Imagine your teacher once told you: “You can’t take the square root of a negative number.”
Then one day, math says: “Actually… yes you can. Let’s just imagine it.” And thus, imaginary numbers were born.
They sound like fantasy, but imaginary numbers are very real—and they’ve transformed everything from electrical engineering to quantum physics. Let’s step into this strange and beautiful world.
What Is an Imaginary Number?
It all starts with a problem:
What is:
No real number gives us a negative when squared. So, mathematicians invented a new number:
This little symbol “i” is the foundation of all imaginary numbers,
Combine them with real numbers and you get complex numbers, like:
3 + 4i, −2 − i, 0 + i
These are called complex because they have two parts: one real, one imaginary.
Are They Real or Not?
“Imaginary” might sound fake—but in math, it’s just a name. Imaginary numbers are just as real as negative numbers once were. Think about it:
People once believed you couldn’t subtract a bigger number from a smaller one. Negative numbers were “nonsense.” Later, square roots of negative numbers seemed impossible. Now we use them to solve real-world problems. Imaginary numbers aren’t just weird—they’re useful.
Where Do We Use Imaginary Numbers?
You might be surprised how often they show up:
- Engineering: Used to analyze alternating current (AC) in electrical circuits.
- Quantum Physics: Complex numbers describe wave functions and probabilities.
- Signal Processing: Imaginary numbers help compress, transform and filter digital signals.
- Fractals: The beautiful Mandelbrot Set is built using complex numbers.
- Imaginary numbers turn out to be the key to solving equations that real numbers can’t touch.
The Complex Plane: A New Way to See Numbers
With imaginary numbers, math steps into two dimensions:
- The x–axis shows real numbers.
- The y–axis shows imaginary numbers.
So a number like 3 + 4i becomes a point in this 2D space: 3 steps right, 4 steps up. This “complex plane” turns math into geometry—where equations become curves, spirals and fractals.
Powers of i: The Magical Cycle
Imaginary numbers have a rhythm of their own:
i1 = i, i2 = −1, i3 = −i, i4 = 1, i5 = i (and so on…)
The powers of i repeat every 4 steps. It’s like the seasons of the math world.
A Real-World Superpower
Despite their name, imaginary numbers are real tools solving real problems in the real world. They help us:
- Model sound waves
- Predict quantum particle behavior
- Create smooth animations
- Build MRI machines
- Understand chaos and order
Without imaginary numbers, we wouldn’t have modern electronics, quantum computing, or fractal art.
Conclusion: Why Imaginary Numbers Matter
- They may seem strange at first. But imaginary numbers are:
- A natural extension of the number system
- A bridge between algebra and geometry
- A powerful tool for science and technology
- They remind us that math isn’t just about what’s obvious—it’s about what’s possible.
Imaginary numbers may not be “real”—but they’re what make reality work.
Name: Govind Sir
Department: Maths Foundation
Narayana Jaipur Center (NIHQ)