Numbers like 12, $\frac{5}{6}$, $\sqrt{2}$ and $\pi$ were known since antiquity and worked well for most purposes. For Euclid the concept of number meant the length of a line segment in geometry and as such he did not work with negative numbers. Diophantus acknowledged negative intermediate results in the third century but dismissed negative answers as absurd on the grounds that one cannot have a physical quantity less than zero. In the seventh century the Indian mathematician Brahmagupta described negative numbers as “debts” and gave the first clear rules for adding, subtracting, multiplying and dividing them. One consequence of his ideas was that positive times positive and negative times negative are both positive and thus $x^2$ can never be negative for any real number $x$. Around 850 CE the last of the notable Jain mathematicians Mahavira Acharya built on Brahmagupta’s work and realized that a negative number is never square and cannot have a square root. Much later in the sixteenth century in Europe the Italian mathematician Cardano met up with square roots of negative numbers while trying to solve cubic equations and was able to manipulate them though he considered them “fictitious” and essentially useless. They appear in his 1545 Latin textbook on Algebra titled Ars Magna meaning the great art. Finally in 1572 Rafael Bombelli developed the rules for working with square roots of negative numbers in his book L’ Algebra. This laid the foundations of what we now call complex numbers. In 1777 Euler introduced the use of the letter $i$ (meaning imaginary) to stand for the square root of minus one. Today complex numbers are an indispensable tool in many fields most notably electrical engineering where they are a great help in simplifying complicated formulas. However the letter $j$ is often used instead of $i$ which represents the current. Complex numbers are also used in quantum mechanics and in this instance they are not merely a calculation aid but an essential ingredient without which the entire theory would collapse. If you want to learn more about these mysterious and powerful entities please see our lessons below.