What are the Square Roots of 9i?

In the realm of complex numbers, the square root of a complex number can be quite intriguing. Let's delve into the problem of finding the square roots of z 9i. This will involve expressing the complex number in polar form and then using De Moivre's theorem to find the roots.

Understanding the Problem

The complex number 9i can be expressed as z 9qquad text{arg z} frac{pi}{2}. Here, z is represented in polar form z re^{i theta} where r |z| and theta text{arg z}. For z 9i, we have r 9 and theta frac{pi}{2}.

Using De Moivre's Theorem

De Moivre's theorem and Euler's formula are powerful tools in solving complex number problems. Euler's formula states that e^{i theta} cos theta i sin theta. We can use this to find the square roots of z.

Deriving the Square Roots

To find the square roots of z 9i, we use the formula for the nth roots of a complex number:

For z re^{i theta}, the n square roots are given by:

z_k sqrt{r} e^{left(frac{theta 2kpi}{n}right)} where k 0, 1, ..., n-1

Step-by-Step Calculation for Square Roots of 9i

Given z 9i, we have r 9 and theta frac{pi}{2}. Therefore, the two square roots are:

w_k sqrt{9} left[cos left(frac{frac{pi}{2} 2kpi}{2}right) isin left(frac{frac{pi}{2} 2kpi}{2}right)right] where k 0, 1

Let's break down the process for each value of k to find w_0 and w_1.

When k 0: w_0 3 left[cos left(dfrac{frac{pi}{2} 0}{2}right) isin left(dfrac{frac{pi}{2} 0}{2}right)right] 3 left[cos left(dfrac{pi}{4}right) isin left(dfrac{pi}{4}right)right] 3 left[frac{1}{sqrt{2}} ifrac{1}{sqrt{2}}right] frac{3}{sqrt{2}} ifrac{3}{sqrt{2}} Calculation for w_1

When k 1: w_1 3 left[cos left(dfrac{frac{pi}{2} 2pi}{2}right) isin left(dfrac{frac{pi}{2} 2pi}{2}right)right] 3 left[cos left(dfrac{5pi}{4}right) isin left(dfrac{5pi}{4}right)right] 3 left[-frac{1}{sqrt{2}} - ifrac{1}{sqrt{2}}right] -frac{3}{sqrt{2}} - ifrac{3}{sqrt{2}}

Conclusion

The square roots of 9i are therefore: {i left{frac{3}{sqrt{2}} ifrac{3}{sqrt{2}}, -frac{3}{sqrt{2}} - ifrac{3}{sqrt{2}} right}

Understanding how to derive these roots is crucial for anyone working with complex numbers in mathematics, engineering, and physics. This method not only illustrates the fundamental concepts of complex numbers but also how they can be manipulated in various applications.