You remember high school math. You're sitting there, maybe bored, maybe confused, and the teacher drops a bomb: "You can't take the square root of a negative number." It makes sense, right? A positive times a positive is positive. A negative times a negative is also positive. There is literally no way to multiply a number by itself and get a negative result. It's impossible.
Then, a week later, they introduce i.
Suddenly, the "impossible" is just a letter. They tell you that i squared equals -1, and they expect you to just move on with your life. But honestly, it feels like cheating. If you feel like mathematicians just made this up because they got stuck, you're actually kind of right. But it's the most productive "cheating" in the history of human thought.
The weird reality of what i squared equals
The core definition is simple: $i^2 = -1$.
In the standard world of real numbers, this is total nonsense. If you try to find a spot for $i$ on the number line between 1 and 2, or back by -5, you’ll never find it. It doesn't live there. Instead, $i$—the imaginary unit—lives on its own axis.
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Think of it this way. The real number line is a single road running East and West. Positive numbers are East, negative numbers are West. Multiplying by -1 is basically a 180-degree U-turn. If you’re at 1 and you multiply by -1, you flip over to -1. If you multiply by -1 again, you flip back to 1.
But what if you only wanted to go halfway through that turn? What if you wanted to rotate 90 degrees? That’s what $i$ is. It’s a 90-degree turn off the road. When you multiply by $i$ twice, you've done two 90-degree turns, totaling 180 degrees. You’re now facing the opposite direction. You’re at -1. That is fundamentally why i squared equals -1. It’s not just a weird rule; it’s a rotation.
Why calling it "imaginary" was a huge mistake
René Descartes, the guy who gave us the coordinate plane, was actually the one who coined the term "imaginary." He didn't mean it as a compliment. He thought the idea of square roots of negative numbers was ridiculous and useless.
Later, Leonhard Euler and Carl Friedrich Gauss—some of the heaviest hitters in math history—realized Descartes was wrong. Gauss actually hated the name "imaginary." He thought we should have called them "lateral" numbers. If we had called them lateral numbers from the start, generations of students wouldn't feel like they were studying make-believe magic.
These numbers are real in the sense that they describe real-world phenomena. They aren't "imaginary" like a unicorn; they're "imaginary" like the wind—you can't see the number itself, but you can definitely see what it does to the things around it.
Where do we actually use this?
It’s easy to think this is just academic torture. It isn’t. Without the fact that i squared equals -1, your phone wouldn't work. The internet wouldn't exist. Your car's engine wouldn't have been designed as efficiently.
Electronics and Signal Processing
Electrical engineers use $i$ (though they often call it $j$ because they already used $i$ for current) to describe alternating current (AC). AC doesn't just flow in one direction; it oscillates. It has a phase. Real numbers are great for measuring "how much," but they suck at measuring "where in the cycle." Complex numbers, which use $i$, allow engineers to calculate voltage and current together without getting lost in a mess of trigonometry.
Quantum Mechanics
At the subatomic level, the universe gets weird. The Schrödinger equation, which is basically the "F=ma" of the quantum world, literally has $i$ in it. You cannot describe how an electron moves or where it is without using the square root of -1. If $i$ were just a figment of our imagination, the fundamental building blocks of reality wouldn't follow its rules.
Fluid Dynamics
If you've ever flown on a plane, you can thank $i$. Engineers use complex analysis to model how air flows over a wing. It turns out that treating 2D air flow as a complex plane makes the math incredibly elegant.
The cycle of powers
One of the coolest things about $i$ is that it's repetitive. It doesn't just go off into infinity like real numbers do when you power them up. It loops.
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- $i^1 = i$
- $i^2 = -1$
- $i^3 = -i$
- $i^4 = 1$
And then it starts over. $i^5$ is $i$ again. It’s a perfect four-step circle. This circular nature is exactly why it’s so good at describing things that spin or vibrate, like sound waves or the vibrations of a guitar string.
Common misconceptions that trip people up
People often try to apply standard square root rules to $i$ and end up breaking math.
A classic "gotcha" looks like this:
Is $1 = \sqrt{1} = \sqrt{-1 \times -1} = \sqrt{-1} \times \sqrt{-1} = i \times i = -1$?
Wait. Does $1 = -1$?
No. The rule $\sqrt{a \times b} = \sqrt{a} \times \sqrt{b}$ only works when $a$ and $b$ aren't both negative. This is a nuance that even smart people miss. It’s a reminder that while $i$ is powerful, you have to respect the boundaries of the system.
Actionable insights for mastering complex numbers
If you're trying to actually use this information—whether for a test, a coding project, or just to satisfy your brain—don't just memorize the formula.
Visualize the rotation.
Whenever you see $i$, stop thinking about "numbers" and start thinking about "turning." Multiplying by $i$ is a 90-degree turn counter-clockwise. This makes complex multiplication feel intuitive rather than like a chore.
Check your software libraries.
If you're a programmer, most modern languages like Python have native support for complex numbers. In Python, you use 1j (e.g., (2 + 3j)). Don't try to build your own complex number class from scratch unless you're doing it for fun; the built-in stuff is optimized for the weirdness of $i$.
Master the complex conjugate.
If you have a number like $3 + 4i$, its conjugate is $3 - 4i$. When you multiply them together, the imaginary parts vanish. This is the "secret sauce" for dividing complex numbers and is used constantly in signal processing to find the magnitude of a wave.
Don't fear the "imaginary."
Accept that $i$ is just an extension of our number system to include a second dimension. We went from counting fingers (natural numbers) to debt (negative numbers) to slices of pizza (fractions). $i$ is just the next logical step: moving from a flat line to a 2D plane. It’s a tool, and once you know how to use it, the "impossible" becomes remarkably useful.
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By understanding that i squared equals -1, you aren't just learning a trivia fact. You're looking at the hidden coordinate system that runs the modern world. It’s a shift in perspective that takes you from a one-dimensional view of math to a multi-dimensional understanding of how things actually oscillate, rotate, and exist.
Next Steps for Deepening Your Knowledge:
- Practice Vector Rotations: Try multiplying a coordinate $(x, y)$ by $i$ on a graph to see the 90-degree shift in action.
- Explore Euler’s Identity: Look into $e^{i\pi} + 1 = 0$, which famously links $i$, $\pi$, and $e$ in one equation.
- Review AC Circuit Basics: See how "impedance" uses $i$ to combine resistance and reactance into one number.