Imaginary number

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

An imaginary number is a real number multiplied by the imaginary unit $i$ , which is defined by its property $i 2 = -1$ .^[1] The square of an imaginary number is less than or equal to zero. For example, $5 i$ is an imaginary number and its square is $-25$ . According to some definitions, zero ( $0 i$ ) is not regarded as an imaginary number, but as a pure real.

An imaginary number $bi$ can be added to a real number $a$ to form a complex number of the form $a + bi$ , where $a$ and $b$ are called, respectively, the real part and the imaginary part of the complex number. Imaginary numbers can therefore be thought of as complex numbers whose real part is zero. The name "imaginary number" was coined in the 17th century as a derogatory term, as such numbers were regarded by some as fictitious or useless, but today they have a variety of essential, concrete applications in science and engineering.

1 History
2 Geometric interpretation
3 Applications of imaginary numbers
4 Powers of
5 Multiplication of square roots
6 See also
7 References
8 Bibliography
9 External links

History

Main article: History of complex numbers

Although Greek mathematician and engineer Heron of Alexandria is noted as the first to have conceived these numbers,^[2]^[3] Rafael Bombelli first set down the rules for multiplication in the 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.^[4] 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 found by Caspar Wessel (1745–1818).^[5]

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.

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 $i\mathbb{R}$ , $\mathbb{I}$ , or simply $Im$ .

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 $x^2 = -1$ — see imaginary unit. 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.

Applications of imaginary numbers

Main article: Complex number#Applications

Imaginary numbers are useful because they allow the construction of non-real complex numbers, which have essential concrete applications in a variety of scientific and related areas such as signal processing, control theory, electromagnetism, fluid dynamics, quantum mechanics, cartography, and vibration analysis.

Powers of $i$

The powers of $i$ repeat in a cycle. This can be expressed with the following pattern where n is any integer:

$i^{4n} = 1\,$

$i^{4n%2B1} = i\,$

$i^{4n%2B2} = -1\,$

$i^{4n%2B3} = -i.\,$

This leads to the conclusion that $i^n = i^{n \bmod 4}\,$ .

Multiplication of square roots

Care must be used in multiplying square roots of negative numbers. For example,^[6] the following reasoning is incorrect:

$i^2 = \sqrt{-1}\sqrt{-1} = \sqrt{(-1)(-1)} = \sqrt{1} = 1$ ?

The fallacy is that the rule $\sqrt{x}\sqrt{y} = \sqrt{xy}$ is generally valid only if at least one of the two numbers x or y is positive, which is not the case here.

References

^ Uno Ingard, K. (1988), Fundamentals of waves & oscillations, Cambridge University Press, p. 38, ISBN 0-521-33957-X, http://books.google.com/books?id=SGVfGIewvxkC , Chapter 2, p 38
^ Hargittai, István (1992). Fivefold symmetry (2nd ed.). World Scientific. p. 153. ISBN 981020600-3. http://books.google.com/books?id=-Tt37ajV5ZgC. , Extract of page 153
^ Roy, Stephen Campbell (2007). Complex numbers: lattice simulation and zeta function applications. Horwood. p. 1. ISBN 190427525-7. http://books.google.com/books?id=J-2BRbFa5IkC.
^ Martinez, Albert A. (2006), Negative Math: How Mathematical Rules Can Be Positively Bent, Princeton: Princeton University Press, ISBN 0691123098 , discusses ambiguities of meaning in imaginary expressions in historical context.
^ Rozenfeld, Boris Abramovich (1988). A history of non-euclidean geometry: evolution of the concept of a geometric space. Springer. p. 382. ISBN 0-387-96458-4. http://books.google.com/books?id=DRLpAFZM7uwC. , Chapter 10, page 382
^ Maxwell, E. A. (1959), Fallacies in mathematics, Cambridge University Press, MR 0099907 . Chapter VI, §I.2}}

Bibliography

Nahin, Paul (1998), An Imaginary Tale: the Story of the Square Root of −1, Princeton: Princeton University Press, ISBN 0691027951 , 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

Number systems

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