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In mathematics, an imaginary number (or purely imaginary number) is a complex number whose square is negative or zero. The term was coined by René Descartes in 1637 in his La Geometrie and was meant to be derogatory: obviously, such numbers were thought not to exist.

1 Definition

Any complex number can be written as , where and are real numbers and is the imaginary unit with the property that

The number is the real part of the complex number, and 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 today specifically means a complex number with real part equal to , i.e. a number of the form . Note that technically, is considered to be a purely imaginary number: is the only complex number which is both real and purely imaginary.

2 Geometric interpretation

Geometrically, we find the imaginary numbers on the vertical axis of the complex number plane. 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 on this -axis, draw a -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 .

In this model, multiplication by corresponds to a reflection about the origin, i.e. a rotation of degrees about the origin. Multiplication by corresponds to a -degree rotation in the "positive" direction (i.e. counter-clockwise), and the equation is interpreted as saying that if we apply -degree rotations about the origin, the net result is a single -degree rotation. Note that a -degree rotation in the "negative" direction (i.e. clockwise) also satisfies this interpretation. This reflects the fact that also solves the equation — see imaginary unit.

In electrical engineering and related fields, the imaginary unit is often written as to avoid confusion with a changing current, traditionally denoted by .

3 Are imaginary numbers "real"?

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 theoryNaive set theory 1 is distinguished from axiomatic set theory by the fact that the former regards sets as collections of objects, called the elements or members of the set, whereas the latter regards sets only as that which satisfies certain axioms. Sets.) One way to see why this is so, is to realize that numbers themselves are abstractions, and we should not be fooled into thinking the abstractions are not real simply because they do not always apply in the real world. For example, fractions such as and 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 and are meaningless when keeping score in a football match, but essential when keeping track of monetary debts 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 processingSignal processing is the processing, amplification and interpretation of signals. Signals may come from various sources. There are various sorts of signal processing, depending on the nature of the signal, as in the following examples. Digital signal proc, control theoryThis article is about an engineering theory called control theory. There is also a sociological theory of deviant behavior that is called control theory. In engineering and mathematics, control theory deals with the behaviour of dynamical systems over tim, electromagnetismElectromagnetism is the physics of the electromagnetic field: a field, encompassing all of space, composed of the electric field and the magnetic field. The electric field is produced by stationary electric charges, and gives rise to the electric force, t, quantum mechanicswavefunctions of an electron in a hydrogen atom possessing definite energy (increasing downward: n 1,2,3,. and angular momentum (increasing across: s p d . Brighter areas correspond to higher probability density for a position measurement. The angular mom, and cartography. They are absolutely indispensable in advanced mathematics.

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