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Imaginary number
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In mathematics, an imaginary number (or purely imaginary number) is a complex
number whose square is a negative real number. Imaginary numbers were defined
in 1572 by Rafael Bombelli. At the time, such numbers were thought not to
exist, much as zero and the negative numbers were sometimes regarded by some as
fictitious or useless. Many other mathematicians were slow to believe in
imaginary numbers at first, including Descartes who wrote about them in his La
Géométrie, where the term was meant to be derogatory.
Contents
* 1 Definition
* 2 Geometric interpretation
* 3 Applications of imaginary numbers
* 4 See also
* 5 External links
[edit]
Definition
Any complex number can be written as a + bi, where a and b are real numbers and
i is the imaginary unit with the property that:
i^2 = -1.\,
<http://upload.wikimedia.org/math/4/7/7/4776d59e5e0c41800db6bedb59793c8a.png>
The number a is the real part of the complex number, and b is the imaginary
part. Although Descartes originally used the term "imaginary number" to mean
what is currently meant by the term "complex number", the term "imaginary
number" today usually means a complex number with a real part equal to 0, that
is, a number of the form bi.
[edit]
Geometric interpretation
Geometrically, imaginary numbers are found on the vertical axis of the complex
number plane, allowing them to be presented orthogonal 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, draw a y-axis with "positive"
direction going up; "positive" imaginary numbers then "increase" in magnitude
upwards, and "negative" imaginary numbers "decrease" in magnitude downwards.
This vertical axis is often called the "imaginary axis" and is denoted
i\mathbb{R}
<http://upload.wikimedia.org/math/0/7/7/077329316fa81769b859f78f90e3bdc8.png> .
In this model, multiplication by − 1 corresponds to a reflection about the
origin, i.e. a rotation of 180 degrees about the origin. Multiplication by i
corresponds to a 90-degree rotation in the "positive" direction (i.e.
counter-clockwise), and the equation i2 = − 1 is interpreted as saying that if
we apply 2 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 x2 = − 1 — see imaginary unit.
In electrical engineering and related fields, the imaginary unit is often
written as j to avoid confusion with a changing current, traditionally denoted
by i.
[edit]
Applications of imaginary numbers
Despite their name, imaginary numbers are just as "real" as real numbers. (See
the definition of complex numbers on how they can be constructed using set
theory.) One way to understand this is by realizing that numbers themselves are
abstractions, and the abstractions can be valid even when they are not
recognized in a given context. For example, fractions such as \frac{3}{4}
<http://upload.wikimedia.org/math/8/a/7/8a763ab2930ce9708e0c5196b9faebe7.png>
and \frac{5}{7}
<http://upload.wikimedia.org/math/e/d/a/eda06dcd157a77397b265e9f45d8df33.png>
are meaningless to a person counting stones, but essential to a person
comparing the sizes of different collections of stones. Similarly, negative
numbers such as − 3 and − 5 are meaningless when keeping score in a football
game, but essential when keeping track of monetary debits and credits.
Imaginary numbers follow the same pattern. For most human tasks, real numbers
(or even rational numbers) offer an adequate description of data, and imaginary
numbers have no meaning; however, in many areas of science and mathematics,
imaginary numbers (and complex numbers in general) are essential for describing
reality. Imaginary numbers have essential concrete applications in a variety of
sciences and related areas such as signal processing, control theory,
electromagnetism, quantum mechanics, and cartography.
For example, in electrical engineering, when analyzing AC circuitry, the values
for the electrical voltage (and current) are expressed as imaginary or complex
numbers known as phasors. These are real voltages that can cause damage/harm to
either humans or equipment even if their values contain no "real part".
Specifically, Euler's formula is used extensively to express signals (e.g.,
electromagnetic) that vary periodically over time as a combination of sine and
cosine functions. Euler's formula accomplishes this more conveniently via an
expression of exponential functions with imaginary exponents. Euler's formula
states that, for any real number x,
e^{ix} = \cos x + i\sin x. \,
<http://upload.wikimedia.org/math/b/6/4/b641e81960a7f37eb9b480e8ab7a732c.png>
[edit]
See also
* Complex number
* Quaternion
* Octonion
[edit]
External links
* Why imaginary numbers really do exist
<http://www.math.toronto.edu/mathnet/answers/imaginary.html>
* Imaginary number calculator
<http://www.algebra.com/algebra/homework/complex/>
Retrieved from "http://en.wikipedia.org/wiki/Imaginary_number";
Category: Complex analysis
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