foundations of mathematics Number systems

While the ancient Greeks were familiar with the positive integers, rationals, and reals, zero (used as an actual number instead of denoting a missing number) and the negative numbers were first used in India, as far as is known, by Brahmagupta in the 7th century ad. Complex numbers were introduced by the Italian Renaissance mathematician and physician Gerolamo Cardano (1501–76), not just to solve equations such as x² + 1 = 0 but because they were needed to find real solutions of certain cubic equations with real coefficients. Much later, the German mathematician Carl Friedrich Gauss (1777–1855) proved the fundamental theorem of algebra, that all equations with complex coefficients have complex solutions, thus removing the principal motivation for introducing new numbers. Still, the Irish mathematician Sir William Rowan Hamilton (1805–65) and the French mathematician Olinde Rodrigues (1794–1851) invented quaternions in the mid-19th century, but these proved to be less popular in the scientific community until quite recently.

Currently, a logical presentation of the number system, as taught at the university level, would be as follows:null → null → null → null → null → null. Here the letters, introduced by Nicolas Bourbaki, refer to the natural numbers, integers, rationals, reals, complex numbers, and quaternions, respectively, and the arrows indicate inclusion of each number system into the next. However, as has been shown, the historical development proceeds differently:null⁺ → null⁺ → null⁺ → null → null → null, where the plus sign indicates restriction to positive elements. This is the development, up to null, which is often adhered to at the high-school level.