Letters can represent any number, formulas with specific meanings, numbers that meet the requirements, and even numbers with certain rules. In short, letters can concisely express quantitative relations. For example: a can represent a set; F(x) represents the function of x and so on.

Elementary algebra refers to the equation theory developed before the first half of19th century. It mainly studies whether an equation [group] is solvable, how to find all the roots (including approximate roots) of the equation, and what properties the roots of the equation have.

Egyptian papyrus, written about 1700 years ago, has recorded the application problem of understanding the linear equation of one yuan, and even earlier Babylonians had solved the quadratic equation of one yuan in clay tablets by pairing method.

However, in ancient times, arithmetic, algebra and geometry were intertwined. In ancient Greece, geometry and mathematics were obviously separated, so that pure arithmetic or algebraic problems were translated into geometric language. For example, quantity was interpreted as length, and the product of two quantities was interpreted as rectangular area. In modern mathematics, quadratic power is still called "square" and cubic power is called "cube", which is the reason.

The ancient Greek mathematician Nicomark [1 century] wrote "Introduction to Arithmetic" around AD 100, which made the science of numbers independent from geometry for the first time. Thus setting an example for the establishment of pure algebra.

Diophantine (about 246-330), a Greek mathematician, published his first algebra book Arithmetic in the third century, including number theory and indefinite equations. He introduced unknowns and some operational symbols in this book, which greatly simplified algebraic expressions. Because the symbols of Diophantine are mostly abbreviations of related items, the algebra of Diophantine is called abbreviated algebra.

After the 4th century AD, Greek mathematics began to decline, but mathematics in India and the Middle East made great progress. Indian mathematicians in the 78th century mainly studied the solution of indefinite equations, and had used abbreviations and some symbols to express unknowns and operations. In Brahmagupta's works, a radical solution of quadratic equation x2+px-q=0 and general solutions of some indefinite equations are also given.