# The Truth Behind Imaginary Numbers: Understanding Their Role

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Imaginary numbers can often seem perplexing, as I discovered while tutoring a psychology student a while back. We were discussing various number systems when I brought up imaginary numbers, prompting her to erupt in laughter.

*“What’s so amusing?”*

*“Imaginary numbers? You must be joking! That’s not real math.”*

When you’re deeply engaged in mathematical concepts, it’s easy to overlook how absurd they might sound to others.

*“But they are real!” I insisted. “And when combined with real numbers, they form complex numbers!”*

*“What? Now numbers can be complex too?!”*

It’s clear that the term “complex” has a vastly different connotation for someone studying psychology...

**The crux of the matter:** Don’t dismiss a mathematical concept based solely on its name. What might initially appear nonsensical could actually represent a groundbreaking idea in mathematics.

## Why Are They Called Imaginary?

### A Common Challenge

During the 16th and 17th centuries, mathematicians were devising methods and formulas to tackle complex algebraic equations. Being a mathematician at that time was quite prestigious, with colleagues engaging in math duels for monetary rewards for solving the most challenging problems.

However, they frequently encountered a frustrating obstacle.

This obstacle is one that algebra students are all too familiar with.

If the solutions to algebraic equations are where the graph intersects the x-axis, what happens when the graph looks like this? Do solutions simply not exist? But shouldn’t quadratics have two solutions?

### Bombelli’s Innovative Approach

Italian mathematician Rafael Bombelli faced the same dilemma, but rather than giving up, he found a workaround.

His thought process was simple: if no number squared equals -1, so be it. Instead of trying to find the square root of -1, he left it unresolved and continued.

Though unconventional, this method proved effective. He applied algebraic laws to the square root of -1, establishing what we now refer to as complex arithmetic.

However, Bombelli’s approach did not gain immediate acceptance.

Until that time, mathematics was grounded in tangible concepts. Problems either had practical applications or could be visualized geometrically or graphically.

The square root of -1 lacked both. It seemed nonsensical.

### Descartes Labels Them Imaginary

Much like the skepticism you might feel toward imaginary numbers, Bombelli’s contemporaries shared similar doubts.

One of these skeptics was Rene Descartes, who introduced the term *imaginary* in his work, La Geometrie:

> “For the rest, neither the false nor the true roots are always real; sometimes they are only imaginary. One may conceive as many as I stated in each equation, but at times there exists no quantity corresponding to those one imagines.” > > — Rene Descartes

Descartes emphasized that this represents an alternative system, a method for solving hypothetical scenarios. These *imaginary* roots, while useful, aren’t real in the sense of being true graph solutions.

They are *imagined* solutions.

### Gauss Clarifies the Confusion

Mathematicians eventually accepted Descartes’ view, and the term *imaginary* persisted. Gradually, they began implementing Bombelli’s rules, replacing the square root of -1 with *i* to highlight its intangible, imaginary aspect.

It took over a century and a prominent mathematician to clarify the confusion surrounding imaginary numbers.

> “The obscure nature of this subject [imaginary numbers] can largely be attributed to poorly suited notation. Had +1, -1, and the square root of -1 been referred to as direct, inverse, and lateral units instead of positive, negative, and imaginary (or impossible), such obscurity would have been avoided.” > > — Friedrich Gauss

Gauss contended that imaginary numbers are not fictitious; they are entirely logical and can be visualized.

What a relief!

The issue lies in our search for them in the wrong context. They do not belong to the set of real numbers. Instead, they exist alongside them, or laterally. You could think of them as another dimension, an extension of the real number line.

## From the Number Line to the Complex Plane

Expanding our number system might not be as bizarre as it sounds.

Mathematicians have been doing this for centuries. It began with the addition of zero to the **natural numbers** to create **whole numbers**. Subsequently, we incorporated **rational numbers**, the concept of negatives, and even **irrational numbers** like ?, *e*, and the square root of 2 to account for numbers without patterns or endpoints.

All of these fit neatly onto our number line, but what about imaginary numbers?

### Where Do Imaginary Numbers Fit?

The quick answer is: They don’t.

Imaginary numbers are an extension of the reals. We represent them by drawing a vertical *imaginary* number line through zero.

Together, these two number lines form the **complex plane**.

### The Key Lies in Rotation

The simplest way to comprehend imaginary numbers is to view multiplication by +1, -1, and ?-1 (or as Gauss describes them: direct, inverse, and lateral units) as rotations within the complex plane.

### Multiplying by +1

Multiplying by +1 can be interpreted as a 0° or 360° rotation around the origin, since either way, you end up right back where you started.

### Multiplying by -1

Multiplying by -1 represents a 180° counterclockwise rotation around the origin. For instance, starting at 2 and multiplying by -1 brings you to -2, which is a 180° rotation. If you multiply -2 by -1, you return to positive 2.

### Multiplying by *i* or ?-1

Here’s where it gets interesting. A 90° rotation is needed to engage with our new axis, which is where imaginary numbers come into play.

If I multiply 2 by *i*, I get 2*i*, resulting in a 90° counterclockwise rotation.

Multiplying 2*i* by *i* gives me 2*i². While it may not seem like -2 at first, recall that ? *i²* is indeed -1.

Thus, 2*i² = 2(-1) or -2, representing another 90° counterclockwise rotation.

Similarly, multiplying -2 by *i* yields -2*i*, another quarter turn.

Finally, -2*i* multiplied by *i* equals -2*i², or -2(-1), which results in 2.

Continuing to multiply by *i* allows us to rotate around the plane, which is why imaginary numbers form a repeating pattern every four iterations.

Ultimately, it turns out that imaginary numbers are indeed logical!

What fascinates me most about imaginary numbers is that it took centuries for humanity to grasp this paradigm-shifting idea. However, due to the relentless efforts of mathematicians to understand what was initially a theoretical concept, we now possess tools crucial for modern disciplines like electrical engineering and quantum mechanics.

So, if imaginary numbers still seem a bit odd and enigmatic, rest assured that you are not alone—many brilliant minds also grappled with this groundbreaking notion.

## Upcoming Topic: Working with Complex Numbers

In my next article, I will demonstrate how basic geometry can be employed to delve deeper into the complex plane and explore rotations that yield complex numbers.

## Further Reading

### The Lost Art of Square Roots

You may have never attempted this before. In fact, it took me a long time to realize that I had no clue how...

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### Why does a² + b² = c²?

Visual Proof of the Pythagorean Theorem

medium.com

### That Common Misconception About Probability

Dependent vs Independent Events

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### Why Does 0.999… Equal 1?!

Close only counts in horseshoes, hand grenades, and… math???

medium.com