Algebra: Rules and Symbols

What is algebra? This question is not as simple as it seems. In this course, we will explore the meaning and extraordinary mathematical power of algebra through a study of its historical development. We’ll begin with so-called classical algebra—or the invented use of letter symbols (variables) to represent numbers and the rules for their manipulation. 

Using variables led to the key concepts such as polynomials and a focus on solving equations–a focus that drove the development of algebra for a millenia, from Babylonian cuneiform tablets to the influential 9th-century treatise “?is?b al-jabr wa’l-muq?bala” (Calculation by Completion and Balancing) by al-Khwarizmi. Next, we’ll explore the development of symbolic algebra (initiated by the Latin translation of al-Khwarizmi), the utilization of cubic equations, which gave rise to complex numbers, and Descartes’s breakthrough invention of an X-Y coordinate system, which tied together for the first time algebra and geometry. 

Finally, we’ll explore the multiplicity of algebras that blossomed in the 19th century till today: linear (matrix) algebra, which arose from solving system of linear equations; “abstract” algebra, which defined algebraic structures such groups, fields, and rings; and Boolean algebra, which treated logic algebraically. Throughout, we’ll ask: how does algebra undergird the ways we interpret, organize and manipulate the world?