An imaginary number is a real number multiplied by the imaginary unit i,^{[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. By definition, 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]

History

An illustration of the complex plane. The imaginary numbers are on the vertical coordinate axis.

Although 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,^[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

90-degree rotations in the complex plane

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 ${\displaystyle i\mathbb {R} ,}$ ${\displaystyle \mathbb {I} ,}$ or ℑ.^[12]

In this representation, multiplication by –1 corresponds to a rotation of 180 degrees about the origin, which is a half circle. Multiplication by i corresponds to a rotation of 90 degrees about the origin which is a quarter of a circle. Both these numbers are roots of ${\displaystyle 1}$: ${\displaystyle (-1)^{2}=1}$, ${\displaystyle i^{4}=1}$. In the field of complex numbers, for every ${\displaystyle n\in \mathbb {N} }$, ${\displaystyle 1}$ has nth roots ${\displaystyle \varphi _{n))$, meaning ${\displaystyle \varphi _{n}^{n}=1}$, called roots of unity. Multiplying by the first ${\displaystyle n}$th root of unity causes a rotation of ${\displaystyle {\frac {360}{n))}$ degrees about the origin.

Multiplying by a complex number is the same as rotating around the origin by the complex number's argument, followed by a scaling by its magnitude.^[13]

Square roots of negative numbers

Care must be used when working with imaginary numbers that are expressed as the principal values of the square roots of negative numbers:^[14]

${\displaystyle 6={\sqrt {36))={\sqrt {(-4)(-9)))\neq {\sqrt {-4)){\sqrt {-9))=(2i)(3i)=6i^{2}=-6.}$

That is sometimes written as:

${\displaystyle -1=i^{2}={\sqrt {-1)){\sqrt {-1)){\stackrel {\text{ (fallacy) )){=)){\sqrt {(-1)(-1)))={\sqrt {1))=1.}$

The fallacy occurs as the equality ${\displaystyle {\sqrt {xy))={\sqrt {x)){\sqrt {y))}$ fails when the variables are not suitably constrained. In that case, the equality fails to hold as the numbers are both negative, which can be demonstrated by:

${\displaystyle {\sqrt {-x)){\sqrt {-y))=i{\sqrt {x))\ i{\sqrt {y))=i^{2}{\sqrt {x)){\sqrt {y))=-{\sqrt {xy))\neq {\sqrt {xy)),}$

where both x and y are positive real numbers.