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Imaginary number

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A number of the form $x+iy$, where $i$ is the imaginary unit, $x$ and $y$ are real numbers and $y\neq0$, that is, a complex number which is not real; an imaginary number of the form $iy$ is called purely imaginary (sometimes only these numbers are called imaginary).

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Usually the phrases "imaginary number" and "complex number" just mean the same, "imaginary" being the historically chosen word and "complex" being more accepted nowadays.

Mathematicians were first confronted with imaginary numbers in the first decades of the 16th century. In the course of solving the equation $x^3=15x+4$ using the newly discovered methods (S. del Ferro (1465–1526), B. Tartaglia (1499–1557)) the number 4 appeared in the form $(2+\sqrt{-121})^{1/3}+(2-\sqrt{-121})^{1/3}$. It was R. Bombelli (1526–1572) who dared to operate with roots of negative numbers just as with "ordinary numbers" . Nevertheless, it was not until the first decades of the 17th century that these so-called "quantitates sophistacae" were more or less accepted, though reluctantly. Even R. Descartes (1596–1650) did not accept them as genuine numbers, but only as "mental buildings". During the 18th century these numbers gained more ground, mainly owing to the contributions of L. Euler (1707–1783), who also introduced the letter $i$ to denote $\sqrt{-1}$ ($i$ for imaginary). The use of the letter $i$ in this connotation was adopted by C.F. Gauss (1777–1855). Around 1800 several mathematicians, including J.R. Argand (1768–1822) and C. Wessel (1745–1818), gave geometrical interpretations of imaginary numbers. A. Girard (1595–1632) had already worked along the same lines. The best known result is the so-called Argand diagram, in which an imaginary number is represented by its modulus and argument (cf. Complex number, vector interpretation).

W.R. Hamilton (1805–1865) introduced imaginary numbers more algebraically as pairs $(a,b)$ of real numbers $a$ and $b$, and defined operations $+$ and $\cdot$ as follows:

$$(a,b)+(c,d)=(a+c,b+d),$$

$$(a,b)\cdot(c,d)=(ac-bd,ad+bc).$$

From these definitions the well-known computation rules for imaginary numbers are derived; it turns out that they form a field with these operations (denoted by $\mathbf C$). It can be proved that the field $\mathbf R$ of real numbers can be isomorphically imbedded in this field of complex numbers, whereby the real number $a$ corresponds to the pair $(a,0)$. In agreement with the multiplication rule: $(0,1)\cdot(0,1)=(-1,0)$, the pair $(0,1)$ may be identified with the imaginary unit $i$. So it follows that the pair $(a,b)$ can be written as

$$(a,b)=(a,0)+(0,b)=(a,0)+(b,0)\cdot(0,1),$$

which is also written as $a+bi$. The intuitive way of working with numbers $a+bi$ is thus given a sound basis.

How to Cite This Entry:
Imaginary number. Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Imaginary_number&oldid=33472