Imaginary number

The powers of $i$ are cyclic:
$\ \vdots$
$\ i^{-2}=-1{\phantom {i}}$
$\ i^{-1}=-i{\phantom {1}}$
$\ \ i^{0}\ ={\phantom {-}}1{\phantom {i}}$
$\ \ i^{1}\ ={\phantom {-}}i{\phantom {1}}$
$\ \ i^{2}\ =-1{\phantom {i}}$
$\ \ i^{3}\ =-i{\phantom {1}}$
$\ \ i^{4}\ ={\phantom {-}}1{\phantom {i}}$
$\ \ i^{5}\ ={\phantom {-}}i{\phantom {1}}$
$\ \vdots$
$i$ is a 4th root of unity

An imaginary number is the product of a real number and the imaginary unit $i$ ,^{[note 1]} which is defined by its property $i 2 = -1$ .^[1]^[2] The square of an imaginary number $bi$ is $- b 2$ . For example, $5 i$ is an imaginary number, and its square is $-25$ . The number 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 Heron 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 $i\mathbb {R} ,$ $\mathbb {I} ,$ or $ℑ$ .^[12]

In this representation, multiplication by $i$ corresponds to a counterclockwise rotation of 90 degrees about the origin, which is a quarter of a circle. Multiplication by $- i$ corresponds to a clockwise rotation of 90 degrees about the origin. Similarly, multiplying by a purely imaginary number $bi$ , with $b$ a real number, both causes a counterclockwise rotation about the origin by 90 degrees and scales the answer by a factor of $b$ . When $b < 0$ , this can instead be described as a clockwise rotation by 90 degrees and a scaling by $| b |$ .^[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] For example, if $x$ and $y$ are both positive real numbers, the following chain of equalities appears reasonable at first glance:

\textstyle {\sqrt {x\cdot y{\vphantom {t}}}}={\sqrt {(-x)\cdot (-y)}}\mathrel {\stackrel {\text{ (fallacy) }}{=}} {\sqrt {-x{\vphantom {ty}}}}\cdot {\sqrt {-y{\vphantom {ty}}}}=i{\sqrt {x{\vphantom {ty}}}}\cdot i{\sqrt {y{\vphantom {ty}}}}=-{\sqrt {x\cdot y{\vphantom {ty}}}}\,.

But the result is clearly nonsense. The step where the square root was broken apart was illegitimate. (See Mathematical fallacy.)

Notes

^ $j$ is usually used in engineering contexts where $i$ has other meanings (such as electrical current)

References

^ Uno Ingard, K. (1988). "Chapter 2". Fundamentals of Waves and Oscillations. Cambridge University Press. p. 38. ISBN 0-521-33957-X.
^ Weisstein, Eric W. "Imaginary Number". mathworld.wolfram.com. Retrieved 2020-08-10.
^ Sinha, K.C. (2008). A Text Book of Mathematics Class XI (Second ed.). Rastogi Publications. p. 11.2. ISBN 978-81-7133-912-9.
^ Giaquinta, Mariano; Modica, Giuseppe (2004). Mathematical Analysis: Approximation and Discrete Processes (illustrated ed.). Springer Science & Business Media. p. 121. ISBN 978-0-8176-4337-9. Extract of page 121
^ Aufmann, Richard; Barker, Vernon C.; Nation, Richard (2009). College Algebra: Enhanced Edition (6th ed.). Cengage Learning. p. 66. ISBN 978-1-4390-4379-0.
^ Hargittai, István (1992). Fivefold Symmetry (2 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 978-1-904275-25-1.
^ Descartes, René, Discours de la méthode (Leiden, (Netherlands): Jan Maire, 1637), appended book: La Géométrie, book three, p. 380. From page 380: "Au reste tant les vrayes racines que les fausses ne sont pas tousjours reelles; mais quelquefois seulement imaginaires; c'est a dire qu'on peut bien tousjours en imaginer autant que jay dit en chasque Equation; mais qu'il n'y a quelquefois aucune quantité, qui corresponde a celles qu'on imagine, comme encore qu'on en puisse imaginer trois en celle cy, x³ – 6xx + 13x – 10 = 0, il n'y en a toutefois qu'une reelle, qui est 2, & pour les deux autres, quoy qu'on les augmente, ou diminue, ou multiplie en la façon que je viens d'expliquer, on ne sçauroit les rendre autres qu'imaginaires." (Moreover, the true roots as well as the false [roots] are not always real; but sometimes only imaginary [quantities]; that is to say, one can always imagine as many of them in each equation as I said; but there is sometimes no quantity that corresponds to what one imagines, just as although one can imagine three of them in this [equation], x³ – 6xx + 13x – 10 = 0, only one of them however is real, which is 2, and regarding the other two, although one increase, or decrease, or multiply them in the manner that I just explained, one would not be able to make them other than imaginary [quantities].)
^ 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.
^ von Meier, Alexandra (2006). Electric Power Systems – A Conceptual Introduction. John Wiley & Sons. pp. 61–62. ISBN 0-471-17859-4. Retrieved 2022-01-13.
^ Webb, Stephen (2018). "5. Meaningless marks on paper". Clash of Symbols – A Ride Through the Riches of Glyphs. Springer Science+Business Media. pp. 204–205. doi:10.1007/978-3-319-71350-2_5. ISBN 978-3-319-71350-2.
^ Kuipers, J. B. (1999). Quaternions and Rotation Sequences: A Primer with Applications to Orbits, Aerospace, and Virtual Reality. Princeton University Press. pp. 10–11. ISBN 0-691-10298-8. Retrieved 2022-01-13.
^ 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

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.
5Numbers programme 4 BBC Radio 4 programme
Why Use Imaginary Numbers? Archived 2019-08-25 at the Wayback Machine Basic Explanation and Uses of Imaginary Numbers

[1] $j$ is usually used in engineering contexts where $i$ has other meanings (such as electrical current)

[2] Uno Ingard, K. (1988). "Chapter 2". Fundamentals of Waves and Oscillations. Cambridge University Press. p. 38. ISBN 0-521-33957-X.

[3] Weisstein, Eric W. "Imaginary Number". mathworld.wolfram.com. Retrieved 2020-08-10.

[4] Sinha, K.C. (2008). A Text Book of Mathematics Class XI (Second ed.). Rastogi Publications. p. 11.2. ISBN 978-81-7133-912-9.

[5] Giaquinta, Mariano; Modica, Giuseppe (2004). Mathematical Analysis: Approximation and Discrete Processes (illustrated ed.). Springer Science & Business Media. p. 121. ISBN 978-0-8176-4337-9. Extract of page 121

[6] Aufmann, Richard; Barker, Vernon C.; Nation, Richard (2009). College Algebra: Enhanced Edition (6th ed.). Cengage Learning. p. 66. ISBN 978-1-4390-4379-0.

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

[8] Roy, Stephen Campbell (2007). Complex Numbers: lattice simulation and zeta function applications. Horwood. p. 1. ISBN 978-1-904275-25-1.

[9] Descartes, René, Discours de la méthode (Leiden, (Netherlands): Jan Maire, 1637), appended book: La Géométrie, book three, p. 380. From page 380: "Au reste tant les vrayes racines que les fausses ne sont pas tousjours reelles; mais quelquefois seulement imaginaires; c'est a dire qu'on peut bien tousjours en imaginer autant que jay dit en chasque Equation; mais qu'il n'y a quelquefois aucune quantité, qui corresponde a celles qu'on imagine, comme encore qu'on en puisse imaginer trois en celle cy, x³ – 6xx + 13x – 10 = 0, il n'y en a toutefois qu'une reelle, qui est 2, & pour les deux autres, quoy qu'on les augmente, ou diminue, ou multiplie en la façon que je viens d'expliquer, on ne sçauroit les rendre autres qu'imaginaires." (Moreover, the true roots as well as the false [roots] are not always real; but sometimes only imaginary [quantities]; that is to say, one can always imagine as many of them in each equation as I said; but there is sometimes no quantity that corresponds to what one imagines, just as although one can imagine three of them in this [equation], x³ – 6xx + 13x – 10 = 0, only one of them however is real, which is 2, and regarding the other two, although one increase, or decrease, or multiply them in the manner that I just explained, one would not be able to make them other than imaginary [quantities].)

[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.

[Meier-12] von Meier, Alexandra (2006). Electric Power Systems – A Conceptual Introduction. John Wiley & Sons. pp. 61–62. ISBN 0-471-17859-4. Retrieved 2022-01-13.

[13] Webb, Stephen (2018). "5. Meaningless marks on paper". Clash of Symbols – A Ride Through the Riches of Glyphs. Springer Science+Business Media. pp. 204–205. doi:10.1007/978-3-319-71350-2_5. ISBN 978-3-319-71350-2.

[14] Kuipers, J. B. (1999). Quaternions and Rotation Sequences: A Primer with Applications to Orbits, Aerospace, and Virtual Reality. Princeton University Press. pp. 10–11. ISBN 0-691-10298-8. Retrieved 2022-01-13.

[15] 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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