Why is i not defined as the square root of 1?

Why isn’t the square root of 1?

It’s a common point of confusion: if we use i to represent the imaginary unit, why do we define it as the square root of -1 instead of 1? The answer lies in the fundamental rules of real numbers.

When you square any real number—positive or negative—the result is always positive. For example, (-3) × (-3) = 9, and 3 × 3 = 9. There’s no real number that, when squared, gives you a negative result. This creates a problem when solving equations like x² = -1. In the world of real numbers, there’s no solution.

That’s where imaginary numbers come in. To bridge this gap, mathematicians introduced i, defining it specifically as the square root of -1. This isn’t a real number, it exists outside the real number line, in what we call the complex plane. So, i² = -1 by definition, not because it’s derived from real arithmetic, but because we needed a consistent way to work with such equations.

If we tried to define i as the square root of 1, we’d just get 1 or -1, both of which are already real numbers. That wouldn’t open up any new mathematical possibilities. The whole point of i is to go beyond the real, to handle cases that real numbers alone can’t solve.

So, i isn’t the square root of 1 because that wouldn’t solve the problem it was created for. Its power comes from being something other, a doorway into complex numbers, where math gains new depth and symmetry. It’s not about defying logic; it’s about expanding it.