The successive enlargements of what counts as a number: the natural counting numbers (1, 2, 3...), extended to include zero and negative numbers (the integers), then ratios of integers (the rationals), then, to fill the gaps a rational number cannot reach, the real numbers, and finally, to give every polynomial a root, the complex numbers, which pair a real number with a multiple of the imaginary unit i. Each extension was historically controversial in its own moment, negative numbers and irrational numbers both took centuries to be accepted as genuine numbers rather than mere calculating devices, and each extension exists because the previous system could not, by itself, answer a question mathematicians actually needed to ask.