Taking a leap of faith into imaginary numbers opens new doors in the real world through complex analysis

To a nonmathematician, having the letter “i” represent a number that does not quite exist and is “imaginary” can be hard to wrap your head around. If you open your mind to this way of thinking, however, a whole new world becomes possible.

I’m a mathematician who studies analysis: an area of math that deals with complex numbers. Unlike the more familiar real numbers – positive and negative integers, fractions, square roots, cube roots and even numbers such as pi – complex numbers have an imaginary component. This means they are made of both real numbers and the imaginary number i: the square root of negative 1.

Remember, a square root of a number represents a number whose square is the original number. A positive number times itself is a positive number. A negative number times itself is a positive number. The imaginary number i depicts a number that somehow when multiplied by itself is negative.

Conversations about imaginary numbers with a nonmathematician often lead to objections like, “But those numbers don’t really exist, do they?” If you are one of these skeptics, you’re not alone. Even mathematical giants found complex numbers difficult to swallow. For one, calling -√1 “imaginary” isn’t doing it any favors in helping people understand that it’s not fantastical. Mathematician Girolamo Cardano, in his 1545 book dealing with complex numbers, “Ars Magna,” dismissed them as “subtle as they are useless.” Even Leonhard Euler, one of the greatest mathematicians, supposedly computed √(-2) √(-3) as √6. The correct answer is -√6.

In high school, you may have encountered the quadratic formula, which gives solutions to equations where the unknown variable is squared. Maybe your high school teacher didn’t want to deal with the issue of what happens when (b² - 4ac) – the expression under the square root in the quadratic formula – is negative. They might have brushed this under the rug as something to deal with in college.

However, if you are willing to believe in the existence of square roots of negative numbers, you will get solutions to a whole new set of quadratic equations. In fact, a whole amazing and useful world of mathematics comes into view: the world of complex analysis.

Complex numbers simplify other areas of math

What do you get for your leap of faith in complex numbers?

For one, trigonometry becomes a lot easier. Instead of memorizing several complicated trig formulas, you need only one equation to rule them all: Euler’s 1740 formula. With decent algebra skills, you can manipulate Euler’s formula to see that most of the standard trigonometric formulas used to measure a triangle’s length or angle become a snap.

Calculus becomes easier, too. As mathematicians Roger Cotes, René Descartes – who coined the term “imaginary number” – and others have observed, complex numbers make seemingly impossible integrals easy to solve and measure the area under complex curves.

Complex numbers also play a role in understanding all the possible geometric figures you can construct with a ruler and compass. As noted by mathematicians Jean-Robert Argand and Carl Friedrich Gauss, you can use complex numbers to manipulate geometric figures such as pentagons and octagons.

Complex analysis in the real world

Complex analysis has many applications to the real world.

Mathematician Rafael Bombelli’s idea of performing algebraic operations such as addition, subtraction, multiplication and division on complex numbers makes it possible to use them in calculus.

From here, much of what scientists use in physics to study signals – or data transmission – becomes more manageable and understandable. For example, complex analysis is used to manipulate wavelets, or small oscillations in data. These are critical to removing the noise in a garbled signal from a satellite, as well as compressing images for more efficient data storage.

Complex analysis allows engineers to transform a complicated problem into an easier one. Thus, it is also an important tool in many applied physics topics, such as studying the electrical and fluid properties of complicated structures.

Once they became more comfortable with complex numbers, famous mathematicians like Karl Weierstrass, Augustin-Louis Cauchy and Bernhard Riemann and others were able to develop complex analysis, building a useful tool that not only simplifies mathematics and advances science, but also makes them more understandable.

This article is republished from The Conversation, a nonprofit, independent news organization bringing you facts and trustworthy analysis to help you make sense of our complex world. It was written by: William Ross, University of Richmond

Read more:

William Ross does not work for, consult, own shares in or receive funding from any company or organization that would benefit from this article, and has disclosed no relevant affiliations beyond their academic appointment.