Imaginary Numbers Unlock New Possibilities in the Real World

As a mathematician specializing in analysis—a branch of mathematics that deals with complex numbers—I work regularly with numbers that have both real and imaginary components. While familiar real numbers include integers, fractions, and square roots, complex numbers extend this by incorporating the imaginary unit i, which is defined as the square root of -1.

To understand this, recall that the square root of a number is the value that, when multiplied by itself, results in the original number. A positive number squared yields a positive result, while squaring a negative number also produces a positive value. The imaginary number i represents a number whose square is negative, specifically i² = -1.

For many outside the mathematical world, the validity of imaginary numbers often comes into question. “Do these numbers really exist?” they ask. Even great mathematicians were initially skeptical. Girolamo Cardano, in his 1545 work Ars Magna, dismissed imaginary numbers as “subtle and useless.” Likewise, renowned mathematician Leonhard Euler once mistakenly calculated the square root of a negative number.

In high school, students often encounter the quadratic formula for solving equations with squared unknown variables. However, teachers may avoid addressing situations where the expression under the square root (b² - 4ac) becomes negative, suggesting that this would be covered in college. But by accepting the existence of square roots of negative numbers, you unlock a whole new category of quadratic equations, leading to the fascinating and practical world of complex analysis.
The Benefits of Complex Numbers in Mathematics

What advantages do complex numbers bring to mathematics? For one, trigonometry becomes more accessible. Instead of memorizing various complex trigonometric formulas, Euler’s formula simplifies many of them into a single equation, making them much easier to handle.

Calculus also becomes more straightforward. Mathematicians like Roger Cotes and René Descartes—who coined the term "imaginary number"—highlight that complex numbers enable the simple calculation of integrals and measurement of areas under complex curves.

Additionally, complex numbers provide a way to manipulate geometric shapes that can be constructed using just a ruler and compass. Mathematicians such as Jean-Robert Argand and Carl Friedrich Gauss used complex numbers to solve geometric problems involving polygons like pentagons and octagons.
Applications of Complex Analysis in the Real World

The power of complex analysis reaches far beyond the classroom, with numerous real-world applications. Through the work of mathematician Rafael Bombelli, who developed algebraic operations involving complex numbers, these numbers became vital tools in calculus.

In modern science, complex analysis plays a crucial role in the study of signals and data transmission. For example, complex analysis is essential in the field of wavelets—small oscillations in data that help remove noise from satellite signals and compress images for more efficient data storage.

Engineers also rely on complex analysis to simplify complex problems. In applied physics, it aids in understanding the electrical and fluid properties of intricate structures, making it an indispensable tool in a variety of industries.

The groundbreaking work of mathematicians like Karl Weierstrass, Augustin-Louis Cauchy, and Bernhard Riemann in advancing complex analysis has helped create a powerful mathematical framework. This framework not only simplifies complex equations but also propels scientific progress, making once-difficult concepts more accessible to students and professionals alike.

In embracing imaginary numbers, we not only unlock new mathematical tools but also open doors to innovations across technology, engineering, and science, demonstrating how abstract concepts can have real, tangible effects on the world around us.

This article was republished from The Conversation under a Creative Commons license. Read the original article here.