a complex number

A complex number is what you get by gluing together an ordinary real number and a multiple of the imaginary unit i. We write it as z = x + i y, where x and y are ordinary real numbers and i obeys i^2 = -1. The x part lives on the usual number line; the i y part points in a brand-new perpendicular direction. So instead of a single number line, you now have a whole flat plane of numbers, and z = x + i y names one point in it — x steps to the right, then y steps up.

Two complex numbers are equal exactly when both their pieces match: x + i y equals a + i b precisely when x = a AND y = b. That is the key structural fact — a complex number carries two independent real quantities, not one, which is why you can think of it as a point (x, y) or as an arrow from the origin to that point. The real numbers are simply the complex numbers with y = 0, sitting along the horizontal axis, and the purely imaginary numbers (like 3i) are the ones with x = 0, sitting along the vertical axis.

Why bother? Because the complex numbers are 'algebraically closed': every polynomial equation, of any degree, with real or complex coefficients, has all its roots inside this one number system (the fundamental theorem of algebra). The reals are not enough for that — x^2 + 1 = 0 has no real root — but once you have z = x + i y, you never need to invent anything new again. That self-contained completeness is what makes complex analysis possible.