Exploring the Concept of Imaginary Numbers

For those less familiar with advanced mathematics, the idea of imaginary numbers can seem confusing. What exactly are they? Are they fictional or invisible, much like imaginary companions? Math enthusiasts, we need your insights!

Defining an Imaginary Number

Imaginary numbers — essentially numbers that, when squared, produce a negative result — were first conceptualized in the 15th and 16th centuries as a solution to particularly challenging equations.

Originally perceived as a mathematical curiosity, imaginary numbers have evolved into a vital tool for understanding complex phenomena. Today, they are indispensable in areas such as electrical engineering and quantum physics.

"Imaginary numbers were created for reasons similar to why negative numbers were introduced," says Cristopher Moore, a physicist at the Santa Fe Institute in New Mexico and coauthor of the 2011 book "The Nature of Computation" with Stephan Mertens.

"Consider basic arithmetic," Moore adds. "What is two minus seven? Without negative numbers, this question has no answer. You can't physically have negative five apples, but you can owe someone five apples or five dollars. The concept became essential with the rise of accounting and bookkeeping."

Similarly, in modern times, we understand that writing checks exceeding our bank balance can result in a negative account balance, a concept now widely accepted.

Working With Complex and Imaginary Numbers

Moore suggests another perspective on negative numbers, which will prove useful later: imagine navigating through a city neighborhood.

If you take a wrong turn and head in the opposite direction of your destination — for instance, five blocks south instead of north — you could consider it as walking five negative blocks north.

"By introducing negative numbers, you expand the mathematical universe, making it possible to discuss concepts that were previously challenging," Moore explains.

Imaginary and complex numbers — those that include an imaginary part — exemplify this kind of innovative thinking. As Moore puts it: "If I ask you what the square root of nine is, it's straightforward, right? The answer is three, though it could also be negative three," since multiplying two negatives yields a positive.

But what about the square root of negative one? Is there a number that, when squared, results in negative one? "At a fundamental level, no such number exists," Moore states.

However, Renaissance mathematicians devised an ingenious solution to this dilemma. "Before the invention of negative numbers, the concept of two minus seven was nonexistent," Moore explains. "So, why not create a number that represents the square root of negative one? Let's name it i."

Once the idea of imaginary numbers was introduced, mathematicians realized they could perform fascinating operations with them. Recall that multiplying a positive by a negative yields a negative, while multiplying two negatives results in a positive.

But what occurs when you multiply i by seven and then by i again? Since i times i equals negative one, the result is negative seven. However, multiplying seven by i four times brings you back to positive seven. "They neutralize each other," Moore observes.

Consider this: by repeatedly incorporating an imaginary number into an equation, you arrive at a tangible, real-world number that is widely used.

Imaginary Numbers Represented as Points on a Plane

It wasn't until the early 19th century that mathematicians, including Mark Levi, found a new way to interpret imaginary numbers: by visualizing them as points on a plane. Levi is a professor and chair of the mathematics department at Penn State University and authored the 2012 book "Why Cats Land on Their Feet: And 76 Other Physical Paradoxes and Puzzles."

By extending the concept of numbers from a line to a two-dimensional plane, "the points on that plane represent imaginary numbers," he explains.

Picture a number line. Negative numbers are positioned 180 degrees opposite to positive numbers. "Multiplying two negative numbers adds their angles: 180 degrees plus 180 degrees equals 360 degrees, which is why the result is positive," Levi clarifies.

The square root of negative one doesn't fit anywhere on the X axis. It simply doesn't work. However, by introducing a Y axis perpendicular to the X axis, you create a space to accommodate it.

While imaginary numbers might appear to be purely abstract, they play a crucial role in modern technology, such as calculating airflow over an airplane wing or determining energy loss from resistance combined with oscillation in electrical systems.

Complex numbers with imaginary parts are also valuable in theoretical physics, according to Rolando Somma, a physicist specializing in quantum computing algorithms at Los Alamos National Laboratory.

"Because of their connection to trigonometric functions, they are ideal for describing periodic functions," Somma explains via email. "These functions emerge as solutions to wave equations, making complex numbers indispensable for modeling waves like electromagnetic waves. Thus, in physics, complex calculus is a powerful tool for simplifying calculations."

Complex numbers also play a significant role in quantum mechanics, a framework that explains the behavior of matter at atomic and subatomic levels.

"In quantum mechanics, the imaginary unit i explicitly appears in Schrödinger's equation," Somma notes. "This suggests that complex numbers hold a more fundamental role in quantum mechanics, beyond being merely a convenient computational tool."

"The state of a quantum system is defined by its wave function," he adds. "As a solution to Schrödinger's equation, this wave function represents a superposition of states, with complex numbers appearing in the coefficients. Quantum interference phenomena, for instance, can be succinctly described using complex numbers."