FP 2 Complex numbers nth roots KUS objectives

FP 2: Complex numbers : nth roots • KUS objectives BAT use De Moivres theorem to find the nth roots of a complex number Starter: write in exponential form

Notes 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

y First you need to express z in the modulus-argument form. Now then find an expression for z in terms of k 1 x We can then solve this to find the roots of the equation above k=0 k=1 TA DA! These are the roots of z 3 = 1 k = -1 k = 2 would cause the argument to be outside the range

WB 18 part 2 Solve the equation z 3 = 1 and represent your solutions on an Argand y diagram The values of k you choose should keep the argument within the range: -π < θ ≤ π x The solutions will all the same distance from the origin The angles between them will also be the same The sum of the roots is always equal to 0 SO the points make the vertices of a regular shape

Notes

WB 19 part 1 Solve the equation z 4 - 2√ 3 i = 2 modulus-argument and exponential forms. By rearranging… z 4 = 2 + 2√ 3 i As before, use an Argand diagram to express the equation in the modulus-argument form Give your answers in both the y r θ 2 2√ 3 x

WB 19 part 2 Solve the equation z 4 - 2√ 3 i = 2 modulus-argument and exponential forms. k=0 Give your answers in both the Solutions in the modulus-argument form k=1 Solutions in the exponential form k = -1 k = -2

As before, use an Argand diagram to express the equation in exponential form y r Equating the modulus and argument on both sides gives k=0 k=1 k = -1 θ x

Notes The nth roots of any complex number lie at the vertices of a regular polygon (n sides) with its centre at the origin

y O The coordinates are shown on the Argand diagram x

KUS objectives BAT use De Moivres theorem to find the nth roots of a complex number self-assess One thing learned is – One thing to improve is –

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