THE THEORY OF EQUATIONS
3 5 8 + 2 + ??1 7 3 + 5I 2 ?3I 8 A + BI A ?BI 9 PROVE THAT THE SUM OF TWO CONJUGATE COMPLEX NUMBERS IS REAL AND THAT THEIR DI?ERENCE IS A PURE IMAGINARY §4 ] GEOMETRICAL REPRESENTATION 10 PROVE THAT THE CONJUGATE OF THE SUM OF TWO COMPLEX NUMBERS IS EQUAL TO THE SUM OF THEIR CONJUGATES DOES THE RESULT HOLD TRUE IF EACH WORD SUM IS REPLACED BY THE WORD DI?ERENCE? 11 PROVE THAT THE CONJUGATE OF THE PRODUCT (OR QUOTIENT) OF TWO COMPLEX NUMBERS IS EQUAL TO THE PRODUCT (OR QUOTIENT) OF THEIR CONJUGATES 12 PROVE THAT, IF THE PRODUCT OF TWO COMPLEX NUMBERS IS ZERO, AT LEAST ONE OF THEM IS ZERO 13 FIND TWO PAIRS OF REAL NUMBERS X, Y FOR WHICH (X + YI)2 = ?7 + 24I AS IN EX 13, EXPRESS AS COMPLEX NUMBERS THE SQUARE ROOTS OF 14 ?11 + 60I 15 5 ?12I 16 4CD + (2C2 ?2D2)I 3 CUBE ROOTS OF UNITY ANY COMPLEX NUMBER X WHOSE CUBE IS EQUAL TO UNITY IS CALLED A CUBE ROOT OF UNITY SINCE X3 ?1 = (X ?1)(X2 + X + 1), THE ROOTS OF X3 = 1 ARE 1 AND THE TWO NUMBERS X FOR WHICH X2 + X + 1 = 0, (X + 4, X + 2 + ? 3I IN VIEW OF THE ORIGIN OF ?, WE HAVE THE IMPORTANT RELATIONS ?2 + ? + 1 = 0, ?3 = 1 SINCE ??? = 1 AND ?3 = 1, IT FOLLOWS THAT ?? = ?2, ? = ??2 4 GEOMETRICAL REPRESENTATION OF COMPLEX NUMBERS USING RECT- ANGULAR AXES OF COÖRDINATES, OX AND OY , WE REPRESENT THE COMPLEX NUMBER A + BI BY THE POINT A HAVING THE COÖRDINATES A, B (FIG 1) COMPLEX NUMBERS [CH I O A X Y ? A B R FIG THE POSITIVE NUMBER R = ? A2 + B2 GIVING THE LENGTH OF OA IS CALLED THE MODULUS (OR ABSOLUTE VALUE) OF A+BI THE ANGLE ? = XOA, MEASURED COUNTER-CLOCKWISE FROM OX TO OA, IS CALLED THE AMPLITUDE (OR ARGUMENT) OF A + BI THUS COS ? = A/R, SIN ? = B/R, WHENCE (1) A + BI = R(COS ? + I SIN ?) THE SECOND MEMBER IS CALLED THE TRIGONOMETRIC FORM OF A + BI FOR THE AMPLITUDE WE MAY SELECT, INSTEAD OF ?, ANY OF THE ANGLES ? ± 360?, ? ± 720?, ETC TWO COMPLEX NUMBERS ARE EQUAL IF AND ONLY IF THEIR MODULI ARE EQUAL AND AN AMPLITUDE OF THE ONE IS EQUAL TO AN AMPLITUDE OF THE OTHER FIG AT THE VERTICES OF AN EQUILATERAL TRIANGLE INSCRIBED IN A CIRCLE OF RADIUS UNITY AND CENTER AT THE ORI- GIN O (FIG 2) THE INDICATED AMPLITUDES OF ? AND ?2 ARE 120?AND 240?RESPECTIVELY, WHILE THE MODULUS OF EACH IS 1 THE MODULUS OF ?3 IS 3 AND ITS AMPLITUDE IS 180?OR 180?PLUS OR MINUS THE PRODUCT OF 360?BY ANY POSITIVE WHOLE NUMBER 5 PRODUCT OF COMPLEX NUMBERS BY ACTUAL MULTIPLICATION, R(COS ? + I SIN ?) R?(COS ? + I SIN ?) = RR? (COS ? COS ? ?SIN ? SIN ?) + I(SIN ? COS ? + COS ? SIN ?) = RR? COS(? + ?) + I SIN(? + ?)], BY TRIGONOMETRY HENCE THE MODULUS OF THE PRODUCT OF TWO COMPLEX NUMBERS IS EQUAL TO THE PROD- UCT OF THEIR MODULI, WHILE THE AMPLITUDE OF THE PRODUCT IS EQUAL TO THE SUM OF THEIR AMPLITUDES FOR EXAMPLE, THE SQUARE OF ? = COS 120?+ I SIN 120?HAS THE MODULUS 1 AND THE AMPLITUDE 120?+ 120?AND HENCE IS ?2 = COS 240?+ I SIN 240? AGAIN, THE §8 ] CUBE ROOTS PRODUCT OF ? AND ?2 HAS THE MODULUS 1 AND THE AMPLITUDE 120?+ 240?AND HENCE IS COS 360?+ I SIN 360?, WHICH REDUCES TO 1 THIS AGREES WITH THE KNOWN FACT THAT ?3 = 1 TAKING R = R? = 1 IN THE ABOVE RELATION, WE OBTAIN THE USEFUL FORMULA (2) (COS ? + I SIN ?)(COS ? + I SIN ?) = COS(? + ?) + I SIN(? + ?) 6 QUOTIENT OF COMPLEX NUMBERS TAKING ? = ? ?? IN (2) AND DI- VIDING THE MEMBERS OF THE RESULTING EQUATION BY COS ? + I SIN ?, WE GET COS ? + I SIN ? COS ? + I SIN ? = COS(? ??) + I SIN(? ??) HENCE THE AMPLITUDE OF THE QUOTIENT OF R(COS ?+I SIN ?) BY R(COS ?+I SIN ?) IS EQUAL TO THE DI?ERENCE ? ?? OF THEIR AMPLITUDES, WHILE THE MODULUS OF THE QUOTIENT IS EQUAL TO THE QUOTIENT R/R OF THEIR MODULI THE CASE ? = 0 GIVES THE USEFUL FORMULA COS ? + I SIN ? = COS ? ?I SIN ? 7 DE MOIVRE’S THEOREM IF N IS ANY POSITIVE WHOLE NUMBER, (3) (COS ? + I SIN ?)N = COS N? + I SIN N? THIS RELATION IS EVIDENTLY TRUE WHEN N = 1, AND WHEN N = 2 IT FOLLOWS FROM FORMULA (2) WITH ? = ? TO PROCEED BY MATHEMATICAL INDUCTION, SUPPOSE THAT OUR RELATION HAS BEEN ESTABLISHED FOR THE VALUES 1, 2, , M OF N WE CAN THEN PROVE THAT IT HOLDS ALSO FOR THE NEXT VALUE M + 1 OF N FOR, BY HYPOTHESIS, WE HAVE (COS ? + I SIN ?)M = COS M? + I SIN M? MULTIPLY EACH MEMBER BY COS ? + I SIN ?, AND FOR THE PRODUCT ON THE RIGHT SUB- STITUTE ITS VALUE FROM (2) WITH ? = M? THUS (COS ? + I SIN ?)M+1 = (COS ? + I SIN ?)(COS M? + I SIN M?), = COS(? + M?) + I SIN(? + M?), WHICH PROVES (3) WHEN N = M + 1 HENCE THE INDUCTION IS COMPLETE EXAMPLES ARE FURNISHED BY THE RESULTS AT THE END OF §5: (COS 120?+ I SIN 120?)2 = COS 240?+ I SIN 240?, (COS 120?+ I SIN 120?)3 = COS 360?+ I SIN 360? COMPLEX NUMBERS [CH I 8 CUBE ROOTS TO ?ND THE CUBE ROOTS OF A COMPLEX NUMBER, WE ?RST EXPRESS THE NUMBER IN ITS TRIGONOMETRIC FORM FOR EXAMPLE, ? 2 + OR R3 IN FACT, FROM K = 3 WE OBTAIN R1, FROM K = 4 WE OBTAIN R2, FROM K = WE OBTAIN R3, FROM K = 6 WE OBTAIN R1 AGAIN, AND SO ON PERIODICALLY EXERCISES 1 VERIFY THAT R2 = ?R1, R3 = ?2R1 VERIFY THAT R1 IS A CUBE ROOT OF 8(COS 45?+ I SIN 45?) BY CUBING R1 AND APPLYING DE MOIVRE’S THEOREM WHY ARE THE NEW EXPRESSIONS FOR R2 AND R3 EVIDENTLY ALSO CUBE ROOTS? 2 FIND THE THREE CUBE ROOTS OF ?27; THOSE OF ?I; THOSE OF ? 3 FIND THE TWO SQUARE ROOTS OF I; THOSE OF ?I; THOSE OF ? 4 PROVE THAT THE NUMBERS COS ?+I SIN ? AND NO OTHERS ARE REPRESENTED BY POINTS ON THE CIRCLE OF RADIUS UNITY WHOSE CENTER IS THE ORIGIN 2HERE, AS ELSEWHERE WHEN THE CONTRARY IS NOT SPECI?ED, ZERO AND NEGATIVE AS WELL AS POSITIVE WHOLE NUMBERS ARE INCLUDED UNDER THE TERM “INTEGER ” §9 ] ROOTS OF COMPLEX NUMBERS 5 IF A+BI AND C+DI ARE REPRESENTED BY THE POINTS A AND C IN FIG 3, PROVE THAT THEIR SUM IS REPRESENTED BY THE FOURTH VERTEX S OF THE PARALLELOGRAM TWO OF WHOSE SIDES ARE OA AND OC HENCE SHOW THAT THE MODULUS OF THE SUM OF TWO COMPLEX NUMBERS IS EQUAL TO OR LESS THAN THE SUM OF THEIR MODULI, AND IS EQUAL TO OR GREATER THAN THE DI?ERENCE OF THEIR MODULI X Y O E F H G S A C FIG X U O A C P FIG