The Square Root of the Imaginary Unit i

Introduction: In mathematics, particularly in complex analysis, the concept of the square root of the imaginary unit i can be expressed in various ways, including using Euler's formula and De Moivre's theorem. This article delves into how to determine the square root of i using these mathematical tools.

Understanding Euler's Theorem

Euler's formula, e^{i theta} cos theta isin theta, can be used to represent complex numbers in polar form. For the imaginary unit i, we have cos theta isin theta i.

Using Euler's formula:

Here, cos theta 0 and sin theta 1 imply that theta frac{pi}{2} 2kpi, where k is an integer. Therefore:

i e^{i (frac{pi}{2} 2kpi)}

Since we are dealing with the square roots, we consider the values of k 0 and k 1 to get:

Note: i e^{i frac{pi}{2}}e^{i frac{5pi}{2}}

The square roots are:

sqrt{i} e^{i frac{pi}{4}}e^{i frac{5pi}{4}} (cos frac{pi}{4} isin frac{pi}{4})(cos frac{5pi}{4} isin frac{5pi}{4}) pm frac{1}{sqrt{2}} ifrac{1}{sqrt{2}}

Verification of the Square Root

To verify the result, we square the value:

(frac{sqrt{2}isqrt{2}}{2})^2 frac{sqrt{2}isqrt{2}}{2} cdot frac{sqrt{2}isqrt{2}}{2} frac{2i^2}{4} frac{2(-1)}{4} -frac{2}{4} -frac{1}{2} i i

Similarly, for the second root:

(frac{-sqrt{2} - isqrt{2}}{2})^2 frac{-sqrt{2} - isqrt{2}}{2} cdot frac{-sqrt{2} - isqrt{2}}{2} frac{sqrt{2}^2 2isqrt{2}sqrt{2} (isqrt{2})^2}{4} frac{2 - 2 i^2 cdot 2}{4} frac{4i}{4} i

Using De Moivre's Theorem

De Moivre's theorem, which states that (cos theta isin theta)^n cos ntheta isin ntheta, can also be applied to find the square root of i:

Given z i cos frac{pi}{2} isin frac{pi}{2}, we can find the square root of z:

sqrt{i} z^{1/2} (cos frac{pi}{2} isin frac{pi}{2})^{1/2} cos left(frac{1}{2} cdot frac{pi}{2}right) isin left(frac{1}{2} cdot frac{pi}{2}right) cos frac{pi}{4} isin frac{pi}{4}

The principal square root is:

boxed{sqrt{i} frac{1}{sqrt{2}} frac{1}{2}i}

The other square root is the negative of the principal value:

-frac{1}{sqrt{2}} - frac{1}{2}i