Descartes meant “imaginary” as an insult—then Euler won
An imaginary number is a real multiple of i, where i² = −1; so (5i)² = −25. René Descartes coined the jab in the seventeenth century; Euler and later Gauss made the idea respectable. Add a real part and you hold a complex number a + bi.
Squaring any number of the form b times i gives minus b squared, so the square of 5i is minus 25, and zero counts as both real and imaginary. To separate them from general complex numbers, such values are often called purely imaginary.
Heron of Alexandria is credited with the first calculation that involved the square root of a negative quantity. Gerolamo Cardano's Ars Magna of 1545 put the idea in print, and in 1572 Rafael Bombelli wrote down the rules for multiplying complex numbers. For a long time these quantities, like negative numbers and once zero itself, were dismissed as fictitious or useless. Descartes coined the word imaginary in La Géométrie precisely to belittle them. Acceptance came with Euler and Gauss, and Caspar Wessel was the first to describe complex numbers as points in a plane. In 1843 William Rowan Hamilton went further, building a four-dimensional system of quaternions in which three axes play the role that the single imaginary axis plays for complex numbers.
Geometrically, the imaginary values occupy the vertical axis of the complex plane, crossing the ordinary number line at zero, with positive multiples rising upward and negative ones running downward. That picture gives multiplication a physical meaning: multiplying by i turns a point a quarter turn counterclockwise about the origin, while multiplying by minus i turns it a quarter turn clockwise. Multiplying by b times i, with b positive, does the counterclockwise quarter turn and also stretches the result by a factor of b.
The familiar rule that the root of a product equals the product of the roots breaks down here. Splitting the square root of 6 into the roots of minus 2 and minus 3 and multiplying produces minus the square root of 6, a false result, because the rule holds only for non-negative reals under the principal complex square root. Paul Nahin's 1998 book An Imaginary Tale traces the whole story.
Source: Imaginary number
