Exploring the Value of 1/i

Introduction

In this article, we delve into the concept of the value of frac{1}{i} where i is the imaginary unit defined as i sqrt{-1}.

Understanding the Value of 1/i with the Imaginary Unit

When i represents the imaginary unit, the value of frac{1}{i} can be computed as follows:

frac{1}{i} frac{1 cdot (-i)}{i cdot (-i)} frac{-i}{-1} i

This simplifies to -i.

When I is an Identity Matrix

In the context where I is the identity matrix, the answer to frac{1}{I} is the multiplicative inverse of the identity matrix. Since the identity matrix's multiplicative inverse is itself, the result is:

I^{-1} I

Thus, the answer is the identity matrix.

Generalization to Complex Numbers

For a more general case involving complex numbers, the expression frac{1}{i} can be evaluated using the property of exponents. Specifically, for any complex number x a bi, the expression x^{-n} is the same as frac{1}{x^n}.

Given that i raised to any integer power follows a repeating pattern:

(i^z)Result (z mod 4 0)(1) (z mod 4 1)(i) (z mod 4 2)(-1) (z mod 4 3)(-i)

Using this property, we can verify that frac{1}{i} -i and understand why this holds true for the imaginary unit.

Conclusion

To summarize, the value of frac{1}{i} when i is the imaginary unit is -i. When I is the identity matrix, the value of frac{1}{I} is the identity matrix itself. This article provides a detailed exploration of these concepts, including the necessary mathematical derivations and generalizations.