In the year 1545, the Italian mathematician Gerolamo Cardano published a groundbreaking book titled Ars Magna, which contained the first known solution to cubic equations. Within these solutions, he encountered a baffling situation where the calculation required the square root of a negative number. Cardano described these quantities as 'as subtle as they are useless' and later referred to the process of using them as 'mental torture.' He did not understand what these numbers were, yet he found that if he ignored them, his calculations for real-world problems involving cubic equations would fail. This was the birth of the complex number, born not from a desire to expand mathematics, but from the desperate need to solve equations that had no real solutions. The equation x squared equals negative one has no answer among the real numbers, because the square of any real number, whether positive or negative, is always non-negative. Yet, when mathematicians like Scipione del Ferro and Niccolò Fontana Tartaglia developed algorithms to solve cubic equations, they discovered that the path to a real solution sometimes required passing through these 'imaginary' numbers. Cardano's initial skepticism gave way to a grudging acceptance, but the true nature of these numbers remained a mystery for nearly three centuries.
The Imaginary Unit Defined
The symbol i, known as the imaginary unit, was coined by René Descartes in 1637 to distinguish these numbers from real ones. Descartes used the term 'imaginary' to stress their unreal nature, believing they existed only in the mind of the mathematician. The defining property of i is that it satisfies the equation i squared equals negative one. This single property allows every complex number to be expressed in the form a plus bi, where a and b are real numbers. The number a is called the real part, and b is called the imaginary part. For example, the number 3 plus 4i has a real part of 3 and an imaginary part of 4. While real numbers form a one-dimensional line, complex numbers form a two-dimensional plane. This plane, known as the complex plane or Argand diagram, allows complex numbers to be visualized as points or vectors. The horizontal axis represents the real part, and the vertical axis represents the imaginary part. A real number like 5 can be viewed as 5 plus 0i, sitting on the horizontal axis, while a purely imaginary number like 4i sits on the vertical axis. This geometric interpretation was not immediately accepted. It was not until the early 19th century that mathematicians like Caspar Wessel and Jean-Robert Argand independently proposed that complex numbers could be represented as points in a plane. Wessel's memoir appeared in the Proceedings of the Copenhagen Academy in 1799, but it went largely unnoticed. Argand published his pamphlet in 1806, providing a rigorous proof of the fundamental theorem of algebra and establishing the modern notation.