What Are Imaginary Numbers?

The square root of 1 is 1 (or -1), the square root of 4 is 2 (or -2), and the square root of 9 is 3 (or -3). But what about the square root of -1? While you may not be able to think of a number that multiplies itself to a negative number, such numbers exist and form an integral part of the number line. Let’s explore what imaginary numbers are in this article.

The unit of the imaginary number, i, is defined as the square root of -1. That means, by definition, i2 = -1. Although it doesn’t look possible that any real numbers are the square root of negative numbers, this thing pops up from time to time in math problems. Therefore, a new type of number was discovered, and we call them imaginary numbers for convenience. This is contrary to real numbers, the type of numbers we usually encounter in everyday contexts.

Interestingly, when you multiply a number by i, you are not amplifying it across any axis on the number line as you would imagine with real numbers. Instead, you’re rotating your number by 90 degrees. Take i itself, for example. By definition, i * i = -1, so you rotate the number 90 degrees counterclockwise. -1 * i = -i, so you rotate it again in the same direction. Multiply by i again, and you get 1, and then i. Thus, i⁴ⁿ⁺¹, where n is an integer, equals to i itself.

When you think of the number line, you may think of it as a single line with negative numbers on the left and positive numbers on the right. However, that’s not the case for imaginary numbers. Instead, they make up the y-axis of the number line, which together make up the complex plane. Now you may be curious about the definition of “complex” in the plane, so we’ll discuss that in the next section.

With the conception of imaginary numbers, the number line is extended to the complex plane, from a 1-dimensional line to a 2-dimensional platform. As you would know, instead of focusing on themselves in two lines, points scatter across the plane. Therefore, there are combinations of real and imaginary numbers. And they’re called complex numbers.

A complex number is expressed as a + bi, where a and b are integers and i is the square root of -1. It composes of a real part, which is its value on the x-axis, and an imaginary part, which is the value on the y-axis. Put them together, and you will get a number in neither the real number line nor the imaginary number line, and instead somewhere in the complex plane.

Some may think that imaginary numbers have no use because it somehow doesn’t connect to the real world. But it’s not the case. In fact, in functions whose output is a real number, imaginary numbers can still show up as a necessary part of the intermediate steps of calculating the value. For example, when you square an imaginary number, you will always get a negative real number, by definition. That cancels out the imaginary parts and renders it a real function.

That means that they can be used to calculate things that simply can’t be solved with real numbers alone. For example, imaginary numbers help us calculate waves, forming some of the most crucial theories in the Universe and affecting some of the products in daily lives by enabling us to represent electricity, among other things, in a precise way.

In this article, we have introduced the concept of imaginary numbers, where they are on the number line, and the fact that those numbers do have daily applications. If you want to learn more about these intriguing numbers, please visit the webpages in the references below.

1. Elaine J. Hom. (2014, January 21). What Are Imaginary Numbers? Retrieved October 8, 2022, from https://www.livescience.com/42748-imaginary-numbers.html
2. (2018, October 26). The (Imaginary) Numbers at the Edge of Reality. Retrieved October 8, 2022, from https://www.quantamagazine.org/the-imaginary-numbers-at-the-edge-of-reality-20181025/
3. (2021, June 30). What are imaginary numbers? Retrieved October 8, 2022, from https://www.geeksforgeeks.org/what-are-imaginary-numbers/
4. (n.d.). Imaginary Numbers. Retrieved October 8, 2022, from https://www.mathsisfun.com/numbers/imaginary-numbers.html
5. (n.d.). Complex Numbers. Retrieved October 8, 2022, from https://www.mathsisfun.com/numbers/complex-numbers.html
6. (2021, March 3). Imaginary Numbers May Be Essential for Describing Reality. Retrieved October 8, 2022, from https://www.quantamagazine.org/imaginary-numbers-may-be-essential-for-describing-reality-20210303/