nth roots

nth Roots of Unity

  • Understand that the nth roots of unity are the solutions to the equation z^n = 1, where z is a complex number
  • Recognise that there are exactly n distinct nth roots of unity
  • Remember that these roots are evenly spaced around the unit circle in the complex plane. They can be expressed in the form cos(2πk/n) + isin(2πk/n), where k ranges from 0 to n-1
  • Be aware of the useful property that the sum of the nth roots of unity is zero

Calculating nth Roots

  • Understand that to calculate the nth roots of a complex number, it’s common to convert this number into polar form
  • Recall that a complex number can be written in polar form as r(cos θ + i sin θ), where r is the modulus of the number and θ is the argument
  • Know that the nth roots of a complex number in polar form, z = r(cos θ + i sin θ), will be s = (r^1/n)[cos((θ+2πk)/n) + i sin((θ+2πk)/n)], where k ranges from 0 to n-1
  • Acknowledge that these roots are also evenly spaced around a circle in the complex plane

De Moivre’s Theorem

  • Be comfortable using De Moivre’s theorem, which states that (cos θ + i sin θ)^n equals cos(nθ) + i sin(nθ)
  • Understand that De Moivre’s theorem is very useful when dealing with powers of complex numbers and can also help to find the nth roots of a complex number
  • Note that the theorem holds true for any real number n, allowing it to be used in finding roots of complex numbers
  • Realise the implications of combining De Moivre’s theorem with Euler’s formula: e^(iθ) = cos θ + i sin θ

Ensure that you’re comfortable solving problems involving nth roots in both cartestian and polar forms. Being able to visualise complex numbers in the complex plane can also be very beneficial for your understanding of this topic.