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{{Math-m}}

{{Math-m}}

The '''complex numbers''' are a set of numbers which have important applications in the analysis of periodic, oscillatory, or wavelike phenomena.  Mathematicians denote the set of complex numbers with an ornate capital letter: <math>\mathbb{C}</math>.  They are the 5^th item in this hierarchy of types of [[number]]s:

The '''complex numbers''' are a set of numbers which have important applications in the analysis of periodic, oscillatory, or wavelike phenomena.  Mathematicians denote the set of complex numbers with an ornate capital letter: <math>\mathbb{C}</math>.  They are the 5th item in this hierarchy of types of [[number]]s:

*The "[[natural number]]s", 1, 2, 3, ...  (There is controversy about whether zero should be included.  It doesn't matter.)

*The "[[natural number]]s", 1, 2, 3, ...  (There is controversy about whether zero should be included.  It doesn't matter.)

While the "invisible" nature of the imaginary component may be disconcerting at first (and the word "imaginary" may be an unfortunate term for it), the complex numbers are just as genuine as the Dedekind cuts and Cauchy sequences that are used in the definition of the "real" numbers.

While the "invisible" nature of the imaginary component may be disconcerting at first (and the word "imaginary" may be an unfortunate term for it), the complex numbers are just as genuine as the Dedekind cuts and Cauchy sequences that are used in the definition of the "real" numbers.

The complex numbers form a [[Field_(mathematics)|field]], with the mathematical operations defined as shown below.

The complex numbers form a [[Field (mathematics)|field]], with the mathematical operations defined as shown below.

==Arithmetic operations==

==Arithmetic operations==