Why Complex Numbers and Complex Analysis Matter

Imaginary numbers and complex numbers

For someone without a mathematics background, the idea that the letter "i" can stand for a number that does not entirely exist and is "imaginary" may be difficult to grasp.

Yet, once you are prepared to think in this way, it opens up an entirely new range of possibilities.

I am a mathematician working in analysis, a branch of mathematics concerned with complex numbers. Whereas the more familiar real numbers include positive and negative integers, fractions, square roots, cube roots and even figures such as pi, complex numbers contain an imaginary part.

They consist of real numbers together with the imaginary number i, which is the square root of negative 1.

A square root is a number which, when squared, produces the original value. Multiplying a positive number by itself gives a positive result, and so does multiplying a negative number by itself. The imaginary number i represents a number which, somehow, gives a negative result when it is multiplied by itself.

When imaginary numbers are discussed with nonmathematicians, the response is often something like, "But those numbers don't really exist, do they?"

Those doubts are far from unusual. Even some of mathematics' greatest figures struggled to accept complex numbers.

The label "imaginary" hardly helps people see that -√1 is not something fantastical. In Ars Magna, his 1545 book about complex numbers, mathematician Girolamo Cardano rejected them as "subtle as they are useless."

Even Leonhard Euler, regarded as one of the greatest mathematicians, is said to have calculated √(-2) √(-3) as √6. The right answer is -√6.

Quadratic equations and negative square roots

At school, you may have met the quadratic formula, which supplies answers for equations in which the unknown variable is squared.

Perhaps your teacher preferred not to address what occurs when (b2 - 4ac), the expression beneath the square root in the quadratic formula, is negative.

It may have been presented as a matter to tackle at university.

But if you are willing to accept that negative numbers can have square roots, you gain solutions to a completely new group of quadratic equations. More importantly, an astonishing and practical mathematical field comes into view: complex analysis.

Complex numbers simplify other areas of maths

What do complex numbers offer in return for this leap of faith?

For a start, they make trigonometry much simpler. Rather than learning several elaborate trigonometric formulae by heart, one equation is enough to govern them all: Euler's 1740 formula.

With sound algebra skills, Euler's formula can be rearranged to show that most standard trigonometric formulae used to calculate a triangle's length or angle become straightforward.

Calculus is simplified as well. Mathematicians Roger Cotes, René Descartes, who introduced the term "imaginary number", and others observed that complex numbers can make apparently impossible integrals simple to solve, allowing the area beneath complex curves to be measured.

Complex numbers are also useful for understanding every possible geometric shape that can be constructed using a ruler and compass. As mathematicians Jean-Robert Argand and Carl Friedrich Gauss noted, complex numbers can be used to manipulate geometric figures including pentagons and octagons.

Complex analysis in the real world

Complex analysis has numerous real-world uses.

Mathematician Rafael Bombelli's approach of carrying out algebraic operations - addition, subtraction, multiplication and division - with complex numbers enables their use in calculus.

This, in turn, makes much of the work scientists do in physics to investigate signals, or data transmission, easier to handle and understand.

For instance, complex analysis can manipulate wavelets, which are small oscillations in data. They are essential for eliminating noise from a distorted satellite signal and for compressing images so that data can be stored more efficiently.

Complex analysis also lets engineers convert a difficult problem into a simpler one. It is therefore a significant tool across many areas of applied physics, including the study of the electrical and fluid characteristics of complicated structures.

After becoming more at ease with complex numbers, prominent mathematicians including Karl Weierstrass, Augustin-Louis Cauchy and Bernhard Riemann were able to develop complex analysis. They created a valuable tool that not only makes mathematics simpler and science more advanced, but also helps make both easier to understand.

William Ross, Professor of Mathematics, University of Richmond

This article is republished from The Conversation under a Creative Commons licence. Read the original article.