# Complex Algebra

Abstract: Complex numbers were developed because there was a need to expand the notion of numbers to include solutions of algebraic equations whose prototype is $x^2 + 1 = 0.$ Such developments are not atypical in the history of mathematics. The invention of irrational numbers occurred because of a need for a number that could solve an equation of the form $x^2-2=0.$ Similarly, rational numbers were the offspring of the operations of multiplication and division and the quest for a number that gives, for example, 4 when multiplied by 3, or, equivalently, a number that solves the equation $3x-4=0.$ There is, however, a crucial difference between complex numbers and all the numbers mentioned above: All rational, irrational, and, in general, real numbers correspond to measurable physical quantities. However, there is no single measurable physical quantity that can be described by a complex number. A natural question then is this: What need is there for complex numbers if no physical quantity can be measured in terms of them? The answer is that although no single physical quantity can be expressed in terms of complex numbers, a pair of physical quantities can be neatly described by a single complex number. For example, a wave with a given amplitude and phase can be concisely described by a complex number. Another, more fundamental, reason is that equations that describe the behavior of subatomic particles are inherently complex.
(The paper)