How to Find Square Root of Complex Number

Here you will learn what is square root and how to find square root of complex number with examples.

Let’s begin –

How to Find Square Root of Complex Number

Let a + ib be a complex number such that \(\sqrt{a + ib}\) = x + iy, where x and y are real numbers.

Then,

\(\sqrt{a + ib}\) = x + iy

\(\implies\) (a + ib) = \((x + iy)^2\)

\(\implies\) a + ib = \((x^2 – y^2)\) + 2ixy

On equating real and imaginary parts, we get

\(x^2 – y^2\) = a                  ………….(i)

and,  2xy = b                        …………..(ii)

Now, \((x^2 + y^2)^2\) = \((x^2 – y^2)\) + \(4x^2y^2\)

\(\implies\) \((x^2 + y^2)^2\) = \(a^2 + b^2\)

\(\implies\) \((x^2 + y^2)\) = \(\sqrt{a^2 + b^2}\)               ………..(iii)

Solving the equations (i) and (iii), we get

\(x^2\) = \((1\over 2)\){\(\sqrt{a^2 + b^2} + a\)}   and  \(y^2\) = \((1\over 2)\){\(\sqrt{a^2 + b^2} – a\)}

\(\implies\)  x = \(\pm\) \(\sqrt{{(1\over 2)}{\sqrt{a^2 + b^2} + a}}\)  and  y = \(\pm\) \(i\sqrt{{(1\over 2)}{\sqrt{a^2 + b^2} – a}}\)

If b is positive, then by equation (ii), x and y are of the same sign.

Hence, \(\sqrt{a + ib}\) = \(\pm\) [\(\sqrt{{(1\over 2)}{\sqrt{a^2 + b^2} + a}}\) + \(i\sqrt{{(1\over 2)}{\sqrt{a^2 + b^2} – a}}\)]

If b is negative, then by equation (ii), x and y are of the different signs.

Hence, \(\sqrt{a + ib}\) = \(\pm\) [\(\sqrt{{(1\over 2)}{\sqrt{a^2 + b^2} + a}}\) – \(i\sqrt{{(1\over 2)}{\sqrt{a^2 + b^2} – a}}\)]

Example : Find the square root of 7 – 24i.

Solution : Let \(\sqrt{7 – 24i}\) = x + iy. Then,

(7 – 24i) = \((x + iy)^2\)

\(\implies\) 7 – 24i = \((x^2 – y^2)\) + 2ixy

On equating real and imaginary parts, we get

\(x^2 – y^2\) = 7                  ……….(i)

and, 2xy = -24                     …………..(ii)

Now,  \((x^2 + y^2)^2\) = \((x^2 – y^2)^2\) + \(4x^2y^2\)

\(\implies\) \((x^2 + y^2)^2\) = 49 + 576 = 625

\(\implies\) \(x^2 + y^2\) = 25            ………..(iii)

On Solving equation (i) and (iii), we get

\(x^2\) = 16   and  \(y^2\) = 9 

\(\implies\) x = \(\pm 4\)   and  y = \(\pm 3\)

From (ii), 2xy is negative. So, x and y are of opposite signs.

\(\therefore\)   (x = 4 and y = -3)  or,  (x = -4 and y = 3)

Hence,  \(\sqrt{7 – 24i}\) = \(\pm\) (4 – 3i)

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