Embracing the concept of imaginary numbers can be challenging, as they represent a number that doesn't exactly exist in the traditional sense.

However, if one opens their mind to this concept, a new realm of possibilities becomes accessible.

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As a mathematician specialising in analysis—a field of mathematics that involves complex numbers— I deal with numbers that consist of both real and imaginary components. Unlike familiar real numbers, which include integers, fractions, square roots, and even pi, complex numbers incorporate the imaginary number i, which is defined as the square root of -1.

To understand this, remember that a square root of a number refers to a number that, when multiplied by itself, equals the original number. While a positive number squared results in a positive number, a negative number squared results in a positive number as well. The imaginary number i represents a number that, when squared, results in a negative number.

Non-mathematicians often question the validity of imaginary numbers, asking, “But do these numbers really exist?” Even great mathematicians struggled with complex numbers. For instance, Girolamo Cardano, in his 1545 work Ars Magna, dismissed them as “subtle as they are useless.” Likewise, Leonhard Euler, one of the most renowned mathematicians, once made an error by calculating the square root of a negative number incorrectly.

In high school, you may have encountered the quadratic formula, which solves equations involving squared unknown variables. Teachers may have avoided addressing situations where the expression under the square root (b² - 4ac) is negative, suggesting this issue would be tackled later in college.

However, if you accept the existence of square roots of negative numbers, you unlock the potential to solve 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 does embracing complex numbers offer? For one, trigonometry becomes much easier. Instead of memorising several complex trigonometric formulas, Euler’s formula provides a single equation that simplifies many of these formulas, making them easier to handle.

Calculus, too, becomes simpler. As noted by mathematicians like Roger Cotes and René Descartes—who coined the term "imaginary number"—complex numbers allow for the straightforward calculation of integrals and the measurement of areas under complex curves.

Complex numbers also assist in understanding geometric shapes that can be constructed with just a ruler and compass. Mathematicians such as Jean-Robert Argand and Carl Friedrich Gauss used complex numbers to manipulate geometric figures like pentagons and octagons.

Applications of complex analysis in the real world

Complex analysis has numerous practical applications in the real world. Through the work of mathematician Rafael Bombelli, who explored algebraic operations with complex numbers, these numbers became useful in calculus.

This approach has made it easier for scientists to study signals and data transmission. Complex analysis, for example, is essential for working with wavelets—small oscillations in data crucial for removing noise from satellite signals and compressing images for more efficient data storage.

Complex analysis is also invaluable to engineers, who use it to simplify complicated problems. It plays a key role in applied physics, such as understanding the electrical and fluid properties of intricate structures.

As mathematicians like Karl Weierstrass, Augustin-Louis Cauchy, and Bernhard Riemann gained confidence with complex numbers, they advanced the development of complex analysis, creating a powerful tool that not only simplifies mathematics and propels scientific progress but also makes these subjects more accessible and understandable.

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This article was republished from The Conversation under a Creative Commons license. Read the original article here: https://theconversation.com/taking-a-leap-of-faith-into-imaginary-numbers-opens-new-doors-in-the-real-world-through-complex-analysis-233965.