Complex Numbers
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Definition
Complex numbers are numbers which take the form , where and are real numbers and .
Such numbers frequently appear in mathematical equations, even in those describing physical systems. Extending our notion of numbers to include complex numbers results in many astounding symmetries and relationships throughout mathematics.
Basic Properties
Complex numbers have two parts: a "real" part represented by and an "imaginary" part represented by ; the factor in the imaginary part forces the two to be separate. The same operations that are used on real numbers, such as addition, subtraction, multiplication, and division, can be used on complex numbers. For example, two complex numbers are added by components, real added to real part and imaginary to imaginary part:
As another example, multiplying two complex numbers is carried out in the same way that we would multiply two real binomials:
Note that because each is the square root of , the product of two terms gives , so .
Visualizing the Complex Numbers
We traditionally visualize the real numbers, such as 2 and 0.5, as points on the number line. We can visualize real numbers this way because all real numbers can be identified by a single value. The real number 5 is unique, and has its own place on the number line. Because complex numbers have two parts, we can think of them as vectors contained in a plane. We call the plane which contains complex numbers the Argand Plane, or the Complex Plane. The yaxis represents the imaginary component of our complex number, and the xaxis the real component. The complex number is shown below:
We can thus speak of the magnitude of a complex number as the length of this vector, which is , as is readily shown by the Pythagorean Theorem.
This vector idea leads to an important relation between trigonometry and the complex numbers. Euler's formula, which can be derived using Taylor Series, tells us that
 .
 This page's main image shows that
 and .
 Substituting gives
 or
 .
Therefore all complex numbers of the form can be expressed with an exponential function.