Post: Complex Numbers

Have you ever heard someone say that you can’t take the square root of a negative number? In some equations, there are situations where you must take the square root of a negative number in order to solve it. That’s what this post is about. The square root of -1 is known as $i$, the imaginary unit. Complex numbers are those that have the imaginary unit $i$, such as the term $3i$.

A Quadratic Equation

Let’s take the quadratic equation $x^2 + 6x + 10 = 0$. We can try to solve it using the formula $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$, but there is a problem. If we plug in the numbers into the formula, we will get $x = \frac{-6 \pm \sqrt{36 - 40}}{2}$. We can’t take the square root of -4 (subtracting 40 from 36 results in -4). It is an imaginary number. From here, we can conclude that the solution of the equation is a complex number. The square root of -4 is $2i$, so if we calculate the rest of the equation we get $x = -3 \pm i$.

The Importance of Complex Numbers

Complex numbers aren’t only used in algebra - they are also common in engineering and physics. $i$ is used in quantum mechanics, such as in the Schrodinger equation (the Schrodinger equation will be saved for another time). In electrical engineering, complex numbers are used for AC circuits. The magnitude and phase of voltage are represented by $i$. Euler’s equation is used to connect magnitude and phase (complex exponentials) to sine and cosine waves. Complex numbers make it easy for scientists and engineers to work with waves.

Conclusion

Complex numbers are all around us even when you don’t notice it. When you pass by an electron, or when you charge you phone. You can see how the square root of a negative number is really more than just an imaginary number - it models the universe. From solving a basic math problem to unifying the fundamental forces, complex numbers will always be there.