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Complex numbers, an extension of real numbers that includes i (the imaginary unit, representing a number which squared is minus 1), are used in physical sciences for some kinds of calculations, such as describing waves and their components, and carrying out some calculations in trigonometry, special relativity and quantum mechanics. With complex numbers, each number has at least one square root, even if the number is negative. A single complex number incorporates the information of a pair of real numbers and can serve as a coordinate pair, describing the position of a point on a plane (and conversely, a complex number can be represented by a point on a plane, a mathematical plane used that way being termed the complex plane). The arithmetic rules of complex numbers, which include the interaction of these two values during arithmetic operations, match the calculation-needs of some physical processes, such as those mentioned above, allowing descriptions of some physical relationships in more concise and familiar-looking formulae. The fact that you can reliably use of them in this way makes the question of whether i "exists" beside the point. Complex number examples:
1 i -4.5+3i 4-πi.
They consist of a real part summed with an imaginary part, the latter consisting of some real number times i. Calculation can be carried out much like treating i as a variable and simplifying algebraically, using the fact that i times i equals minus one (-1). A useful definition of exponentiation by i (e.g., 2i), has long been worked out which preserves consistency with real numbers and is encapsulated in Euler's formula.
Complex numbers can be represented by other means, such as using polar coordinates (an angle from an axis with a distance from the origin) to indicate the position of the complex number on the complex plane. In such a representation, arithmetic on the numbers can be more involved, yet much easier for some specific operations, providing a trade-off; specific calculations can be made easier by choosing the favorable representation, and it can be helpful to convert between representations (one or more times) in the midst of a calculation.