# Help With Complex Numbers

A **complex number** is a number of the form a+bia+bi, where aa and bb are real numbers and ii is the imaginary unit, the square root of−1−1.

In a complex number z=a+biz=a+bi, aa is called the “real part” of zz and bb is called the “imaginary part.” If b=0b=0 , the complex number is a real number; if a=0a=0 , then the complex number is “purely imaginary.”

We can graph a complex number on the Cartesian plane, using the horizontal axis as the real axis and the vertical axis as the imaginary axis. When we use the Cartesian plane this way, we call it the **complex plane**.

The complex number a+bia+bi can be plotted as the ordered pair (a,b)(a,b) on the complex plane.

The *absolute value *or *modulus *of a complex number z=a+biz=a+bi can be interpreted as the distance of the point (a,b)(a,b) from the origin on a complex plane.

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Using the Distance Formula,

|z|=|a+bi| =(a−0)2+(b−0)2−−−−−−−−−−−−−−−√ =a2+b2−−−−−−√|z|=|a+bi| =(a−0)2+(b−0)2 =a2+b2

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