Imaginary numbers had no use when it was invented except for intellectual exploration until it was linked to coordinate geometry. The complex number can be plotted as the ordered pair where is the real part and is the imaginary part. Therefore, the complex numbers , , , and can be plotted as , , , and in a “coordinate plane” respectively as shown below. In the figure, we can see that the origin is and that , the complex conjugate of , is its reflection along the x-axis (Can this be generalized?).
The plane where the complex numbers are plotted above is called the complex plane.
In the complex plane, we can observe that all numbers of the form are real numbers and these numbers are represented by the horizontal axis. The numbers of the form are all imaginary numbers and it is represented by the vertical axis. Thus, we can call them the real axis and the imaginary axis respectively.
In addition, we can conclude that all real numbers are complex numbers since for all real numbers , . So, the set of real numbers is a subset of the set of complex numbers.
This is the third post in the Complex Numbers Primer. In the previous post in this series, we have learned that complex numbers can be added and multiplied just like binomials. We did not discuss subtraction since it follows from addition; that is, if and are complex numbers, then
Since we have completed the three fundamental operations on complex numbers, the next logical question would be how do we divide complex numbers? For example, how do we find the quotient of the expression
Before we answer that question, recall that in simplifying rational expressions with radicals in the denominator, we multiply the numerator and the denominator of the expression by the ‘conjugate’ of the denominator. For instance, to simplify
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