# Imaginary number

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An imaginary number is a square root of a nonpositive real number. Imaginary numbers have the form bi where b is a real number and i is the imaginary unit, defined as the square root of −1.[1]

An imaginary number bi can be added to a real number a to form a complex number of the form a + bi, where a and b are called respectively, the "real part" and the "imaginary part" of the complex number. Imaginary numbers can therefore be thought of as complex numbers where the real part is zero, and vice versa. The name "imaginary number" was originally coined in the seventeenth century as a derogatory term as such numbers were regarded by some as fictitious or useless, but today they have essential, concrete applications in a variety of scientific and related areas.

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### History

Although Greek mathematician and engineer Heron of Alexandria is noted as the first to have observed these numbers, imaginary numbers were defined in AD 1572 by Rafael Bombelli. At the time, such numbers were regarded by some as fictitious or useless, much as zero and the negative numbers once were. Many other mathematicians were slow to adopt the use of imaginary numbers, including René Descartes who wrote about them in his La Géométrie, where the term was meant to be derogatory.[2]

Descartes was the first to use the term “imaginary” number in 1637. However, the concept was invented much earlier by Gerolamo Cardano in the 16th century, but it 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 found by Caspar Wessel (1745–1818).[3]

In 1843 a mathematical physicist, William Rowan Hamilton, extended the idea of an axis of imaginary numbers in the plane to a three-dimensional space of quaternion imaginaries.

With the development of quotients of polynomial rings, the concept behind an imaginary number became more substantial, but then one also finds other imaginary numbers such as the j of tessarines which has a square of +1. This idea first surfaced with the articles by James Cockle beginning in 1848.