Imaginary number

From Academic Kids

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

Contents

Definition

Any complex number can be written as <math>a+ib<math>, where <math>a<math> and <math>b<math> are real numbers and i is the imaginary unit with the property that

<math>i^2 = -1\,<math>

The number <math>a<math> is the real part of the complex number, and <math>b<math> 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 <math>0<math>, i.e. a number of the form ib. Note that technically, <math>0<math> is considered to be a purely imaginary number: <math>0<math> is the only complex number which is both real and purely imaginary.

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 <math>0<math> on this <math>x<math>-axis, draw a <math>y<math>-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 <math>i\mathbb{R}<math>.

In this model, multiplication by <math>-1<math> corresponds to a reflection about the origin, i.e. a rotation of <math>180<math> degrees about the origin. Multiplication by i corresponds to a <math>90<math>-degree rotation in the "positive" direction (i.e. counter-clockwise), and the equation i2 = -1 is interpreted as saying that if we apply <math>2<math> <math>90<math>-degree rotations about the origin, the net result is a single <math>180<math>-degree rotation. Note that a <math>90<math>-degree rotation in the "negative" direction (i.e. clockwise) also satisfies this interpretation. This reflects the fact that -i also solves the equation <math>x^2 = -1<math> — 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.

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 theory.) 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 <math>\frac{3}{4}<math> and <math>\frac{5}{7}<math> 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 <math>-3<math> and <math>-5<math> are meaningless when keeping score in a football game, 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 processing, control theory, electromagnetism, quantum mechanics, and cartography. They are absolutely indispensable in advanced mathematics.

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. There is, however, nothing imaginary (in the non-mathematical sense) about these voltages and they can cause actual damage/harm to either humans or equipment even if their values contain no "real part".

See also

External link

de:Imaginre Zahl es:Nmero imaginario fr:Nombre imaginaire pur ko:허수 is:vertala he:מספר מרוכב nl:Imaginair getal ja:虚数単位 pl:Liczby urojone pt:Nmero imaginrio fi:Imaginaariluku zh:虚数

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