Complex Numbers (Math Lair)

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A complex number is a number of the form x + iy, where x and y are real numbers, and i is the square root of −1. The set of complex numbers if often denoted by C.

During the Renaissance in Europe (and earlier in India and the Muslim world to a certain extent), there was significant interest in algebra and in solving equations, including polynomial equations of degree 2, degree 3, degree 4, and even higher (the "degree" of a polynomial equation refers to the largest exponent of the variable in the equation). These equations can produce solutions such as √−9, which seemed to be nonsensical. Bhaskara, writing in India in the twelfth century, considered such answers but outlawed them, since the squares of both negative and positive numbers were positive and so a negative number could have no square root. During the sixteenth century, Cardano wrote down solutions to equations such as (5 ± √−15), but noted that they were meaningless and fictitious.

During the seventeenth century mathematicians realized (but did not yet prove) what would later become known as the fundamental theorem of algebra. The fundamental theorem of algebra states that an equation of degree n always has n solutions, or roots, and that these roots have to obey certain relationships between them. One of the people who studied these relationships was René Descartes. He called negative solutions to equations "false" and solutions involving negative square roots "imaginary".

The fundamental theorem of algebra also states that any root of a polynomial equation can be represented as a complex number. There is no need to go beyond the complex numbers to find solutions to polynomial equations (although such extensions, such as quaternions, can be useful for other purposes). During the next few centuries, complex numbers were useful in the theory of equations, but were slow to gain acceptance in other areas of mathematics. In the 18^th century, Euler showed how to treat imaginary numbers in exponents and recommended the general use of imaginary numbers.

Around this time, it also started to become common to represent complex numbers as points on the Cartesian plane. Real numbers were represented along the x-axis and imaginary numbers along the y-axis, and so any complex number can be represented as a point in the complex plane, similar to how real numbers can be represented as points on the real number line. This is called an Argand diagram, named after a book-keeper from Paris who was one of several people to independently discover it in the 19^th century.