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Square Roots of a Complex Number (CIE A Level Maths: Pure 3): Revision Note
Square Roots of a Complex Number
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
- i.e. if
- 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)
- Equating the real components:
- 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 (see 1.1.5 Further Solving Quadratic Equations (Hidden Quadratics))
- In general, we can rearrange (2) to make
- 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
- i.e.
and
- i.e.
Worked example
Examiner Tip
- Most calculators used at A-Level can handle complex numbers.
- Once you have found the square roots algebraically; use your calculator to square them and make sure you get the number you were originally trying to square-root!
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