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To understand complex numbers, it's important to know they consist of two parts: the real part and the imaginary part. The real part is just like any other real number, while the imaginary part includes the imaginary unit, denoted as *i*, where *i* is the square root of -1. For instance, in the complex number \(-\frac{1}{5} + \frac{2}{7}i\), \(-\frac{1}{5}\) is the real part and \(\frac{2}{7}i\) is the imaginary part.

Similarly, in the complex number \(\frac{3}{7} - \frac{3}{4}i\), \(\frac{3}{7}\) is the real part and \(-\frac{3}{4}i\) is the imaginary part. Identifying these parts is the first step towards adding, subtracting, or performing any operation on complex numbers.

Add the real parts together first and then the imaginary ones. This way, things stay clear and organized. Remember, when dealing with the real parts or imaginary parts separately, treat them just like regular numbers.

When adding or subtracting fractions, finding a common denominator is crucial. This makes the fractions easier to add or subtract. A common denominator is a shared multiple of the denominators. For example, in \(-\frac{1}{5} + \frac{3}{7}\) we need to get both fractions to have the same denominator.

The denominators 5 and 7 can be made the same by finding their least common multiple (LCM), which in this case is 35. Therefore:
\[-\frac{1}{5} = -\frac{7}{35}\]
\[\frac{3}{7} = \frac{15}{35}\]

Now they have a common denominator, making it easy to add them:
\[-\frac{7}{35} + \frac{15}{35} = \frac{8}{35}\]

We do the same for the imaginary parts \(\frac{2}{7}i\) and \(-\frac{3}{4}i\), whose denominators are 7 and 4. The common denominator, in this case, is 28:
\[\frac{2}{7}i = \frac{8}{28}i\]
\[-\frac{3}{4}i = -\frac{21}{28}i\]

Then, we add them:
\[\frac{8}{28}i - \frac{21}{28}i = -\frac{13}{28}i\]

Graphing complex numbers helps visualize their real and imaginary parts. Each complex number can be represented as a point in the complex plane. The complex plane is similar to a coordinate plane but with the real part on the x-axis and the imaginary part on the y-axis.

To graph the complex number \(-\frac{1}{5} + \frac{2}{7}i\), plot the point \(-\frac{1}{5}\) on the x-axis and \(\frac{2}{7}\) on the y-axis. Similarly, for \(\frac{3}{7} - \frac{3}{4}i\), plot \(\frac{3}{7}\) on the x-axis and \(-\frac{3}{4}\) on the y-axis. The resultant complex number \(\frac{8}{35} - \frac{13}{28}i\) is plotted by locating \(\frac{8}{35}\) on the x-axis and \(-\frac{13}{28}\) on the y-axis.

When plotting these, the points give a visual way of understanding how complex numbers add up. It becomes clear how the real and imaginary parts affect the position of each point in the complex plane.

By practicing graphing, you’ll get a better intuition for the operations involving complex numbers, making them easier to grasp and manipulate.