CardanBombeliEvaluated.nb 1 Math 421 è Fall 2006 Birth of complex numbers in solving cubic equations Revised 9/6/06 Copyright © 2006 by Murray Eisenberg. All rights reserved. The problem To solve the cubic equation x3 + Ax2 + Bx + K = 0. Strategy: è change-of-variable Ø new cubic with no quadratic term; è solve new cubic; è use the solutions of that to solve the original. Reduction of cubic to depressed cubic (Anonymous, end of 14th century) Temporarily replace x by u and rename the constant term K: In[1]:= cubic = u3 + Au2 + Bu + K Out[1]= K + Bu+ Au2 + u3 Make linear substitution 1 u = x - ÅÅÅÅ3 A 1 In[2]:= depressed = cubic .u→ x − A 3 A A 2 A 3 Out[2]= K + B − + x + A − + x + − + x 3 ê3 3 Collect coefficients of the powers of x: I M I M I M CardanBombeliEvaluated.nb 2 In[3]:= Collect depressed, x 2A3 AB A2 Out[3]= − + K + − + B x + x3 27 3 3 @ D That is the depressed cubic: no x2 term, so of form J N x3 + bx + c where A3 2 A3 AB b =-ÅÅÅÅÅÅÅ3 + B, c = ÅÅÅÅÅÅÅÅÅÅÅ27 - ÅÅÅÅÅÅÅÅÅ3 + K. In[4]:= depressedCubic = x3 + bx + c Out[4]= c + bx+ x3 1 Exercise. The linear substitution just used was u = x - ÅÅÅÅ3 A. Among all possible linear substitutions u = x - cst, 1 why use cst = ÅÅÅÅ3 A? del Ferro & Tartaglia solution of depressed cubic (Scipione del Ferro, 1515, and Niccolò Fontana aka "Tartaglia") Scipione del Ferro and Niccolò Tartaglia discovered formula for a root of a depressed cubic x3 + bx + c. In Mathematica: c c2 b3 c c2 b3 In[5]:= delFerroTartagliaRoot b_, c_ := 3 − + + + 3 − − + 2 4 27 2 4 27 In[6]:= delFerroTartagliaRoot@b, c D &''''''''''''''''''''''''''''''''$%%%%%%%%%%%%%%%%''''''''%%%%%%%''''' &''''''''''''''''''''''''''''''''$%%%%%%%%%%%%%%%%''''''''%%%%%%%''''' 1 3 1 3 c b3 c2 c b3 c2 Out[6]= − − + + − + + 2 27 4 @ 2 D 27 4 ê ê i y i y j z j z j $%%%%%%%%%%%%%%%%%%%%% z j $%%%%%%%%%%%%%%%%%%%%% z The del Ferro-Tartagliaj solutionz will appearj simpler if in thez depressed cubic you take b = 3 p and c = 2 q to obtain the form: k { k { x3 + 3 px + 2 q In[7]:= delFerroTartagliaRoot b, c . b −> 3 p, c → 2 q 1 3 1 3 Out[7]= −q − p3 + q2 + −q + p3 + q2 @ Dê 8 < ê ê Nicer formula from…è!!!!!!!!!!!!!!! è!!!!!!!!!!!!!!! I M I M CardanBombeliEvaluated.nb 3 In[8]:= niceDepressedCubic = depressedCubic . b → 3 p, c → 2 q Out[8]= 2q+ 3px+ x3 ê 8 < …by new Mathematica function: In[9]:= niceDelFerroTartagliaRoot p_, q_ := delFerroTartagliaRoot b, c . b −> 3 p, c → 2 q So del Ferro-Tartaglia formula for root of a depressed cubic of form x3 + 3 px + 2 q is: @ D @ Dê 8 < In[10]:= niceDelFerroTartagliaRoot p, q 1 3 1 3 Out[10]= −q − p3 + q2 + −q + p3 + q2 @ D ê ê è!!!!!!!!!!!!!!! è!!!!!!!!!!!!!!! I M I M Cardan's general solution of the cubic (Girolamo Cardano, Ars magna, 1545) Cardan uses the linear substitution that reduces a general cubic to a depressed cubic along with the del Ferro-Tartaglia formula for one solution of the depressed cubic to obtain a general formula for solving any cubic. Mathematica can do it, too: In[11]:= Solve x3 + Ax2 + Bx + K 0, x TraditionalForm Out[11]//TraditionalForm= A 1 x Ø-ÅÅÅÅÅÅ@+ ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ -2 A3 + 9 BA- 27 K + 3Dêê3 4 KA3 - B2 A2 - 18 BKA+ 4 B3 + 27 K2 ^ 1 3 - 3 3 3 2 3 2 3 B - A2 3 -2 A3 + 9 BA- 27 K + 3 3 4 KA3 - B2 A2 - 18 BKA+ 4 B3 + 27 K2 ^ 1 3 , è!!!! è!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! !!!!!!!! !!!!!!!! !!!!! 99 A è!!!!1 II M H ê LM x Ø-ÅÅÅÅÅÅ - ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ 1 - 3 -2 A3 + 9 BA- 27 K + 3 3 4 KA3 - B2 A2 - 18 BKA+ 4 B3 + 27 K2 ^ 1 3 + è!!!!3 6 3 2 è!!!! è!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! I H LM ë I I M H ê LM= 1 + 3 3 B - A2 322 3 -2 A3 + 9 BA- 27 K + 3 3 4 KA3 - B2 A2 - 18 BKA+ 4 B3 + 27 K2 ^ 1 3 , è!!!! è!!!! è!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! 9 II MI M H ê LM A è!!!!1 3 x Ø-ÅÅÅÅÅÅ - ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ 1 + 3 -2ê A + 9 BA- 27 K + 3 3 4 KA3 - B2 A2 - 18 BKA+ 4 B3 + 27 K2 ^ 1 3 + 3 è!!!!6 3 2 è!!!! è!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! II MH LM ë I I M H ê LM= 1 - 3 3 B - A2 322 3 -2 A3 + 9 BA- 27 K + 3 3 4 KA3 - B2 A2 - 18 BKA+ 4 B3 + 27 K2 ^ 1 3 è!!!! è!!!! è!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! !!!!!!!! !!!!!!!!!!!!! 9 è!!!! II MI M H ê LM ê è!!!! è!!!! è!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! !!!!!!!! !!!!!!!! !!!!! II MH LM ë I I M H ê LM== CardanBombeliEvaluated.nb 4 Bombeli: use square-roots of negative numbers to obtain real roots of cubics (Rafael Bombeli, L'algebra, 1572) ü Paradox è A depressed cubic always has a real solution: x3 + 3 px + 2 q = 0 is equivalent to x3 =-3 px - 2 q, and the cube function y = x3 always intersects the line y =-3 px - 2 q in at least one point, no matter what p and q are! In[12]:= Plot x3, x, −5, 5 , PlotStyle → Red ; @ 8 < 2 D 1 -4 -2 2 4 -1 -2 1 3 1 3 è Yet del Ferro-Tartaglia formula -q - p3 + q2 + -q + p3 + q2 involves square-roots of negative numbers when q2 ê<-p3. ê è!!!!!!!!!!!!!!! è!!!!!!!!!!!!!!! I M I M ü Example In the depressed cubic equation x3 + 3 px + 2 q, take p = 5 and q =-2: x3 - 15 x - 4 = 0 CardanBombeliEvaluated.nb 5 In[13]:= eqn = x3 − 15 x − 4 0 Out[13]= −4 − 15 x + x3 0 There must be a solution—the cubic y = x3 and the line y = 15 x + 4 must intersect: In[14]:= Plot x3,15 x + 4 , x, −7, 7 , PlotStyle → Red, Blue ; 75 @8 < 8 < 8 <D 50 25 -6 -4 -2 2 4 6 -25 -50 -75 In fact, x = 4 is a solution: In[15]:= eqn .x→ 4 Out[15]= True ê ü Bombeli's "wild thought" In[16]:= niceDelFerroTartagliaRoot p, q 1 3 1 3 Out[16]= −q − p3 + q2 + −q + p3 + q2 @ D ê ê The depressed cubicè!!!!!!! has!!!!!!!! p =-5, q =-è!!!!!!!2. So…!!!!!!!! I M I M In[17]:= p3 + q2 . p →−5, q →−2 Out[17]= −121 ê 8 < … then del Ferro-Tartaglia solution in this example is: 3 3 3 3 2 + -121 + 2 - -121 = 2 + 11 -1 + 2 - 11 -1 The known solution x = 4 could be recovered from the del Ferro-Tartaglia if the two terms on the right have the forms "###############è!!!!!!!#### ####!!!!!### "###############è!!!!!!!#### ####!!!!!### "###############è######## !!!!!!!#### "###############è######## !!!!!!!#### 3 3 2 + 11 -1 = m + n -1, 2 - 11 -1 = m - n -1. (*) for suitable"############### m and n.è####### # !!!!!!!#### è!!!!!!! "###############è######## !!!!!!!#### è!!!!!!! How? Well, the sum of these two solutions has the form… CardanBombeliEvaluated.nb 6 In[18]:= m + n −1 + m − n −1 Out[18]= 2m è!!!!!!! è!!!!!!! I M I M …so here we must have: 2 m = 4 (the known solution). Thus to obtain the solution x = 4, Bombeli wants to find n for which (*) holds with m = 2, in other words, 3 3 2 + n -1 = 2 + 11 -1, 2 - n -1 = 2 - 11 -1 Pretend usual rules of algebra hold for expressions involving è!!!!!!! è!!!!!!! è!!!!!!! è!!!!!!! I M I M Â= -1, and assume the special rule è!!!!!!! 2 Â2 = -1 =-1. 3 Then multiply out to calcluate 2 + n -1 : è!!!!!!! I M 3 In[19]:= theCube Expand 2 n 1 = +è!!!!!!! − I M Out[19]= 8 + 12 n − 6n2 −n3 è!!!!!!! AI M E To compare result with 2 + 11 -1 , separate the "real part" from the part involving Â: In[20]:= ComplexExpand theCube è!!!!!!! Out[20]= 8 − 6n2 + 12 n − n3 @ D That is supposed to equal 2 + 11 -1: H L In[21]:= Solve 8 − 6 n2,12 n − n3 2, 11 ,n è!!!!!!! Out[21]= n → 1 @8 < 8 < D Exercise.88 The "usual<< rules of algebra" include such identities as: x + y + u + v = x + u + y + v , x + y u + v = xu + yv + xv + yu, k x + y = kx + ky, kH xyL = Hkx y L= x Hky, L H L Hx + yLH+ z =L x + y + z . These identitiesH Lhold for real numbers x, y, u, v, k, z. Assume suchH LidentitiesH L hold forH "complexL numbers" as well—for numbers of the form a + b  where a and b are real. And stillH assumeL that Â2 =ÂÂ=-H 1.L Then put each of the following into the form y=u +Âv with u and v real: a + b  + c + d  , a + b  c + d  H L H L H LH L CardanBombeliEvaluated.nb 7 ü The moral Square-roots of negative numbers are useful (essential?) in obtaining real roots of certain cubic equations. (But what are such "complex" numbers? That's what's next in this course!) Appendices ü Appendix 1: Obtaining real imaginary parts of complex numbers involving symbolic quantities How in Mathematica can you solve for n the equation 8 - 6 n2 +Â12 n - n3 = 2 + 11 -1 without copying and pasting, or reading off and retyping, the real and imaginary parts (as was done above)? è!!!!!!! H L Use Re and Im. In[22]:= ?Re Re z gives the real part of the complex number z. More… In[23]:= ?Im @ D Im z gives the imaginary part of the complex number z.