BINOMIAL SURDS.] 



MATHEMATICS. ALGEBRA. 



483 



J (a-{- *JV) occur in the results of algebraical problems: 

 if they be left iu this form, and theu when a and b are 

 interpreted, the operations of arithmetic be applied to 

 them, you see that if 6 be irrational, we shall have to 

 find the square root of 6 to a certain number of decimals, 

 and then to extract the square root of a + */ b, a quan- 

 tity necessarily involving decimals. The square root of 6, 

 may be taken out of a table of square roots to several 

 decimal places, when 6 consists of not more than four or 

 five figures ; but the square root of a number consisting 

 of three or four figures, followed by five or six decimals, 

 cunnot be found to any degree of nicety by existing 

 tables ; so that, in such a case, the arithmetical opera- 

 tion must be executed. As we have told you before, 

 algebraists seek such a form for their results as will give 

 arithmeticians the least trouble in the numerical com- 

 putation of them ; and this is one reason why they have 

 sou"ht to convert V (a + V b), into the simpler form 



In order to show how tliis is done, a theorem or two 

 respecting binomial surds must first be established. 



1. The square root of a quantity cannot be partly 

 rational and partly a quadratic surd : that is to say, the 

 condition A /P = 2 + v r ' s impossible, provided p and 

 r are not themselves squares. 



For, assuming this equation to be possible, we should 

 have, by squaring each side .p = 2* -f- 2 3 ^ j- -f- r, so 



V = ^ -J ' a rational quantity, which is 

 2q 



<iry to the supposition, as it shows that r must be 

 , "(ore of the s<eoad member of this equation. , 



2. In every equation of the form a-f- >J 6 = a; -f- ^ y, 

 where o and x are rational quantities, and */ b irrational 

 or surd, a must be equal to x, and therefore 6 will 

 equal y. For if a and x had any difference, then by 

 transposing the a, we should have the square root of a 

 quantity, not itself a square, expressed either by a 

 rational quantity, which is absurd, or by a quantity 

 )>:irtly rational and partly irrational, which hits been 

 hown above to be impossible. It follows that */y must 



iu:i:il as well as ^/ 6. 



3. If v/(a + ,/6)=x + y: then must J (a JV) 

 = x y; where ^/b is a quadratic surd, and x, y, one or 

 both, also quadntu: surds. 



For by squaring the first equation a + J b = j? -f- 2xy 

 -f- y 2 , where y? -f- y- is necessarily rational, and . . 2j:y 

 irrational (by 2 above) being equal io */ b: that ia 

 (by 2), 



By help of this latter principle, combined with theo. 

 2 above, tlie extraction of the square root of a binomial, 

 surd consisting of a rational term, and of a quadratic 

 irrational term, may be eti'ected, as in the following 

 examples : 



1. Extract the square root of 7 + 2 J 10. 

 Put v'(7 + 2, v /10)=z + y, .'. ^(7 2-v/lO) 



x y; 

 . . taking the product, ^ (49 40) = x* y 3 , . . 



3=x 2 y. 

 But 7 + 2 



..z = v/5, y = ^2, .- 



You will at once see that the change of ^/ (7 + 2 ^/ 10) 

 into ^/ 5 + */ 2 facilitates the arithmetical computation : 

 a table of square roots gives us ^ 5 = 2 -2360680, and 

 N / 2 = 1-4142136 ; so that the root sought is 3 "6502816. 

 If the unchanged form be used, we should have, from the 

 table, V 10 = 3-1022777 ; twice this increased by 7 is 

 13-3245554, and the square root of this we should have 

 to find by actual extraction, on account of the necessarily 

 linited extent of the tables. 



2. Find the square root of 10 v/96. 

 Put V(10 + V90)=x + y, .-. V(10 V 90)= x y. 



90) =x 2 y* = 2. > 

 = lOi 

 6 2. 



3. Find general expressions for the square roots of 

 a + J 6, and a / 6. 



Putting x + y, and a; y, for the required roots, we 

 have 



By adding and subtractin 



Alsox 2 y a = ^/(a 2 I 



, = 



therefore x + y, x y, that is 

 Y/ b) are respectively 



(<* + 



,/ (a 



These general expressions show that, whenever a? 

 & is not a sijuare, nothing is gained by changing ^/ (a + 

 s/ 6) into the new form, but in fact something lost in the 

 way of simplification : the changed form is less complex 

 than the original only when o 2 6 ia a square : it is 

 prudent, therefore, to ascertain whether or not this be 

 the case, before entering upon the proposed transfor- 

 mation ; and to leave the form unchanged if a- b be 

 not a square. 



EXAMPLES FOE EXERCISE. 



2. 

 3. 

 4. 

 5. 

 6. 



(8+ 

 ,(76 32^3) = 8 



6) = 6+ 



5. 



On Multipliers which render Binomial Surds rational. 



It has already been noticed (page 466), that it is iu 

 general inconvenient to have the result of an algebraic 

 operation in the form of a fraction with a surd denomi- 

 nator ; and when the surd so occurring is monomial, the 

 means of removing it have been explained. We are now 

 to show how, in like manner, a fraction having a binomial 

 surd Y/a+Y/frorv/a ^/6 for denominator, may be 

 converted into an equivalent fraction with a rational 

 (U-iinminutnr ; in other words, to explain the method of 

 rationalising binomial surds of the above form. This ia 

 very easy, the rationalising multiplier being at once 

 suggested from the binomial surd itself : if it bo ^/ a -f- 

 v/6, the multiplier is obviously J 'a J b ; and if it be 

 ^/ a Y/ 6, the multiplier is V a + vf " > au d vou thus 

 have a useful application of the principle that the sum 

 multiplied by the difference of two quantities gives the 

 difference of their squares. Suppose, for example, our 



Q 



fraction is j~- a form which is inconvenient for 



Vi> + V ' 



computation, because after getting /5 + ^/ 7 we should 

 have the troublesome operation of dividing 3 by a num- 

 ber consisting of many figures. But by multiplying 

 numerator and denominator by x/5 ^/ 7, we change 



the fraction into 3 (\/ 5 >/^)= J (^/7 ^/5), by which 



5 i 

 change the long division spoken of is avoided. 



As a second example, let r-TZ ^ P r P osed - The 



rationalising multiplier here is 5 + \/ 3, and the changed 

 fraction i. U-*V+jJS> = 1+^ 



Sometimes the binomial surd to be rationalised is of 

 the form -^/ a + \/ b : in this case the suitable multiplier 

 will be trinomial : it will consist of the squares of both 

 the irrational terms, and of their product with changed 

 sign, as actual multiplication will show ; for 



The rationalising multiplier is therefore easily remem- 



