Imaginary number
All integer powers of i assume values from blue area |
i−3 = i |
i−2 = −1 |
i−1 = −i |
i0 = 1 |
i1 = i |
i2 = −1 |
i3 = −i |
i4 = 1 |
i5 = i |
i6 = −1 |
i is a 4th root of unity |
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. 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
Although the Greek
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
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 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 x and y both positive real numbers:
See also
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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.
- ISBN 978-81-7133-912-9.
- ISBN 978-1-4390-4379-0.
- ISBN 981-02-0600-3.
- 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].)
- ISBN 0-691-12309-8, discusses ambiguities of meaning in imaginary expressions in historical context.
- ISBN 0-387-96458-4.
- ISBN 0-471-17859-4. Retrieved 2022-01-13.
- ISBN 978-3-319-71350-2.
- ISBN 0-691-10298-8. Retrieved 2022-01-13.
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? Basic Explanation and Uses of Imaginary Numbers