Why Complex Numbers?

Complex Analysis · Axiom Academy

Some equations have no answer on the number line. Watch what happens when we stop saying "impossible" and invent the one number that fixes everything. One equation the real numbers can't answer You can solve x^2 = 9 in your head: x = 3 or x = -3 . So try the cousin of that problem — x^2 = -1 . You need a number whose square is negative. But squaring kills every minus sign: a positive squared is positive, and a negative squared is also positive. On the entire number line, nothing works. The equation x^2 + 1 = 0 has no real solution at all. Watch why. The sweep traces the parabola y = x^2 + 1 across the plane. Its lowest point sits at height 1 — the whole curve floats above the x -axis and never crosses it. No crossing means no real root. So instead of giving up, mathematicians did something bold: they defined a brand-new number, i , to be the answer — the number with i^2 = -1 — and dropped it into the gap. The curve clears the axis by exactly 1 , so there is no real x with x^2 + 1 = 0 . We invent one: i , with i^2 = -1 . From a line to a plane: every number is a + bi Once i exists, a number can carry two parts: a real part a and an imaginary part b , written a + bi . Real numbers don't disappear — they're the numbers with b = 0 , sitting on the horizontal axis. Drag the dot. Slide it along that axis and you have an ordinary real number; lift it off the line and you've reached a place the old number line could never go.

This is the written version of the interactive lesson above. See the full Complex Analysis course.