Complex Numbers
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See all →- How do I find a power of any complex number, like (1 + i√3)⁶, using De Moivre's theorem? In one line Convert the number to polar form r(cosθ+isinθ) ; then its n th power is simply rⁿ(cos nθ+isin nθ) . Explained simply Multiplying... 1
- How do I find all the nth roots of a complex number and why are there exactly n of them? In one line Write the number as r[cos(θ+2kπ)+isin(θ+2kπ)] and apply De Moivre's theorem with power 1/n ; taking k=0,1,dots,n-1 gives the n d... 1
- How do I solve equations like x⁴ + 1 = 0 using roots of unity? In one line Rewrite the equation as xⁿ= a complex number, express that number in polar form with 2kπ added, and take the n th roots. Explain... 1
- What are the Cartesian, polar and exponential forms of a complex number and how do I convert between them? In one line A complex number can be written as x+iy (Cartesian), r(cosθ+isinθ) (polar) or re^iθ (exponential); all three describe the same p... 1
- How do I add, multiply and divide complex numbers and use modulus, argument and conjugate? In one line Complex numbers follow ordinary algebra with the single extra rule i²=-1 ; the conjugate bar z=x-iy is the tool that removes i f... 1
- What does De Moivre's theorem state and how does it simplify powers of cos θ + i sin θ? In one line De Moivre's theorem says (cosθ+isinθ)ⁿ=cos nθ+isin nθ : raising this number to the power n simply multiplies the angle by n . Ex... 1