Imaginary number

An accepted version of this page, accepted on 16 July 2016, was based on this revision.

$...$ (repeats the pattern from blue area)
$i -3 = i$
$i -2 = -1$
$i -1 = - i$
$i 0 = 1$
$i 1 = i$
$i 2 = -1$
$i 3 = -i$
$i 4 = 1$
$i 5 = i$
$i 6 = -1$
$i n = i n(mod 4)$

An imaginary number^{[note 1]} is a complex number that can be written as a real number multiplied by the imaginary unit $i$ ,^{[note 2]} which is defined by its property $i 2 = -1$ .^[1] The square of an imaginary number $bi$ is $- b 2$ . For example, $5 i$ is an imaginary number, and its square is $-25$ . Zero is considered to be both real and imaginary.^[2]

Originally coined in the 17th century as a derogatory term and regarded as fictitious or useless, the concept gained wide acceptance following the work of Leonhard Euler and Carl Friedrich Gauss.

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.^[3]^{[note 3]} Some authors use the term pure imaginary number to denote what is called here an imaginary number, and imaginary number to denote any complex number with non-zero imaginary part.^[4]

History

Although Greek mathematician and engineer Heron of Alexandria is noted as the first to have conceived these numbers,^[5]^[6] Rafael Bombelli first set down the rules for multiplication of complex numbers in 1572. The concept had appeared in print earlier, for instance in work by Gerolamo Cardano. At the time, such numbers were poorly understood and 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 imaginary was used and meant to be derogatory.^[7] 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).^[8]

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 quotient rings 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.^[9]

Geometric interpretation

Geometrically, imaginary numbers are found on the vertical axis of the complex number plane, allowing 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 this $x$ -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" and is denoted $i ℝ$ , $\scriptstyle \mathbb {I}$ , or $ℑ$ .

In this representation, multiplication by $-1$ corresponds to a rotation of 180 degrees about the origin. Multiplication by $i$ corresponds to a 90-degree rotation in the "positive" direction (i.e., counterclockwise), and the equation $i 2 = -1$ is interpreted as saying that if we apply two 90-degree rotations about the origin, the net result is a single 180-degree rotation. Note that a 90-degree rotation in the "negative" direction (i.e. clockwise) also satisfies this interpretation. This reflects the fact that $- i$ also solves the equation $x 2 = -1$ . In general, 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.

Square roots of negative numbers

Care must be used when working with imaginary numbers expressed as the principal values of the square roots of negative numbers. For example:^[10]

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

Sometimes this is written as:

-1=i^{2}={\sqrt {-1}}{\sqrt {-1}}{\stackrel {\text{ (fallacy) }}{=}}{\sqrt {(-1)(-1)}}={\sqrt {1}}=1,

where the fallacy is that the rule ${\sqrt {xy}}={\sqrt {x}}{\sqrt {y}}$ can fail when the variables are not suitably constrained, in this case when they are not both positive.

Notes

^ This is the definition adopted in this article.
^ j is often used in Engineering
^ Both the real part and the imaginary part are defined as real numbers.

References

^ Uno Ingard, K. (1988). "Chapter 2". Fundamentals of waves & oscillations. Cambridge University Press. p. 38. ISBN 0-521-33957-X.
^ Sinha, K.C. A Text Book of Mathematics XI. Rastogi Publications. p. 11.2. ISBN 8171339123.
^ Aufmann, Richard; Barker, Vernon C.; Nation, Richard (2009). College Algebra: Enhanced Edition (6th ed.). Cengage Learning. p. 66. ISBN 1-4390-4379-5.
^ C.L. Johnston, J. Lazaris, Plane Trigonometry: A New Approach, Prentice Hall, 1991, p. 247.
^ Hargittai, István (1992). Fivefold symmetry (2nd ed.). World Scientific. p. 153. ISBN 981-02-0600-3.
^ Roy, Stephen Campbell (2007). Complex numbers: lattice simulation and zeta function applications. Horwood. p. 1. ISBN 1-904275-25-7.
^ Martinez, Albert A. (2006), Negative Math: How Mathematical Rules Can Be Positively Bent, Princeton: Princeton University Press, ISBN 0-691-12309-8, discusses ambiguities of meaning in imaginary expressions in historical context.
^ Rozenfeld, Boris Abramovich (1988). "Chapter 10". A history of non-euclidean geometry: evolution of the concept of a geometric space. Springer. p. 382. ISBN 0-387-96458-4.
^ James Cockle (1848) "On Certain Functions Resembling Quaternions and on a New Imaginary in Algebra", London-Dublin-Edinburgh Philosophical Magazine, series 3, 33:435–9 and Cockle (1849) "On a New Imaginary in Algebra", Philosophical Magazine 34:37–47
^ Nahin, Paul J. (2010). An Imaginary Tale: The Story of "i" [the square root of minus one]. Princeton University Press. p. 12. ISBN 978-1-4008-3029-9. Extract of page 12

Bibliography

Nahin, Paul (1998). An Imaginary Tale: the Story of the Square Root of −1. Princeton: Princeton University Press. ISBN 0-691-02795-1., explains many applications of imaginary expressions.

External links

How can one show that imaginary numbers really do exist? – an article that discusses the existence of imaginary numbers.
In our time: Imaginary numbers Discussion of imaginary numbers on BBC Radio 4.
5Numbers programme 4 BBC Radio 4 programme

[1] This is the definition adopted in this article.

[2] s often used in Engineering

[6] Both the real part and the imaginary part are defined as real numbers.

[3] Uno Ingard, K. (1988). "Chapter 2". Fundamentals of waves & oscillations. Cambridge University Press. p. 38. ISBN 0-521-33957-X.

[4] Sinha, K.C. A Text Book of Mathematics XI. Rastogi Publications. p. 11.2. ISBN 8171339123.

[5] Aufmann, Richard; Barker, Vernon C.; Nation, Richard (2009). College Algebra: Enhanced Edition (6th ed.). Cengage Learning. p. 66. ISBN 1-4390-4379-5.

[7] C.L. Johnston, J. Lazaris, Plane Trigonometry: A New Approach, Prentice Hall, 1991, p. 247.

[8] Hargittai, István (1992). Fivefold symmetry (2nd ed.). World Scientific. p. 153. ISBN 981-02-0600-3.

[9] Roy, Stephen Campbell (2007). Complex numbers: lattice simulation and zeta function applications. Horwood. p. 1. ISBN 1-904275-25-7.

[Martinez-10] Martinez, Albert A. (2006), Negative Math: How Mathematical Rules Can Be Positively Bent, Princeton: Princeton University Press, ISBN 0-691-12309-8, discusses ambiguities of meaning in imaginary expressions in historical context.

[11] Rozenfeld, Boris Abramovich (1988). "Chapter 10". A history of non-euclidean geometry: evolution of the concept of a geometric space. Springer. p. 382. ISBN 0-387-96458-4.

[12] James Cockle (1848) "On Certain Functions Resembling Quaternions and on a New Imaginary in Algebra", London-Dublin-Edinburgh Philosophical Magazine, series 3, 33:435–9 and Cockle (1849) "On a New Imaginary in Algebra", Philosophical Magazine 34:37–47

[13] Nahin, Paul J. (2010). An Imaginary Tale: The Story of "i" [the square root of minus one]. Princeton University Press. p. 12. ISBN 978-1-4008-3029-9. Extract of page 12

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v t e Number systems
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