A complex number is an expression of the form  x + iy, where  x  and  y  are real numbers, and  i  is an imaginary number such that  i2  = –1.

Usually, a complex number is denoted by a single letter, e.g.,

z = x + i y.

The numbers  x  and  y  are called the real and imaginary parts of  z. They are also symbolized as

x = Re z,
y = Im z.

Note once more that both numbers,  Re z  and  Im z, are real numbers.

The set of all complex numbers is denoted by the symbol  C.

Any real number x can be considered as a complex number whose imaginary part equals zero. It means that the set of complex numbers includes the set of all real numbers as a subset.

The set of real numbers is a proper subset of the set of complex numbers:

.

If  Re z = 0  then a number  z = iy  is said to be purely imaginary.

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