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Complex Numbers: Introduction and Operations

Definition:

In mathematics, a complex number is a number of the form .

Where a and b are real numbers, and i is the imaginary unit, with the property i 2 = −1. The real number a is called the real part of the complex number, and the real number b is the imaginary part. Real numbers may be considered to be complex numbers with imaginary part set to zero; that is, the real number a is equivalent to the complex number a+0i.

If z = a + bi, the real part (a) is denoted Re(z), and the imaginary part (b) is denoted Im(z).

Complex numbers can be added, subtracted, multiplied, and divided like real numbers. They cannot be ordered, but have other elegant properties.

Equality:

Two complex numbers are equal if and only if their real parts are equal and their imaginary parts are equal.   That is, a + bi = c + di if and only if a = c and b = d.

Notation and Operations:

The set of all complex numbers is usually denoted by . Complex numbers are added, subtracted, and multiplied by formally applying the associative, commutative and distributive laws of algebra, together with the equation i2 = −1:   In order to add, substract, multiply or divide two complex numbers use our complex numbers calculator!