What is the square root of 2i?
Understand the Problem
The question is asking for the square root of the complex number 2i. To solve this, we need to express the number in polar form and then apply the formula for finding square roots of complex numbers.
Answer
The square roots of \( 2i \) are \( 1 + i \) and \( -1 - i \).
Answer for screen readers
The square roots of the complex number ( 2i ) are ( 1 + i ) and ( -1 - i ).
Steps to Solve
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Convert to Polar Form
First, let's express the complex number ( 2i ) in polar form. A complex number can be written as ( r(\cos \theta + i \sin \theta) ), where ( r ) is the magnitude and ( \theta ) is the argument (angle).For ( 2i ):
- The magnitude ( r ) is given by ( r = |2i| = \sqrt{(0)^2 + (2)^2} = 2 ).
- The argument ( \theta ) can be found as ( \theta = \frac{\pi}{2} ), since it lies on the positive imaginary axis.
Therefore, the polar form of ( 2i ) is:
$$ 2i = 2 \left( \cos\frac{\pi}{2} + i \sin\frac{\pi}{2} \right) $$ -
Apply the Square Root Formula
To find the square root of a complex number in polar form, we can use the formula:
$$ \sqrt{r} \left( \cos\frac{\theta + 2k\pi}{2} + i \sin\frac{\theta + 2k\pi}{2} \right) $$
where ( k = 0, 1 ).Substituting ( r = 2 ) and ( \theta = \frac{\pi}{2} ):
- Compute ( \sqrt{r} = \sqrt{2} ).
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Calculate the Two Roots
Now, we calculate both roots by substituting ( k = 0 ) and ( k = 1 ):-
For ( k = 0 ): $$ \text{Root}_0 = \sqrt{2} \left( \cos \frac{\frac{\pi}{2}}{2} + i \sin \frac{\frac{\pi}{2}}{2} \right) = \sqrt{2} \left( \cos \frac{\pi}{4} + i \sin \frac{\pi}{4} \right) $$ $$ = \sqrt{2} \left( \frac{\sqrt{2}}{2} + i \frac{\sqrt{2}}{2} \right) = 1 + i $$
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For ( k = 1 ): $$ \text{Root}_1 = \sqrt{2} \left( \cos \frac{\frac{\pi}{2} + 2\pi}{2} + i \sin \frac{\frac{\pi}{2} + 2\pi}{2} \right) = \sqrt{2} \left( \cos \frac{5\pi}{4} + i \sin \frac{5\pi}{4} \right) $$ $$ = \sqrt{2} \left( -\frac{\sqrt{2}}{2} - i \frac{\sqrt{2}}{2} \right) = -1 - i $$
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Final Roots
Thus, the two square roots of ( 2i ) are:
$$ 1 + i \quad \text{and} \quad -1 - i $$
The square roots of the complex number ( 2i ) are ( 1 + i ) and ( -1 - i ).
More Information
Finding the square root of complex numbers often involves converting to polar form, which allows us to apply trigonometric functions to calculate the roots. This technique is essential for working with complex numbers in higher math.
Tips
- Students sometimes forget to include both roots when calculating the square root of a complex number.
- Others may incorrectly convert the complex number to polar form, so it’s essential to check the steps while finding the magnitude and argument.
