In this chapter, we further develop operations with products, quotients, and fractions and show how they can be utilized to solve certain types of equations. These algebraic methods are extremely useful in many applied problems as well as in higher-level mathematics.

The development of the symbols now used in algebra in itself led to advances in mathematics and science. Until about 1500, most problems and their solutions were stated in words, which made them very lengthy and often difficult to follow.

As time went on, writers abbreviated some of the words in a problem, but they still essentially wrote out the solution in words. Eventually, some symbols did start to come into use. For example, the $\hspace{0.17em}+\hspace{0.17em}$ and $\hspace{0.17em}-\hspace{0.17em}$ signs first appeared in a published book in the late 1400s, the square root symbol was first used in 1525, and the $\hspace{0.17em}=\hspace{0.17em}$ sign was introduced in 1557. Then, in the late 1500s, the French lawyer François Viète wrote several articles in which he used symbols, including letters, to represent numbers. He so improved the symbolism of algebra that he is often called “the father of algebra.” By the mid-1600s, the notation being used was reasonably similar to that we use today.

The use of symbols in algebra made it more useful in all fields of mathematics. It was important in the development of calculus in the 1600s and 1700s. In turn, this gave scientists a powerful tool that allowed for much greater advancement in all areas of science and technology.

Although the primary purpose of this chapter is to develop additional algebraic methods, many technical applications of these operations will be shown. The important applications in optics are noted by the picture of the Hubble telescope shown below. Other areas of application include electronics, mechanical design, thermodynamics, and physics.