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^ Descartes, René, Discours de la méthode (Leiden, (Netherlands): Jan Maire, 1637), appended book: La Géométrie, book three, p.Complex Numbers: lattice simulation and zeta function applications. College Algebra: Enhanced Edition (6th ed.).

Mathematical Analysis: Approximation and Discrete Processes (illustrated ed.).

^ Giaquinta, Mariano Modica, Giuseppe (2004).

A Text Book of Mathematics Class XI (Second ed.). This vertical axis is often called the "imaginary axis" and is denoted i R, 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. 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. 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. 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. The geometric significance of complex numbers as points in a plane was first described by Caspar Wessel (1745–1818). The use of imaginary numbers was not widely accepted until the work of Leonhard Euler (1707–1783) and Carl Friedrich Gauss (1777–1855). 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.

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. The concept had appeared in print earlier, such as in work by Gerolamo Cardano. The imaginary numbers are on the vertical coordinate axis.Īlthough the Greek mathematician and engineer Hero of Alexandria is noted as the first to present a calculation involving the square root of a negative number, it was Rafael Bombelli who first set down the rules for multiplication of complex numbers in 1572. Originally coined in the 17th century by René Descartes 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).Īn 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. By definition, zero is considered to be both real and imaginary. For example, 5 i is an imaginary number, and its square is −25. The square of an imaginary number bi is − b 2. For the 2013 EP by The Maine, see Imaginary Numbers (EP).Īn imaginary number is a real number multiplied by the imaginary unit i, which is defined by its property i 2 = −1.