Imaginary number

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In mathematics, an imaginary number is a number whose square is negative. The term was coined by René Descartes in the seventeenth century and was meant to be derogatory: obviously such numbers don't exist. Nowadays we find the imaginary numbers on the vertical axis of the complex number plane. Every imaginary number can be written as where is a real number and the imaginary unit with the property that

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.

Imaginary numbers are critically important in particle physics, where eigenvalues are generally expressed in terms of them.

See the definition of complex numbers on how they can be constructed.

(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.) Every complex number can be written uniquely as a sum of a real number and an imaginary number (its imaginary part).

Despite their name, imaginary numbers are just as "real" as real numbers (or just about as real as a number, which is an abstract concept, can be). Imaginary numbers have concrete applications in a variety of sciences and related areas such as electromagnetism, quantum mechanics, and cartography.

The powers of i repeat in a cycle:

This can be expressed with the following pattern where n is any integer:

i and Euler's Formula

Euler's formula , substituting π/2 in for x, has the form

If both sides are raised to the power of i, remembering that , we obtain:

and, therefore, the following astounding property of the imaginary number i:

Imaginary number

i and Euler's Formula

See also