alone which is recognized in ArithmetiArithmetical Algebra, and which may therefore be called the cal square arithmetical root. root. Arithmeti 646. We have already had occasion to notice these am- Occurrence of ambigubiguous square roots in Arithmetical Algebra (Art. 383) in ous square deducing the square roots of squares, such as x2-2ax+a2 and roots in a2 - 2 ax + x2, which are identical in their arithmetical value, cal Algebra. though different in the arrangement of their terms. If the relation of the values of the symbols x and a be known, the Rule for the extraction of the square root of these expressions, which is given in Arithmetical Algebra, would require the terms of the square, and therefore of the root to be arranged in the order of their magnitude, and consequently no ambiguity could exist with respect to the arithmetical root, which would be x-a if was greater than a, and a -x if x was less than a: but if the relation of those values be unknown, as where x is an unknown number to be determined from the solution of the equation which leads to the formation of the square, it is uncertain or ambiguous, whether the root be x-a or a-x, until that relation is For-(a - b)=b-a, and therefore − ( a − b ) × − ( a − b ) = ( b − a) × (b − a). The symbolical rived by the same process as in Arith metical Algebra. Use of the double sign ±. Rule for extracting root. assumed or determined*. In this case, however, the ambiguity of the roots originates in the ambiguity of the problem proposed, and not in the independent use of the signs in Symbolical Algebra. 647. In extracting the square root, we follow, as in all other operations, the same process both in Symbolical and in Arithmetical Algebra, assuming the proper relation of the symbols: and the negative is at once found from the positive root, by merely changing its sign. It is not unusual, likewise, to denote the double root by prefixing the double sign to it: thus a means equally a ora, one or both: (a - b) means equally ab and b-a: and similarly in other cases. 648. The following is the Rulet for extracting the square root in Symbolical Algebra: Arrange the terms of the square according to some symbol of the square reference (Art. 576); obliterate the first term of the square, and make its square root the primary term of the root to be found: divide the first remaining term of the square by double the prịmary term of the root, making the quotient the second term of the root: add this second term, with its proper sign, to double the primary, to form the divisor: multiply the last term of the root into the divisor, and subtract their product from the remainder of the square: if there be any remainder, repeat the same process, considering the terms already found in the root as constituting the single primary term: and so on continually until there is no remainder, or until the process becomes obviously interminable. The second root is found by changing the sign of all the terms of the first. Examples of terminable square roots. 649. The following are Examples in which the process terminates. We do not assume, in Arithmetical Algebra, that x-a or a-r, are equally the roots of x2-2ax+a2 when the relation of values of x and a is unknown, but that xa is always the root of a2 - 2 ax+a2, and a-r the root of a2 −2 ax + x2: and it is only when the relation of values of x and a is unknown, that there is nothing to guide us in the selection of one of those forms of the square in preference to the other. †The Rule for extracting the square root in Arithmetical Algebra has not been formally stated apart from the corresponding Rule in Arithmetic (Art. 218.) (1) To extract the square root of a2 – 2ab + b2. a2 - 2ab+b2 (a−b: the second root is 2a-b) - 2ab+b2 -(a - b) or b-a. We obliterate a2 and make its square root a, the primary term of the root: we double a (2a), and we divide - 2ab by it: the quotient - b is the second term of the root: we add (Art. 547) -b to 2a, making the divisor 2a-b: and we subtract the product (2a - b) × - b from the first remainder - 2ab+b2, and there is no second remainder. (2) To find the square root of a1 – 2 a3x + 3a2x2 − 2 ax3 + x1. a-2a3x+3a2x2-2ax3+x+, 2a2 — ax) − 2 a3x + 3 a2x2 − 2 ax3 +x1 (a2 -· ax+x2: the In forming the second divisor, we consider a2- ax as the single primary term of the root, the double of which is 2a2-2ax: we divide the first term 2a2x2 of the second remainder by the first term 2a2 of 2 a2 - 2ax, and the quotient + is the third term of the root: we add +x2 to 2a2-2ax to form the second and final divisor. (3) To extract the square root of 4a2+962 +16c2 - 12ab+ 16ac - 24bc. Making a the symbol of reference, this expression becomes 4a2 (12b-16c) a +962 - 24 b c +16c2 (2a-(3b-4c), 4a-(3b-4c)} - (12b-16c) a +9b2 – 24bc + 16c2 Examples of inter minable square roots. We divide (12b-16c) a, which is the first term of the first and only remainder, by double the primary term of the root or 4a: we thus get (3b-4c), which forms the second term of the root*. 650. In the following Examples the process leads to an indefinite series. (1) To extract the square root of a2 + x2. Inasmuch as the number of terms in the subtrahend is always greater than in the remainder, the process can never terminate. It follows, therefore, that If we reverse the order of the terms in the square, we shall and diver The only arrangement of the terms in the square which Convergent would be recognized in Arithmetical Algebra, is that in which gent series. they follow the order of their magnitude, Art. 646: thus, if a be greater than x, it is the first (1) of these series only, which is convergent: if a be less than x, it is the second (2): if this order be reversed, the same series are divergent, and no approximation is made to the value of the roots by the aggregation of any number of their terms. (Art. 587.) It will be observed, that in both these series, the signs of the terms after the first are alternately positive and negative. (2) To extract the square root of a2 — x2. The process followed in the last Example will give us an indefinite series, where all the terms after the first, are negative. root of a is greater than x. This series is convergent or divergent, according as a is greater The square or less than x: in the first case, both the square and its posi- q2 - 13 tive root are arithmetical quantities, and we can approximate both when indefinitely to the value of the latter by the aggregation of and less the successive terms of the series*: but in the second case, neither the square nor its roots are recognized in Arithmetical Algebra, and we approximate to no definite arithmetical value, by the aggregation of any number of the terms of the resulting divergent series. It is hardly necessary to observe, that the general rule given in Art. 648, is equally applicable to the extraction of the square root of a2- x2, both when a is greater and less than x. 651. But if a, in the expression a2x2, be less than x, Square we may replace x2 by a+b2, which gives us roots of negative symbols. √(25 − 1) = √√/24 = 5 ~ .1 - .001.00002.0000005 where the sum of two terms is 4.9, of three terms 4.899, of four terms is 4.89898, of five terms 4.8989795, the last of which differs from the true value of the |