Finding Nth Roots of Complex Numbers and Representing Solutions on an Argand Diagram

Teachings for Teachings for Exercise 1F Exercise 1F

?(???? + ?????)?= ??????? + ?????? ? ??: ? = ? cos(? + 2??) + ????(? + 2??) Complex Numbers ??: ? = ? ???? + ????? You can use De Moivre s theorem to find the nth roots of a complex number You already know how to find real roots of a number, but now we need to find both real roots and imaginary roots! We need to apply the following results: 1) If: ? = ? ???? + ????? Then: where k is an integer ? = ? cos(? + 2??) + ????(? + 2??) This is because we can add multiples of 2 to the argument as it will end up in the same place (2 = 360 ) 2) De Moivre s theorem ?(???? + ?????)?= ??????? + ?????? 1F

?(???? + ?????)?= ??????? + ?????? ? ??: ? = ? cos(? + 2??) + ????(? + 2??) Complex Numbers ??: ? = ? ???? + ????? Im In this case the modulus and argument are simple to find! You can use De Moivre s theorem to find the nth roots of a complex number Solve the equation z3 = 1 and represent your solutions on an Argand diagram. ? = 1 1 Re ? = 0 First you need to express z in the modulus-argument form. Use an Argand diagram. ?3= 1 ???0 + ????0 Apply the rule above Cube root (use a relevant power) Apply De Moivre s theorem Now we know r and we can set z3 equal to this expression, when written in the modulus-argument form ?3= cos(0 + 2??) + ????(0 + 2??) 1 3 ? = cos 0 + 2?? + ????(0 + 2??) We can then find an expression for z in terms of k 0 + 2?? 3 0 + 2?? 3 ? = cos + ???? We can then solve this to find the roots of the equation above 1F

?(???? + ?????)?= ??????? + ?????? ? ??: ? = ? cos(? + 2??) + ????(? + 2??) Complex Numbers ??: ? = ? ???? + ????? 0 + 2?? 3 0 + 2?? 3 ? = ??? + ???? You can use De Moivre s theorem to find the nth roots of a complex number k = 0 Sub k = 0 in and calculate the cosine and sine parts Solve the equation z3 = 1 and represent your solutions on an Argand diagram. ? = ??? 0 + ???? 0 ? = 1 We now just need to choose different values for k until we have found all the roots k = 1 2? 3 2? 3 ? = ??? + ???? Sub k = 1 in and calculate the cosine and sine parts The values of k you choose should keep the argument within the range: - < ? = 1 3 2+ ? 2 k = -1 Sub k = -1 in and calculate the cosine and sine parts (k = 2 would cause the argument to be outside the range) So the roots of z3 = 1 are: ? = ??? 2? + ???? 2? 3 3 ? = 1, 1 2 and 1 3 3 2+ ? 2 ? 2 ? = 1 3 2 ? (these are known as the cube roots of unity (ie 1)) 2 1F

?(???? + ?????)?= ??????? + ?????? ? ??: ? = ? cos(? + 2??) + ????(? + 2??) Complex Numbers ??: ? = ? ???? + ????? Im You can use De Moivre s theorem to find the nth roots of a complex number ? ? ?+ ? ? Solve the equation z3 = 1 and represent your solutions on an Argand diagram. 2 3? ? Re 2 3? We now just need to choose different values for k until we have found all the roots 2 3? ? ? ? ? The values of k you choose should keep the argument within the range: - < ? The solutions will all the same distance from the origin So the roots of z3 = 1 are: The angles between them will also be the same ? = 1, 1 2 and 1 3 3 2+ ? 2 ? 2 The sum of the roots is always equal to 0 1F

?(???? + ?????)?= ??????? + ?????? ? ??: ? = ? cos(? + 2??) + ????(? + 2??) Complex Numbers ??: ? = ? ???? + ????? ? = 1 3 2+ ? You can use De Moivre s theorem to find the nth roots of a complex number 2 Square 2 ?2= 1 3 2+ ? b) Show that the three cube roots of 1 can be written as 1 + ? + ?2 where 1 + ? + ?2= 0 2 Expand bracket ?2=1 3 4+ ?23 3 4 ? 4 ? 4 The roots of z3 = 1 are: Simplify ? = 1, 1 2 and 1 3 3 ?2=1 4 3 3 2+ ? 2 ? 4 2? 2 4 Simplify more ?2= 1 3 2 ? Let ? = 1 3 2 (ie the second of the roots) 2 2+ ? 1 + ? + ?2= 0 Replace terms 1 + 1 2 1 3 3 2+ ? 2 ? = 0 2 1F

?(???? + ?????)?= ??????? + ?????? ? ??: ? = ? cos(? + 2??) + ????(? + 2??) Complex Numbers ??: ? = ? ???? + ????? You can use De Moivre s theorem to find the nth roots of a complex number This result will be very important next lesson! 2??? ? for ? = 1,2..? 2?? ? 2?? ? In general, the solutions to ??= 1 are ? = ??? + ???? = ? These are known as the nth roots of unity If ? is a positive integer, there will be a root of unity ? such that: The roots can be written as 1,?,?2 ?? 1 1 + ? + ?2+..+?? 1= 0 (we just showed an example of this) The roots 1 + ? + ?2 will form the vertices of a regular polygon (more on this next lesson!) 1F

?(???? + ?????)?= ??????? + ?????? ? ??: ? = ? cos(? + 2??) + ????(? + 2??) Complex Numbers ??: ? = ? ???? + ????? Im Find the modulus and argument You can use De Moivre s theorem to find the nth roots of a complex number r 2 22+ 2 3 2 3 ? = 4 ? = Solve the equation ?4 2 3? = 2 Re ? = ??? 12 3 ? =? 2 2 3 Give your answers in both the modulus- argument and exponential forms. ? 3 ? 3 ?4= 4 ??? By rearranging ?4= 2 + 2 3? + ???? Apply the rule above Take the 4th root of each side ? 3+ 2?? + ???? ? 3+ 2?? ?4= 4 ??? As before, use an argand diagram to express the complex number in the modulus-argument form 1 4 ? 3+ 2?? + ???? ? 3+ 2?? 4 ? 3+ 2?? 4 ? 3+ 2?? ? 3+ 2?? 4 ? 3+ 2?? 4 ? = 4 ??? De Moivre s Theorem Work out the power at the front 1 4 ??? ? = 4 + ???? Then choose values of k until you have found all the solutions ? = 2 ??? + ???? 1F

?(???? + ?????)?= ??????? + ?????? ? ??: ? = ? cos(? + 2??) + ????(? + 2??) Complex Numbers ??: ? = ? ???? + ????? ? 3 4 ? 3 4 You can use De Moivre s theorem to find the nth roots of a complex number Sub k = 0 in and simplify (you can leave in this form) ? = 2 ??? + ???? k = 0 ? 12 ? 12 ? = 2 ??? + ???? Solve the equation ?4 2 3? = 2 ? 3+ 2? 4 ? 3+ 2? 4 ? = 2 ??? + ???? k = 1 Give your answers in both the modulus- argument and exponential forms. 7? 12 7? 12 ? = 2 ??? + ???? Choose values of k that keep the argument between and By rearranging ?4= 2 + 2 3? ? 3 2? 4 ? 3 2? 4 ? = 2 ??? + ???? k = -1 2 ??? 5? + ???? 5? ? = As before, use an argand diagram to express the complex number in the modulus-argument form 12 12 ? 3 2? 4 ? 3 2? 4 ? = 2 ??? + ???? ? 3+ 2?? 4 ? 3+ 2?? 4 k = -2 ? = 2 ??? + ???? 2 ??? 11? + ???? 11? ? = 12 12 Then choose values of k until you have found all the solutions 1F

?(???? + ?????)?= ??????? + ?????? ? ??: ? = ? cos(? + 2??) + ????(? + 2??) Complex Numbers ??: ? = ? ???? + ????? Solutions in the modulus-argument form You can use De Moivre s theorem to find the nth roots of a complex number 2 ??? 5? + ???? 5? ? 12 ? 12 ? = ? = 2 ??? + ???? 12 12 2 ??? 11? + ???? 11? 7? 12 7? 12 Solve the equation ?4 2 3? = 2 ? = ? = 2 ??? + ???? 12 12 Give your answers in both the modulus- argument and exponential forms. Solutions in the exponential form ? 12? 2? 5? 2? 11? 12? ? = 2? ? = By rearranging ?4= 2 + 2 3? 7? 12? 12? ? = 2? ? = As before, use an argand diagram to express the complex number in the modulus-argument form ? 3+ 2?? 4 ? 3+ 2?? 4 ? = 2 ??? + ???? Then choose values of k until you have found all the solutions 1F

?(???? + ?????)?= ??????? + ?????? ? ??: ? = ? cos(? + 2??) + ????(? + 2??) Complex Numbers ??: ? = ? ???? + ????? Im Find the modulus and argument You can use De Moivre s theorem to find the nth roots of a complex number 2+ 4 2 2 ? = 8 ? = 4 2 ???? = 3? Solve the equation: 4 2 Re 4 ?3+ 4 2 + 4? 2 = 0 4 2 r Rearrange ?3= 4 2 4? 2 Sketch the complex number on an argand diagram For the angle, since the lengths are equal, ? = 45 =? 4 So the argument will be equal to 3? 4 You can also solve these problems using the exponential form ?3= 4 2 4? 2 Write using the exponential form ?3= 8? 3? 4? 1F

?(???? + ?????)?= ??????? + ?????? ? ??: ? = ? cos(? + 2??) + ????(? + 2??) Complex Numbers ??: ? = ? ???? + ????? ?3= 8? 3? 4? You can use De Moivre s theorem to find the nth roots of a complex number Apply the rule above 3? 4+2?? ? ?3= 8? Cube root Solve the equation: 1 3 3? 1 3? 4+2?? ? ? = 8 Simplify ?3+ 4 2 + 4? 2 = 0 3? 4+2?? 3 ? ? = 2? ? = 0 ? = 1 ? = 1 3? 3 3? 3? 4+2? 3 4 2? 3 4 ? ? ? ? = 2? ? = 2? ? = 2? ? = 2? ? ? = 2? 11? 5? 12? 4? 12? ? = 2? Remember to ensure that the arguments are between ? and ? 1F