imaginary number
An imaginary number is what you get when you ask a forbidden question: what number, multiplied by itself, gives a negative result? No ordinary number works — a positive times a positive is positive, and so is a negative times a negative. So mathematicians simply invented one. They called it i, defined by the rule that i × i = −1, making it a square root of minus one. From this single seed grow all the imaginary numbers: 2i, −5i, and so on.
Far from being a useless curiosity, i turns out to be one of the most practical tools ever discovered. Pair it with the ordinary numbers and you get the complex numbers — written like 3 + 4i — which behave like a flat plane rather than a single line. Engineers lean on them constantly to describe alternating electrical current, radio waves, and vibrations; modern electronics and quantum physics would be almost impossible to write down without them.
The one thing to unlearn is the name. "Imaginary" was originally a sneer, hurled by skeptics who thought these numbers were fake. They aren't. They are exactly as real and as useful as the −1 that lets your bank account go into the red. The word stuck, but the insult was wrong.
The single rule that defines i — every imaginary number is built from it.
The dismissive label came from René Descartes, who used the French phrase "nombres imaginaires" in La Géométrie (1637). The symbol i for a square root of minus one was used by Leonhard Euler (in a paper written around 1777, published posthumously in 1794) and was popularized by Carl Friedrich Gauss in 1801. Today engineers often write j instead of i, to avoid clashing with i for electric current.