Roots of Complex Numbers
How do I find the square root of a complex number?
- The square roots of a complex number will themselves be complex:
- i.e. if then
- We can then square () and equate it to the original complex number (), as they both describe :
- Then expand and simplify:
- As both sides are equal we are able to equate real and imaginary parts:
- Equating the real components: (1)
- Equating the imaginary components: (2)
- These equations can then be solved simultaneously to find the real and imaginary components of the square root
- In general, we can rearrange (2) to make and then substitute into (1)
- This will lead to a quartic equation in terms of d; which can be solved by making a substitution to turn it into a quadratic
- The values of can then be used to find the corresponding values of , so we now have both components of both square roots ()
- Note that one root will be the negative of the other root
- g. and
How do I use de Moivre’s Theorem to find roots of a complex number?
- De Moivre’s Theorem states that a complex number in modulus-argument form can be raised to the power of n by
- Raising the modulus to the power of n and multiplying the argument by n
- When in modulus-argument (polar) form de Moivre’s Theorem can then be used to find the roots of a complex number by
- Recall that adding 2π to the argument of a complex number does not change the complex number
- Therefore we must consider how different arguments will give the same result
- Taking the nth root of the modulus and dividing the argument by n
- If then
- This can be rewritten as
- This can be written in exponential (Euler’s) form as
- For ,
- The nth root of complex number will have n roots with the properties:
- The five roots of a complex number raised to the power 5 will create a regular pentagon on an Argand diagram
- The eight roots of a complex number raised to the power 8 will create a regular octagon on an Argand diagram
- The n roots of a complex number raised to the power n will create a regular n-sided polygon on an Argand diagram
- The modulus is for all roots
- There will be n different arguments spaced at equal intervals on a circle centred about the origin
- This creates some geometrically beautiful results
Examiner Tip
- de Moivre's theorem makes finding roots of complex numbers very easy, but you must be confident converting from Cartesian form into Polar and Euler's form first
- You can use your calculator to convert between forms
Worked example
a)
Find the square roots of 5 + 12i, giving your answers in the form a + bi.
b)
Solve the equation giving your answers in the form r (cosθ + isinθ).