Imaginary number

A (purely) imaginary number (also imaginary number , Latin numerus imaginarius ) is a complex number whose square is a non-positive real number . Equivalently, one can define the imaginary numbers as those complex numbers whose real part is zero.

The term “imaginary” was first used by René Descartes in 1637 , but for non-real solutions to algebraic equations .

General

Representation of a complex number in the Gaussian plane

As the real numbers emerge from the unit 1, the imaginary numbers are based on the imaginary unit , a non-real number with the property ${\displaystyle \mathrm {i} }$

${\displaystyle \mathrm {i} ^{2}=-1.}$

Multiplying the imaginary unit with a real factor results in with ${\displaystyle \mathrm {i} }$ ${\displaystyle b}$

${\displaystyle b\cdot \mathrm {i} }$

always an imaginary number. And vice versa, every imaginary number is a real multiple of the imaginary unit. In the Gaussian plane (see picture) the imaginary numbers form the straight line labeled Im, which intersects the real number line Re at the common number 0 at a right angle.

application

Equations that cannot have real solutions can be solved in the imaginary numbers. For example, the equation has

${\displaystyle x^{2}-4=0}$

as a solution two real numbers, namely 2 and −2. But the equation

${\displaystyle x^{2}+4=0}$

cannot have a real solution, since real number squares are never negative, so there is no real number whose square is −4. The solution to this equation is two imaginary numbers, and . ${\displaystyle +2\mathrm {i} }$ ${\displaystyle -2\mathrm {i} }$

A preoccupation with square roots of negative numbers was necessary when solving cubic equations in the case of the case irreducibilis .

In the complex AC calculation, the symbol for the imaginary unit is used instead of a in order to avoid confusion with the instantaneous value of the current . This name goes back to Charles P. Steinmetz . It is permitted as a symbol in accordance with DIN 1302, DIN 5483-3 and ISO 80000-2 . ${\displaystyle \mathrm {i} }$ ${\displaystyle \mathrm {j} }$ ${\displaystyle i(t)}$

Calculation rules

Sums or differences between two imaginary numbers are always imaginary:

${\displaystyle b\mathrm {i} -c\mathrm {i} =(b-c)\cdot \mathrm {i} }$

Products or quotients of two imaginary numbers are always real:

${\displaystyle b\mathrm {i} \cdot c\mathrm {i} =bc\cdot \mathrm {i} ^{2}=-bc}$

Potencies

${\displaystyle {\begin{aligned}\mathrm {i} ^{-1}&={\frac {1}{\mathrm {i} }}={\frac {\mathrm {i} }{\mathrm {i} ^{2}}}={\frac {\mathrm {i} }{-1}}=-\mathrm {i} \\\mathrm {i} ^{0}&=1\\\mathrm {i} ^{1}&=\mathrm {i} \\\mathrm {i} ^{2}&=-1\\\mathrm {i} ^{3}&=\mathrm {i} ^{2}\cdot \mathrm {i} =-\mathrm {i} \\\mathrm {i} ^{4}&=\mathrm {i} ^{2}\cdot \mathrm {i} ^{2}=(-1)^{2}=1\end{aligned}}}$

general:

${\displaystyle {\begin{aligned}\mathrm {i} ^{4n}&=1\\\mathrm {i} ^{4n+1}&=\mathrm {i} \\\mathrm {i} ^{4n+2}&=-1\\\mathrm {i} ^{4n+3}&=-\mathrm {i} \\\mathrm {i} ^{2n}&=(\mathrm {i} ^{2})^{n}=(-1)^{n}\end{aligned}}}$

for everyone . ${\displaystyle n\in \mathbb {Z} }$

Complex numbers

→ Main article : Complex number

The imaginary unit allows the extension of the field of real numbers to the field of complex numbers. ${\displaystyle \mathrm {i} }$

Today imaginary numbers are understood as special complex numbers. Any complex number can be represented as the sum of a real number and a real multiple of the imaginary unit . ${\displaystyle \mathrm {i} }$

Algebraically is defined as a zero of the polynomial and the complex numbers as the resulting body extension . The second zero is then . You can only distinguish between the two zeros when you have designated one of the two with . There are no distinguishing features for the two zeros. It doesn’t matter which “zero” is now called. (If, however, as usual, the complex number range is represented on the structure of the defined instead of just with its help, then one can very well distinguish the possible zeros and obviously selects instead of the equally possible .) ${\displaystyle \mathrm {i} }$ ${\displaystyle x^{2}+1}$ ${\displaystyle -\mathrm {i} }$ ${\displaystyle \mathrm {i} }$ ${\displaystyle \mathrm {i} }$ ${\displaystyle \mathbb {R} ^{2}}$ ${\displaystyle \mathrm {i} :=(0,1)^{\mathrm {T} }}$ ${\displaystyle \mathrm {i} :=(0,-1)^{\mathrm {T} }}$

All complex numbers can be represented in the Gaussian plane , an extension of the real number line . The complex number with real numbers has the real part and the imaginary part . Due to the arithmetic rules of complex numbers, the square of a number whose real part is 0 is a non-positive real number: ${\displaystyle a+\mathrm {i} \cdot b}$ ${\displaystyle a,b}$ ${\displaystyle a}$ ${\displaystyle b}$

${\displaystyle (b\,\mathrm {i} )^{2}=-b^{2}}$

additional

Extensions represent the hyper- complex numbers, which, in addition to the complex numbers, have several imaginary units. For example, there are three imaginary units in the four-dimensional quaternions , and there are seven imaginary units in the eight-dimensional octonions .

In Euler's identity , a concise, simple connection is established between the imaginary unit and three other fundamental mathematical constants , namely with Euler's number , the circle number and the real unit 1: ${\displaystyle \mathrm {i} }$ ${\displaystyle \mathrm {e} }$ ${\displaystyle \mathrm {\pi } }$

${\displaystyle \mathrm {e} ^{\mathrm {i} \,\pi }=-1}$

literature

• Ilja N. Bronstein, KA Semendjajew, Gerhard Musiol, Heiner Muehlig: Pocket book of mathematics . 7th edition. Harri Deutsch, 2008, ISBN 978-3-8171-2007-9 .

Web links

Wikibooks: Imaginary and Complex Numbers  - Learning and Teaching Materials

Wikibooks: Complex Numbers  - Learning and Teaching Materials

Individual evidence

1. ↑ Eric W. Weisstein : Imaginary Number . In: MathWorld (English).
2. ↑ Helmuth Gericke : History of the number concept. Bibliographisches Institut, Mannheim 1970, p. 66.
3. ^ Kurt Jäger, Friedrich Heilbronner: Lexicon of electrical engineers . 2nd Edition. VDE Verlag, 2010, ISBN 978-3-8007-2903-6 , p. 418 .