Late Modern Era
Matrix Theory
As noted in Chapter 5, Chinese mathematicians dabbled with using matrices to solve a system of equations. In the 19th century, Arthur Cayley and James Joseph Sylvester, two British mathematicians, established the matrix as a simple way of understanding and computing a system of equations (using inverse matrices). Not only did this result in new ways of thinking about solving equations, but it also created a new offshoot in mathematics: linear algebra – the study of geometry and space with matrices and algebra.
Complex Numbers
Although mathematicians knew about complex numbers since their introduction in Europe during the Renaissance, they widely ignored them due to their bizarre nature. That all changed when the Norwegian Mathematician Caspar Wessell introduced the idea of the complex plane. In his research paper, Wessell defined "i" geometrically as a line segment of length one perpendicular to the x-axis. He then logically concluded that i^2 is perpendicular to the vertical (imaginary) axis and therefore equal to -1 (180° rotation of a segment of length one on the positive x-axis is represented as -1 on the negative x-axis). Later, Sir William Rowan Hamilton, an Irish mathematician, expanded upon this idea and formally established the convention of communicating complex numbers with real and imaginary parts.
Abstract Algebra
Until this point, Algebra had always been about solving an unknown quantity. Civilizations developed processes to solve one or two unknowns, and eventually, algebra became about solving equations involving polynomials and other functions. However, with the introduction of Galois theory and advancements in the classification of numbers and sets, a new field emerged: Abstract Algebra. Instead of focusing on solving equations, this new field of algebra concentrated on the structures and properties of classical algebra.