Complex Numbers - Trigonometry

To solve this problem, we must first identify the conjugate of this complex number. The conjugate keeps the real portion of the number the same, but changes the sign of the imaginary part of the number. Therefore the conjugate of is . Now, we need to multiply these together using distribution, combining like terms, and substituting .

To solve this problem, we must understand what a complex conjugate is and how it relates to a complex number. The conjugate of a number is . Therefore the conjugate of is .

To add complex numbers, we must combine like terms: real with real, and imaginary with imaginary.

In order to solve this problem, we must combine real numbers with real numbers and imaginary numbers with imaginary numbers. Be careful to distribute the subtraction sign to all terms in the second set of parentheses.

To solve this problem, make sure you set it up to multiply the entire parentheses by itself (a common mistake it to try to simply distribute the exponent 2 to each of the terms in the parentheses.)

(recall that )

Please note that while the answer choice is not incorrect, it is not fully simplified and therefore not the correct choice.

While these terms may not look like they follow the typical format of , don't let them fool you! We can read 5 as and we can read 3i as . Now recalling that the complex conjugate of is , we can see that the complex conjugate of is just and the complex conjugate of is

To perform division on complex numbers, multiple both the numerator and the denominator of the fraction by the complex conjugate of the denominator. This looks like:

To represent complex numbers graphically, we treat the x-axis as the "axis of reals" and the y-axis as the "axis of imaginaries." To plot , we want to move 6 units on the x-axis and -3 units on the y-axis. We can plot the point P to represent , but we can also represent it by drawing a vector from the origin to point P. Both representations are in the diagram below.

We can take any complex number and graph it as a vector, measuring units in the x direction and units in the y direction. Therefore . Likewise, . Then, we can add these two vectors together, summing their real parts and their imaginary parts to create their resultant vector . Therefore the correct answer is .

The above image is a graphic representation of subtraction of complex numbers (which are represented by vectors and . We can take any complex number and graph it as a vector, measuring units in the x direction and units in the y direction. Therefore . Likewise, .

To help us visualize subtraction, instead of thinking about taking , we should instead visualize . The below figure shows with a dotted line. Visually, the resultant vector lies in between the vectors and . Algebraically, we get or . Either way you think about it, the resulting vector is