Imaginary number

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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 Ren� Descartes in 1637 in his ''La Geometrie'' and was meant to be derogatory: obviously, such numbers were thought not to exist. Nowadays, we find the imaginary numbers on the vertical axis of the complex number plane. A complex number can be written as <math>a+ib</math> where <math>b</math> is a real number and <math>i</math> the imaginary unit with the property that :<math>i^2 = -1\,</math> and therefore :<math>i = \sqrt{-1}</math> 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 zero on this ''x'' axis, draw a ''y'' axis; "positive" imaginary numbers then increase in magnitude upwards, and "negative" imaginary numbers increase in magnitude downwards. According to this model, multiplying by -1 moves us 180 degrees (between positive and negative). If we want to stop off at the half way point (at the vertical axis) we must have multiplied by the square root of -1. Note that technically, zero is considered to be a purely imaginary number: zero is the only complex number which is both real and purely imaginary. This vertical axis is often called the "imaginary axis" and is denoted <math>i\mathbb{R}</math>. Every complex number can be written uniquely as a sum of a real number and an imaginary number: the latter is equal to <math>i</math> times its imaginary part: :<math> a + ib\,</math> It follows that every purely real number is a sub-set of this, i.e: :<math> a + i0\,</math> and every purely imaginary number is also sub-set of this, i.e: :<math> 0 + ib\,</math> In electrical engineering and related fields, the imaginary unit is often written as <math>j</math> to avoid confusion with a changing current, traditionally denoted by <math>i</math>. ==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 3/4 and 5/7 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 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 harmonic analysis, electromagnetism, quantum mechanics, and cartography. They are absolutely indispensable in advanced mathematics. Another way to think about the term is to consider these numbers imaginary because they can be imagined in your mind. ==See also== *Complex number *Quaternion *Octonion ==External link== *Why imaginary numbers really do exist de:Imagin�re Zahl es:N�mero imaginario fr:Nombre imaginaire pur ja:虚数単位 nl:Imaginair getal pl:Liczby urojone zh:虚数