K. 1 Introduction to complex numbers 5 wv Use the imaginary number \( i \) to rewrite the expression below as a complex number. Simplify all radicals. \( \sqrt{-25} \) \begin{tabular}{l} i \\ \hline\end{tabular}

The expression \(\sqrt{-25}\) can be rewritten using the imaginary unit \(i\). Since \(\sqrt{-1} = i\), we can express \(\sqrt{-25}\) as \(\sqrt{25} \cdot \sqrt{-1} = 5i\). Therefore, the simplified complex number is \(5i\). The concept of complex numbers, comprising real and imaginary parts, first gained traction in the 16th century with mathematicians like Gerolamo Cardano. Initially frowned upon, these numbers now play a critical role in diverse fields, from engineering to quantum physics, showcasing their indispensable nature in solving equations that lack real solutions. Complex numbers are not just theoretical; they're utilized in signal processing, fluid dynamics, and electrical engineering. In engineering, for example, alternating current (AC) circuits are often analyzed using complex numbers, allowing engineers to efficiently handle phase shifts and impedances in a simplified manner. They make calculations less cumbersome and help design more effective systems!