CHAPTER XIV. IMAGINARY AND COMPLEX QUANTITIES.

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1 CHAPTER XIV. SURDS. IMAGINARY AND COMPLEX QUANTITIES Definitions. A surd is a root of au arithmetical number which can only be found approximately. An algebraical expression such as s]a is also often called a surd, although a may have such a value that *Ja is not in reality a surd. Surds are said to be of the same order when the same root is required to be taken. Thus */2 and */6 are called surds of the second order, or quadratic surds; also ^/4 is a surd of the third order, or a cubic surd ; and tya is a surd of the nth order. Two surds are said to be similar when they can be reduced so as to have the same irrational factors. Thus V8 and yl8 are similar surds, for they are equivalent to 2*J2 and 3\/2 respectively. The rules for operations with surds follow at once from the principles established in the previous chapter. Note. It should be remarked that when a root symbol is placed before an arithmetical number it denotes only the arithmetical root, but when the root symbol is placed before an algebraical expression it denotes one of the roots. Thus \Ja has two values but ^2 is only supposed to denote the arithmetical root, unless it is written + \j%

2 214 SURDS Any rational quantity can be written in the form of a surd. For example, 2 = ^/4 = ^8 = #2", and a = Z/a 2 =$a 3 = ya n. Also, since *Ja x *Jb = \/a& [Art. 165], we have 2V2 = ^4 x ^2 = V(* x 2) = V8, and 5 /3 = ^5 3 x 4/3 = ^(5 3 x 3) = #375, atfab = *Ja«x ^a6 -?j(a n x at) = j/a^1! Conversely, we have V18 = V(9 x 2 ) = V9 x \/2 = 3^2, and ^135 + ^40 = #(3 8 x 5) + ^(2 3 x 5) = 3^5 + 2^5 = 5^ Any two surds can be reduced to surds of the same order. For if the surds be $a and ^Jb f we have %a = n 7a m, and yb = "76" [Art. 165]. Ex. Which is the greater, 4/14 or 4/6? The surds must be reduced to equivalent surds of the same order. Now ^14 = 4/14 2 =^196, and ^6=4/63=^216. Hence, as ^216 is greater than 4/196, 4 / 6 must be greater than,yi4. Thus we can determine which is the greater of two surds without finding either of them The product of two surds of the same order can be written down at once, for we have ya x yb = tyab. Hence, in order to find the product of any number of surds, the surds are first reduced to surds of the same order: their product is then given by the formula Z/axybx #c...= #a&c... Ex. 1. Multiply V5 by 4/2. s/5 x 4/2 = 4/53 x 4/22=4/(5 3 x 2 2 ) = 4/500. Ex. 2. Multiply 3^/5 by 24/2. 3^5 x 24/2 = 3 x 2 x ^5 x 4/2 = 6 x flp x 4/2- = 64/500.

3 MULTIPLICATION OF SURDS. 215 Ex. 3. Multiply </2 by */2. s/2 x ^2=^/23 x $&=$Wx& =^32. Orthna: > /2x j y2=2ix2* = 2i + 4 = 2*= J y 5. Ex. 4. Multiply V2 + V 3 b y x/3 + v/5. (x/3 + x/2)( x /3 + x/5)= N /3 x V3W2 x^3 + ^/3 x ^5 + ^/2 x 5 = 3 + ^/6 + ^15 + ^/10. Ex. 5. Divide ^4 by */& The determination of the approximate value of an expression containing surds is an arithmetical rather than an algebraical problem ; but an expression containing surds must always be reduced to the form most suitable for arithmetical calculation. For this reason when surds occur in the denominators of fractions, the denominators must be rationalized. [See Art. 169.] The following examples will illustrate the process: 2_ = 2x^5 = 2 v/5 ",J5 x ^/5 ~~ 5 * 3 _ 3(J5 + 1) _3 x/o-l~( N /5-l)( N /5 + l)-4 W a + h 1 X 1 + ^3 + V5 + x/15 ~~ (1 + sjwi 1 + \/ 5 ) 4 = 1(^3-1)^5-1) The product and the quotient of two similar quadratic surds are both rational. This is obvious; for any two similar quadratic surds can be reduced to the forms a%/b and c*jb. Conversely, if the product of the quadratic surds sja and *Jb is rational and equal to x, we have x \/a x *Jb; therefore x>jb = A/& x V& x y/b = b\ja, which shews that the surds are similar. So also, if \/a -f- sjb is rational, the surds must be similar. 8 '

4 216 SURDS The following theorem is important. Theorem. If a + \/b = x + sly, where a and x are rational, and *Jb and \Jy are irrational; then will a = x t and b y. For we have a x+*jb = «Jy. Square both sides; then, after transformation, we have 2(a x)sjb = y-b {a x) 2. Hence, unless the coefficient of \Jb is zero, we must have an irrational quantity equal to a rational one, which is impossible. The coefficient of *Jb in the last equation must therefore be zero, so that a = x. And w T hen a = x, the given relation shews that \Jb = \jy, and therefore b y. As a particular case of the above, si a 4= b + \]c, unless 6 = 0 and a = c. Hence \ja + \]c can only be rational when it is zero. Ex. 1. Shew that v/a + /v/fe + v/c4=0» unless the surds are all similar. For we should have tja+*jb=-*jc; and therefore a + b + 2 Ja^Jb = c. Hence *Ja*Jb is rational, which shews [Art. 176], that ^/a and *Jb are similar surds The expressions a + \Jb and a sib are said to be conjugate quadratic surd expressions. It is clear that the sum and the product of two conjugate quadratic surd expressions are both rational. Conversely, if the sum and the product of the expressions a + \/b and c + \Jd are both rational, then a = c and sjb + sjd = 0, so that the two expressions are conjugate. For a + c + s/b + \]d can only be rational when s/b + \Jd is zero. [Art. 177.] And, when \]d \Jb, the product (a + s/b) (c + \Jd) = ac + (c a) sib b, which cannot be rational unless c = a In the expression ax n + bx cat* + + ft, where a, b, c, k are all rational, let a-\- s//3 be substi-

5 SURDS. 217 tuted for x\ and let P be the sum of all the rational terms in the result and Q *J/3 the sum of all the irrational terms. Then the given expression becomes P + Q Vi& Since P and Q are rational, they contain only squares and higher even powers of VA and hence P and Q will not be changed by changing the sign of V/3- Therefore when a *Jft is substituted for x in the given expression the result will be P - Q Vj8. If now the given expression vanish when a + V/3 is substituted for x, we have P + QV = 0. Hence, as P and Q are rational and \//3 is irrational, we must have both P = 0 and Q = 0; and therefore P-QV/3=0. Therefore if the given expression vanish when a + V/3 is substituted for x it will also vanish when a V/3 is substituted for x. Hence [Art. 88], if x a \//3 be a factor of the given expression, x a + ^//3 will also be a factor. Thus, if a rational and integral expression be divisible by either of two conjugate quadratic surd expressions it will also be divisible by the other The square root of a binomial expression which is the sum of a rational quantity and a quadratic surd can sometimes be found in a simple form. The process is as follows. To find *J(a -I- ^b), where *Jb is a surd. Let \/( a + V&) = V# + *Jy> Square both sides; then a + *Jb = x+y-\- 2jxy. Now, since \/b is a surd, we can [Art. 177] equate the rational and irrational terms on the different sides of the last equation; hence x + y = a, and 4*xy = 6.

6 218 SURDS. Hence x and y are the roots of the equation and these roots are x 2 ax H- 7 = 0, 4 i {a + V(^2-6)} and J {a - \/(a 2 - &)}. rn //, /IN /a + *J(a 2 -b) /a->j{a 2 -b) Thus V(a +V&) = iy/ 1 '' + y ^ ' It is clear that, unless \J(a 2 b) is rational, the right side of the last equation is less suitable for calculation than the left. Thus the above process fails entirely unless a 2 b is a square number; and as this condition will not often be satisfied, the process has not much practical utility. It should be remarked that if x and y are really rational, they can generally be written down by inspection. Ex. 1. Find ^(6 + 2^/5). Let s/{6 + 2 s /5)=: s /x + s/y. Then, by squaring, we have 6 + 2^/5 =x + y- s r2 s jxy. Hence, equating the rational and irrational parts, x + y = 6 and xy 5. Whence obviously x = 1 and y = 5. Thus s/(q + 2 s /5) = l+ s /5. Ex. 2. Find ^(28-5^/12). Let s/{28-5»j12)= s /x- ls fy. Then, as before, to/ = 25xl2, or xy = 75 and x+y=28; whence x = 25 and?/ = 3. Thus ^ (28-5 v/12) = 5 - ^3. [If we had taken x = 3 and 2/ = 25 we should have had the negative root, namely ^3-5.] Ex.3. Find ^( ^/3). In this case,j(a 2 - b) is irrational and therefore the required root cannot be expressed in the form sjx +,jy where x and y are rational. The root can however be expressed in the form /x + f/y; for x/( x /3) = v/{ v /3( N /3)} =4/3x^/(12 + 6^3) = 4/3x (3 +J3) =4/243 + ^/27. Ex.4. Find V(10 + 2^/6 + 2^/10 + 2^/15). Assume ^(10 + 2^/6 + 2^/10 + 2^/15) = ^/.r + Jy + *Jz\ then ^6 + 2 v /10 + 2^15 = ;r +?/ ^^ + 2^ars + 2*Jyz. We have now to find, if possible, rational values of x t y, z such that xy = &, xz = 10,?/2 = 15 and x + y + 2 = 10. The first three equations are satisfied by the values.t = 2, y = S, 2=5, and these values satisfy x + y + z=10. Hence N/( N /6 + 2 x /10 + 2V15) = N/2 + N/3 + ^5.

7 IMAGINARY AND COMPLEX QUANTITIES. 219 Ex. 5. Prove that, if $(a + s/b) = x +,Jy; then will /(a -,Jb) = x-^jy. We have a+>jb={x +»Jy)z=xS + Sxy + s/y(sxz + y). Hence, equating the rational and irrational parts, we have a=x 3 + 3xy, and tjb=,sjy(dx 2 + y). Hence a-*jb = x 3 + Sxy - <Jy (3xz +?/);.*. sl(a-sjb)=x- s jy. EXAMPLES XVIII. Simplify the following : 1 J*zl 2J5 ' ^3 + 1* J6 + J T73 + 7^ (2-^-^(2^3)-. 5. _3 j y2_ ijs J6 J3 + J6 ^6+^2 J-2 + JS' - (6-2V7)(3 + V5)(ll + 4 N /7) ' 7. L «J l L 1 JQ + j-2l - J10 - J3o N/2 + ^/3 +,/5- ' ^72-1 JlO + Jli + Jlo + Jll /2^i +^2Tr 12- W^i + J/ , J * l + (/2 + ^4- x " ^2+4/6 + ^18

8 220 IMAGINARY AND COMPLEX QUANTITIES ( ) ( ) {ll + 2(l + 75)(l+77)} { v/(16-873)}. 19. «//o» 7(97-56 K«73). m 20. 9n N/(3 + V2) (3-272) 21. J* + J** V /2 + J(7-2Jl0)' J3 + J2 v/3 - J2 x/2 + ^(2 4-^/3) ^-V(2 + x/3r V(5 + 2 N /6)- x /(5-2 v /6) ^(5 + 2^/6)4-^(5-2^6)* 24. ^{6 + 2^24-2^3 + 2^/6}. 25. V/{ V2 4-4 V3 + 2 */6}. 26. V{ V2-4 V3-4 «/6-4 v'5-2 V */30}. 27. Shew that = 0. V(12-V140) V(8-V60) ^(10W84) 28. Shew that V(l 1-2 v/30) ^(7-2 V10) x/(8 + 4 ^3) = 0. IMAGINARY AND COMPLEX QUANTITIES We have already seen that in order that the formula obtained in Art. 81 for the factors of a quadratic expression may be applicable to all cases, it is necessary to consider expressions of the form J a, where a is

9 IMAGINARY AND COMPLEX QUANTITIES. 221 positive, and to assume that such expressions obey all the fundamental laws of algebra. Since all squares, whether of positive or of negative quantities, are positive, it follows that J a cannot represent any positive or negative quantity; it is on this account called an imaginary quantity. Also expressions of the form a + bj l where a and b are real, are called complex quantities The question now arises whether the meanings of the symbols of algebra can be so extended as to include these imaginary quantities. It is clear that nothing would be gained, and that very much would be lost, by extending the meanings of the symbols, except it be possible to do this consistently with all the fundamental laws remaining true. Now we have not to determine all the possible systems of meanings which might be assigned to algebraical symbols, both to the symbols which have hitherto been regarded as symbols of quantity and to the symbols of operation, subject only to the restriction that the fundamental laws should be satisfied in appearance whatever the symbols may mean: our problem is the much simpler and more definite one of finding a meaning for the imaginary expression J a which is consistent with the truth of all the fundamental laws We already know that 1 is an operation which performed upon any quantity changes it into a magnitude of a diametrically opposite kind. And, if we suppose that J 1 obeys the law expressed by 1 x J- 1 x N/^T= 1, it follows that J 1 must be an operation which when repeated is equivalent to a reversal. Now any species of magnitude whatever can be represented by lengths set off along a straight line; and, when a magnitude is so represented, we may consider the

10 222 COMPLEX QUANTITIES. operation J 1 to be a revolution through a right angle, fur a repetition of the process will turn the line in the same direction through a second right angle, and the line will then be directly opposite to its original direction. Hence, when magnitudes are represented by lengths measured along a straight line, we see that J 1, regarded as a symbol of operation, has a perfectly definite meaning. The symbol J 1 is generally for shortness denoted by i, and the operation denoted by i is considered to be a revolution through a right angle counter-clockwise, i denoting revolution through a right angle in the opposite direction It is clear that to take a units of length and then rotate through a right angle counter-clockwise gives the same result as to rotate the unit through a right angle counter-clockwise and then multiply by a. Thus ai = ia. Again, to multiply ai by hi is to do to ai what is done to the unit to obtain bi, that is to say we must multiply by b and then rotate through a right angle; we thus obtain ab units rotated through two right angles, so that ai x bi ab= abii. From the above we see that the symbol i is commutative with other symbols in a product. Since (ai) x (ai) = aaii = a 2 ( 1) = a 2, it follows that J a 2 = ai', it is therefore only necessary to use one imaginary expression, namely J With the above definition of J 1 or i, namely that it represents the operation of turning through a right angle counter-clockwise, magnitudes being represented by lengths measured along a straight line, the truth of the fundamental laws of algebra for imaginary and complex expressions can be proved. Some simple cases have been considered in the previous Article: for a full discussion see De Morgan's Double Algebra; see also Clifford's Common Sense of the Exact Sciences, Chapter iv. 12 and 13.

11 CONJUGATE COMPLEX EXPRESSIONS If a + bi = 0, where a and b are real, we have a = bi. But a real quantity cannot be equal to an imaginary one, unless they are both zero. Hence, if a 4- bi = 0, we have both a = 0 and 6 = 0. Note. In future, when an expression is written in the form a + bi, it will always be understood that a and 6 are both real If a + bi = c + di, we have a c + (b d) i 0 ; and hence, from Art. 186, a c = 0 and 6 d = 0. Thus, two complex expressions cannot be equal to one another, unless the real and imaginary parts are separately equal The expressions a 4- bi and a bi are said to be conjugate complex expressions. The sum of the two conjugate complex expressions a + bi and a bi is a + a + (b b) i = 2a; also their product is aa + obi abi 6V = a 2 + b\ Hence the sum and the product of two conjugate complex expressions are both real. Conversely, if the sum and the product of two complex expressions are both real, the expressions must be conjugate. ^ For let the expressions be $+ bi and c + di. The sum isa + 6i + c + &' = ft + c+ (b + d) i } which cannot be real unless b + d = 0. Again, (a + bi) (c + di) = ac-\-bci-\-adi + bdi 2 = ac bd + (be + ad)i f which cannot be real unless bc + ad = 0. Now, if b + d = 0 and also bc + ad = 0, we have 6 (c a) = 0 ; whence a = c or b = 0. If b = 0, d is also zero, and both expressions are real; and, if 6 + 0, we have a = c, which with b = d, shews that the expressions are conjugate Definition. The positive value of the square root of a 2 + 6' 2 is called the modulus of the complex

12 224 MODULUS OF A COMPLEX EXPRESSION. quantity a + bi, and is written mod (a + bi). Thus mod (a + bi) = + J a It is clear that two conjugate complex expressions have the same modulus; also, since (a + bi) (a bi) = a 2 +b 2 [Art. 188], the modulus of either of two conjugate complex expressions is equal to the positive square root of their product. Since a and b axe both real, a 2 + b 2 will be zero if, and cannot be zero unless, a and b are both zero. Thus the modulus of a complex expression vanishes if the expression vanishes, and conversely the expression will vanish if the modulus vanishes. If in mod (a + bi) = + J a 2 + b 2 we put b = 0, we have mod a = + J a 2, so that the modulus of a real quantity is its absolute value The product of a + bi and c 4- di is ac + bci + acfo' + bd? = ac bd + (be + ad) i. Hence the modulus of the product of a + bi and c 4- di is V{(ac - bd) 2 + (be + ad) 2 } = >J{(a 2 + b 2 ) (c 2 + d 2 )} = V(a )xv(c 2 + d 2 ). Thus the modulus of the product of two complex expressions is equal to the product of their moduli. The proposition can easily be extended to the case of the product of more than two complex expressions; and, since the modulus of a real quantity is its absolute value, we have the following Theorem. The modulus of the product of any number of quantities whether real or complex, is equal to the product of their moduli Since the modulus of the product of two complex expressions is equal to the product of their moduli, it follows conversely that the modulus of the quotient of two expressions is the quotient of their moduli. This may also be proved directly as follows :

13 MODULUS OF A PRODUCT. 225 (a / +, oi) i.'\ -f- / (c -f-. di) ^7-\ = a bi x c c + di c di _ac + bd + (be ad) i Hence mod «±"l = Vt(" + ^ + (fa-«ot V{o } _ mod (a + fa) " V{c" + d*} ~ mod (c + ^) ' 192. It is obvious that in order that the product of any number of real factors may vanish, it is necessary and sufficient that one of the factors should be zero, and, by means of the theorem of Art. 190, the proposition can be proved to be true when all or any of the factors are complex quantities. For, since the modulus of a product of any number of factors is equal to the product of their moduli, and since the moduli are all real, it follows that the modulus of a product cannot vanish unless the modulus of one of its factors vanishes. Now if the product of any number of factors vanishes its modulus must vanish [Art. 189]; therefore the modulus of one of the factors must vanish, and therefore that factor must itself vanish. Conversely, if one of the factors vanishes, its modulus will vanish; and therefore the modulus of the product and hence the product itself must vanish In the expression ax n + bx 11 ' 1 + cx n ' &, where a, b, c,... k are all real, let a. + /5i be substituted for #, and let P be the sum of all real terms in the result, and Qi the sum of all the imaginary terms. Then the given expression becomes P + Qi. Since P and Q are both real, they can contain only

14 226 MODULUS OF A PRODUCT. squares and higher even powers of {, and hence P and Q will not be changed by changing the sign of i. Therefore when a fii is substituted for x in the given expression the result will be P Qi. If now the given expression vanishes when a + /3i is substituted for x, we have P + Qi = 0. Hence, as P aod Q are real, we must have both P = Q and Q = 0, and therefore P Qi = 0. Hence if the given expression vanishes when a + /3i is substituted for x, it will also vanish when is substituted for x. Therefore [Art. 88] if x a fti is a factor of the given expression, x a + fii w 7 ili also be a factor. Thus, if any expression rational and integral in x, and with all its coefficients real, be divisible by either of two conjugate complex expressions it will also be divisible by the other.