Ise De Moivres Theorem to determine the square root of $$ z=y $$.

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De Moivre's Theorem states that for any complex number $$ z $$ expressed in polar form as $$ z = r(\cos \theta + i \sin \theta) $$, the $$ n $$-th power of $$ z $$ can be expressed as: $$ z^n = r^n (\cos(n\theta) + i \sin(n\theta)) $$ To find the square root of a complex number $$ z $$, we can set $$ n = \frac{1}{2} $$. The square root of $$ z $$ can be expressed as: $$ \sqrt{z} = z^{1/2} = r^{1/2} \left( \cos\left(\frac{\theta}{2}\right) + i \sin\left(\frac{\theta}{2}\right) \right) $$ ### Steps to Determine the Square Root of $$ z = y $$: 1. **Express $$ y $$ in Polar Form**: - If $$ y $$ is a real number, we can express it as: - If $$ y > 0 $$: $$ y = y(\cos(0) + i \sin(0)) $$ - If $$ y < 0 $$: $$ y = |y|(\cos(\pi) + i \sin(\pi)) $$ - If $$ y = 0 $$: The square root is simply $$ 0 $$. 2. **Calculate the Magnitude $$ r $$**: - For $$ y > 0 $$: $$ r = y $$ - For $$ y < 0 $$: $$ r = |y| $$ 3. **Determine the Argument $$ \theta $$**: - For $$ y > 0 $$: $$ \theta = 0 $$ - For $$ y < 0 $$: $$ \theta = \pi $$ 4. **Apply De Moivre's Theorem**: - For $$ y > 0 $$: $$ \sqrt{y} = \sqrt{y} \left( \cos\left(0\right) + i \sin\left(0\right) \right) = \sqrt{y} (1 + 0i) = \sqrt{y} $$ - For $$ y < 0 $$: $$ \sqrt{y} = \sqrt{|y|} \left( \cos\left(\frac{\pi}{2}\right) + i \sin\left(\frac{\pi}{2}\right) \right) = \sqrt{|y|} (0 + i) = i\sqrt{|y|} $$ ### Summary: - If $$ y > 0 $$: $$ \sqrt{y} = \sqrt{y} $$ - If $$ y < 0 $$: $$ \sqrt{y} = i\sqrt{|y|} $$ - If $$ y = 0 $$: $$ \sqrt{y} = 0 $$ This gives you the square root of $$ z = y $$ using De Moivre's Theorem. - If $$ y > 0 $$: $$ \sqrt{y} = \sqrt{y} $$ - If $$ y < 0 $$: $$ \sqrt{y} = i\sqrt{|y|} $$ - If $$ y = 0 $$: $$ \sqrt{y} = 0 $$

Ise De Moivres Theorem to determine the square root of .

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