The history of imaginary numbers is not a tale of a single moment or mathematician, but a centuries-long process of reluctant discovery and gradual acceptance. These numbers initially appeared to be mathematical fictions, arising from the need to solve algebraic problems that seemed to require only real-number answers. Their journey from mathematical nuisances to fundamental tools required the efforts of multiple brilliant minds across Europe. The story is less about invention and more about the slow realization that the number system needed to be expanded beyond the familiar real numbers.
Defining the Imaginary Concept
An imaginary number is defined by a simple yet unusual property: its square is a negative value. In the world of real numbers, squaring any number always results in a positive number or zero. The concept of taking the square root of a negative number was therefore long considered an algebraic impossibility.
The foundational element of this system is the imaginary unit, denoted by the lowercase letter \(i\). This unit is formally defined by the equation \(i^2 = -1\). Any number that is a real number multiplied by \(i\), such as \(5i\) or \(-0.5i\), is an imaginary number.
When an imaginary number is combined with a standard real number, the result is a complex number, which takes the form \(a + bi\). Here, \(a\) represents the real part and \(bi\) represents the imaginary part. These numbers are fully consistent mathematical constructs that extend the number line into a two-dimensional plane. They provide solutions to equations that are insoluble using only real numbers.
The First Encounters: Solving the Cubic Equation
The first encounter with these numbers occurred in 16th-century Italy, driven by the practical problem of solving cubic equations. The general solution for the cubic equation was a highly sought-after mathematical prize. Scipione del Ferro first found a method for a specific type of cubic equation, but kept his discovery secret.
The solution was later rediscovered by Niccolò Fontana Tartaglia, who shared it under an oath of secrecy with Gerolamo Cardano. Cardano published the method in his 1545 work, Ars Magna, crediting both men. This formula sometimes produced square roots of negative numbers, even when the final answer was a simple real number.
This scenario, known as the casus irreducibilis, forced mathematicians to confront the square root of a negative number as a necessary intermediate step. Cardano referred to these strange roots as “sophistic” and “useless.” Rafael Bombelli, in his 1572 work L’Algebra, first developed the rules for manipulating these square roots. Bombelli showed that treating these quantities as actual numbers allowed the intermediate steps to cancel out, yielding the correct real solution.
Formal Acceptance and Notation
The numbers remained controversial for centuries, viewed by many as mere algebraic tricks. The name that ultimately stuck was given by René Descartes in the 17th century. In his 1637 work La Géométrie, Descartes coined the term “imaginary” for roots that did not correspond to points on the real number line.
Descartes’s term was often used pejoratively, reflecting the view that these numbers lacked tangible reality. The true legitimization of the concept waited until the 18th century, largely due to the work of Leonhard Euler. Euler was instrumental in standardizing much of modern mathematical notation, which elevated the status of imaginary numbers.
Around 1777, Euler introduced the universal symbol \(i\) to denote the imaginary unit, \(\sqrt{-1}\). This standardization simplified complex algebraic manipulations and led to wider acceptance in mathematical analysis. Euler’s work, including his famous identity connecting \(i\), \(e\), and \(\pi\), demonstrated their utility and deep connections within mathematics.
Visualizing the Complex Plane
Despite the algebraic utility established by Euler, imaginary numbers still lacked a clear visual interpretation, which hindered their full acceptance. The breakthrough that made them tangible was their geometric representation in the form of the complex plane. This concept provided a visual context, showing that complex numbers were a logical extension of the real number system.
The complex plane is a two-dimensional coordinate system where the horizontal axis represents the real numbers and the vertical axis represents the imaginary numbers. A complex number \(a + bi\) can be plotted as a single point with coordinates \((a, b)\). This geometric view was first introduced by the Norwegian surveyor Caspar Wessel in 1797, though his work went largely unnoticed.
A similar representation was independently developed and published by Jean-Robert Argand in 1806, leading to the name Argand diagram. The interpretation was finally popularized by Carl Friedrich Gauss in the early 19th century. Gauss cemented the idea that multiplication by \(i\) corresponds to a rotation of 90 degrees.