Animaginary number is the product of areal number and theimaginary uniti,[note 1] which is defined by its propertyi2 = −1.[1][2] Thesquare of an imaginary numberbi is−b2. For example,5i is an imaginary number, and its square is−25. The numberzero is considered to be both real and imaginary.[3]
An imaginary numberbi can be added to a real numbera to form acomplex number of the forma +bi, where the real numbersa andb are called, respectively, thereal part and theimaginary part of the complex number.[5]
An illustration of the complex plane. The imaginary numbers are on the vertical coordinate axis.
Although the Greekmathematician andengineerHeron of Alexandria is noted as the first to present a calculation involving the square root of a negative number,[6][7] it wasRafael Bombelli who first set down the rules for multiplication ofcomplex numbers in 1572. The concept had appeared in print earlier, such as in work byGerolamo 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, includingRené Descartes, who wrote about them in hisLa Géométrie in which he coined the termimaginary and meant it to be derogatory.[8][9] The use of imaginary numbers was not widely accepted until the work ofLeonhard Euler (1707–1783) andCarl Friedrich Gauss (1777–1855). The geometric significance of complex numbers as points in a plane was first described byCaspar 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 ofquaternion imaginaries in which three of the dimensions are analogous to the imaginary numbers in the complex field.
Geometrically, imaginary numbers are found on the vertical axis of thecomplex number plane, which allows them to be presentedperpendicular to the real axis. One way of viewing imaginary numbers is to consider a standardnumber line positively increasing in magnitude to the right and negatively increasing in magnitude to the left. At 0 on thex-axis, ay-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 orℑ.[12]
In this representation, multiplication by i corresponds to a counterclockwiserotation 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 numberbi, withb a real number, both causes a counterclockwise rotation about the origin by 90 degrees and scales the answer by a factor ofb. Whenb < 0, this can instead be described as a clockwise rotation by 90 degrees and a scaling by|b|.[13]
Care must be used when working with imaginary numbers that are expressed as theprincipal values of thesquare roots ofnegative numbers.[14] For example, ifx andy are both positive real numbers, the following chain of equalities appears reasonable at first glance:
But the result is clearly nonsense. The step where the square root was broken apart was illegitimate. (SeeMathematical fallacy.)
^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.
.svg)
