In fact, that was how the Arabic mathematician who wrote the first book of algebra came up with the idea: he had to describe the rules of arithmetic because he wanted to use the new Hindu numerals, including zero. Al-Khwarizmi first wrote a book to explain the new numerals, and then he went on to show the usefulness of general rules for reducing equations. The new methods for the first time produced a general solution for all quadratic equations — other mathematicians were very impressed.
Since that time, the middle of the ninth century A.D. [the recently politically correct designation is Common Era, C.E.], algebra has become one of the largest and widely-used branches of mathematics. It hasn’t produced finite algorithms for the solutions to polynomials of arbitrary degree: indeed, the impossibility of the solution of an arbitrary polynomial equation of quintic degree in a finite number of steps was proved at the beginning of the modern period of mathematics, in the first third of the nineteenth century.
Algebraic investigation has shown how we can expand the definition of number to include ordered pairs [a subset of which are the complex numbers, with a Real and an Imaginary part, which is to say an ordered pair, with special rules of multiplication], quartets [the technical term for which quadruples is quaternions; they are the device or vision of William Rowan Hamilton on 16 October 1843 and are, as with the Hamiltonian of the same prolific Irish astronomer/mathematician, fundamental to the expression of quantum-mechanical relations, in which form they are termed the Pauli Spin Matrices, after Wolfgang], and octets; yet still retain reasonable rules of addition, subtraction, multiplication, and division. The quartets are used in quantum electrodynamics, and the octets are useful in string theory. Al-Khwarizmi’s innovation has become the central discipline in the modeling of the world according to mathematical law.
From a philosophical point of view, the creation of algebra may have initiated the Scientific Revolution. Certainly the general solution of all polynomials of cubic degree, constructed in 1535 by the North Italian Nicolo Fontana, “Tartaglia” (‘the Stammerer”), marked a significant advance upon antiquity; its rapid succession, by the same group of savants, by the elaboration of the general solution to any polynomial of quartic degree, together with the first discussion of what later became generally termed imaginary numbers, argues for a progressive quickening of innovation in mathematical modeling of reality.
Just that sixteenth century which the American historian of experimental science Lynn Thorndike found to be missing a change of mental frame of reference, which 19th-century art historians began the habit of calling the Renaissance, just that time in northern Italy saw a golden era in algebraic mathematics. And with the following half-century we find the arrival of Galileo and the conceptual foundations put in place of modern scientific enquiry.
It is the considered opinion of the writer of these lines that another such Golden Era of Algebra took place in mid-19th-century Ireland.