Imaginary number

From Wikipedia, the free encyclopedia

In mathematics, an imaginary number (or purely imaginary number) is a complex number whose square is a negative real number or zero. 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 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.^[1]

1 Definition
- 1.1 Corollary
2 Geometric interpretation
3 Applications of imaginary numbers
4 History
5 See also
6 References
7 External links

[edit] Definition

Any complex number, z, can be written as

$z = x + iy \$ ,

where $x\$ and $y\$ are real numbers and $i\$ is the imaginary unit, which has the defined property that:

$i^2 = -1.\$

The number $x\$ , defined by

$x = \operatorname{Re}(z)$

is the real part of the complex number, $z\$ , and $y\$ , defined by

$y = \operatorname{Im}(z)$

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 i y. Zero (0) is the only number that is both real and imaginary.

[edit] Corollary

$i^3 = i^2 i = (-1) i = -i \ ,$

$i^4 = i^3 i = (-i) i = -(i^2) = -(-1) = 1 \ ,$ ,

$i^5 = i^4 i = (1) i = i \ ,$

...

and so on.

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

In this model, 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. counter-clockwise), and the equation $i 2 = - 1$ 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 $x 2 = - 1$ — 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 as "real" as real numbers.^[2] (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 ⅔ and ⅛ 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 US football game, but essential when keeping track of monetary debits and credits^[1] (or yards gained on a play in the same football game).

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". The study of AC (alternating current) entails introduction to electricity governed by trigonometric (i.e. oscillating) functions. From calculus, one knows that differentiating or integrating either "+/- sin(t)" or "+/- cos(t)" four times (with respect to "t," of course) results in the original function "+/- sin(t)" or "+/- cos(t)." From complex algebra, one knows that multiplying the imaginary unit quantity "i" by itself four times will result in the number 1 (identity). Thus, calculus can be represented by the algebraic properties of the imaginary unit quantity (this was exploited by Charles Proteus Steinmetz).

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,

$e^{ix} = \cos x + i\sin x. \,$

Some programming languages also have built-in support for imaginary numbers. For example, in the Python interpreter, one may use them by appending a lowercase or uppercase J to the number:^[3]

>>> (5+2j) * (8+5j)
(30+41j)

[edit] History

Descartes was the first to use the term “imaginary” number in 1637. However, imaginary numbers were invented much earlier by Gerolamo Cardano in the 1500s but they were not widely accepted until the work of Leonhard Euler (1707–1783) and Carl Friedrich Gauss (1777–1855).

[edit] See also

[edit] References

^ ^a ^b Alberto A. Martinez, Negative Math: How Mathematical Rules Can Be Positively Bent (Princeton University Press, 2005), discusses ambiguities of meaning in imaginary expressions in historical context.
^ Paul Nahin, An Imaginary Tale: the Story of the Square Root of -1 (Princeton University Press, 1998), explains many applications of imaginary expressions.
^ Note that the leading angle brackets in the first line are part of the interpreter's syntax and are not part of the equation.