Square Roots of Complex Numbers

Square Roots of Complex Numbers:

The square roots of complex numbers can be determined either by expressing the complex number in the form of a perfect square or by expressing the square roots in the form of a + ib and then evaluating the real numbers ‘a’ and ‘b’ by squaring both the sides and then equating the real and the imaginary parts.

Let us assume, √(x + iy) = a + ib
Then, x + iy = (a + ib)2 = a2 – b2 + 2abi
∴ Equating real and imaginary parts, we get-
a2 – b2 = x ………(i) and 2ab = y ……..(ii)
Now, (a2 + b2)2 = (a2 – b2)2 + 4a2b2
⇒ (a2 + b2)2 = x2 + (2ab)2 = x2 + y2

∴ a2 + b2 = +√(x2 + y2) [the negative sign is disregarded as x, y ∈ R]

Thus, a2 – b2 = x and a2 + b2 = √(x2 + y2)
square root of a complex number formula

It is clear from (ii) that both a and b must be of the same sign (either both positive or both negative) when y > 0 and a and b will have opposite signs if y < 0.

Thus, if y > 0, then the square roots of x + iy are-

square roots of x + iy formula

Again, if y < 0, then the square roots of x + iy are-

Formula of square roots of x + iy
Example- If a2 + b2 = 1, show that (1 + ai + b)/(1 – ai + b) = b + ai

Solution- The given equation is,
(1 + ai + b)/(1 – ai + b) = b + ai

Taking L.H.S.,
= (1 + ai + b)/(1 – ai + b)
= [(1 + b) + ai]/[(1 + b) – ai]
= [(1 + b) + ai]/[(1 + b) – ai] x [(1 + b) + ai]/[(1 + b) + ai]
=[(1 + b) + ai]2/[(1 + b)2 – (ai)2]
= [(1 + b)2 + (ai)2 + 2 (1 + b) (ai)]/[1 + b2 + 2b + a2]
= [1 + b2 + 2b – a2 + 2 (1 + b) (ai)]/[1 + a2 + b2 + 2b]
We know that a2 + b2 = 1, therefore equation becomes-
= [a2 + b2 + b2 + 2b – a2 + 2 (1 + b) (ai)]/[1 + 1 + 2b]
= [2b2 + 2b + 2ai (1 + b)]/[2 + 2b]
= [2b (b + 1) + 2ai (1 + b)]/[2 (1 + b)]
= [2 (1 + b) (b + ai)]/[2 (1 + b)]
= b + ai
Hence L.H.S. = R.H.S.

Complex Numbers OperationsAtomic Orbital and Bond Order
Circumcircle and Inscribed Circle of a TriangleShapes of Atomic Orbitals
Addition and Subtraction Formulas in TrigonometryMolecular Orbital Theory
Periodic Function in TrigonometryPeriodic Table and Periodicity in Properties– NIOS

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