An imaginary number is a
real number multiplied by the
imaginary uniti,[note 1] which is defined by its property i2 = −1.[1][2] The
square of an imaginary number bi is −b2. For example, 5i is an imaginary number, and its square is −25. The number
zero is considered to be both real and imaginary.[3]
Originally coined in the 17th century by
René Descartes[4] as a derogatory term and regarded as fictitious or useless, the concept gained wide acceptance following the work of
Leonhard Euler (in the 18th century) and
Augustin-Louis Cauchy and
Carl Friedrich Gauss (in the early 19th century).
An imaginary number bi can be added to a real number a to form a
complex number of the form a + bi, where the real numbers a and b are called, respectively, the real part and the imaginary part of the complex number.[5]
Although the Greek
mathematician and
engineerHeron of Alexandria is noted as the first to present a calculation involving the square root of a negative number,[6][7] it was
Rafael Bombelli who first set down the rules for multiplication of
complex numbers in 1572. The concept had appeared in print earlier, such as in work by
Gerolamo Cardano. At the time, imaginary numbers and negative numbers were poorly understood and were regarded by some as fictitious or useless, much as zero once was. Many other mathematicians were slow to adopt the use of imaginary numbers, including
René Descartes, who wrote about them in his La Géométrie in which he coined the term imaginary and meant it to be derogatory.[8][9] The use of imaginary numbers was not widely accepted until the work of
Leonhard Euler (1707–1783) and
Carl Friedrich Gauss (1777–1855). The geometric significance of complex numbers as points in a plane was first described by
Caspar Wessel (1745–1818).[10]
In 1843,
William Rowan Hamilton extended the idea of an axis of imaginary numbers in the plane to a four-dimensional space of
quaternion imaginaries in which three of the dimensions are analogous to the imaginary numbers in the complex field.
Geometric interpretation
Geometrically, imaginary numbers are found on the vertical axis of the
complex number plane, which allows them to be presented
perpendicular to the real axis. One way of viewing imaginary numbers is to consider a standard
number line positively increasing in magnitude to the right and negatively increasing in magnitude to the left. At 0 on the x-axis, a y-axis can be drawn with "positive" direction going up; "positive" imaginary numbers then increase in magnitude upwards, and "negative" imaginary numbers increase in magnitude downwards. This vertical axis is often called the "imaginary axis"[11] and is denoted or ℑ.[12]
In this representation, multiplication by i corresponds to a counterclockwise
rotation of 90 degrees about the origin, which is a quarter of a circle. Multiplication by −i corresponds to a clockwise rotation of 90 degrees about the origin. Similarly, multiplying by a purely imaginary number bi, with b a real number, both causes a counterclockwise rotation about the origin by 90 degrees and scales the answer by a factor of b. When b < 0, this can instead be described as a clockwise rotation by 90 degrees and a scaling by |b|.[13]
^Descartes, René, Discours de la méthode (Leiden, (Netherlands): Jan Maire, 1637), appended book: La Géométrie, book three, p. 380.
From page 380:"Au reste tant les vrayes racines que les fausses ne sont pas tousjours reelles; mais quelquefois seulement imaginaires; c'est a dire qu'on peut bien tousjours en imaginer autant que jay dit en chasque Equation; mais qu'il n'y a quelquefois aucune quantité, qui corresponde a celles qu'on imagine, comme encore qu'on en puisse imaginer trois en celle cy, x3 – 6xx + 13x – 10 = 0, il n'y en a toutefois qu'une reelle, qui est 2, & pour les deux autres, quoy qu'on les augmente, ou diminue, ou multiplie en la façon que je viens d'expliquer, on ne sçauroit les rendre autres qu'imaginaires." (Moreover, the true roots as well as the false [roots] are not always real; but sometimes only imaginary [quantities]; that is to say, one can always imagine as many of them in each equation as I said; but there is sometimes no quantity that corresponds to what one imagines, just as although one can imagine three of them in this [equation], x3 – 6xx + 13x – 10 = 0, only one of them however is real, which is 2, and regarding the other two, although one increase, or decrease, or multiply them in the manner that I just explained, one would not be able to make them other than imaginary [quantities].)
^Martinez, Albert A. (2006), Negative Math: How Mathematical Rules Can Be Positively Bent, Princeton: Princeton University Press,
ISBN0-691-12309-8, discusses ambiguities of meaning in imaginary expressions in historical context.
^Rozenfeld, Boris Abramovich (1988).
"Chapter 10". A History of Non-Euclidean Geometry: Evolution of the Concept of a Geometric Space. Springer. p. 382.
ISBN0-387-96458-4.