Basic Operations with Complex Numbers - Algebra 2

Distribute the minus sign:

Combine like terms:

can be restated in standard complex number form as

.

The complex conjugate of a complex number is , so the complex conjugate of is , which is equal to . Therefore, is the complex conjugate of .

can be restated in standard complex number form as

.

The complex conjugate of a complex number is , so the complex conjugate of is , which is also equal to . Therefore, itself is the complex conjugate of .

When adding complex numbers, add the real parts and the imaginary parts separately to get another complex number in standard form.

Adding the real parts gives , and adding the imaginary parts gives .

This is essentially the following expression after translation:

Now add the real parts together for a sum of , and add the imaginary parts for a sum of .

The first step is to distribute which gives us:

which is in standard form.

Multiply both the numerator and the denominator by the conjugate of the denominator which is which results in

The numerator after simplification give us

The denominator is equal to

Hence, the final answer in standard form =

Multiply both the numerator and the denominator by the conjugate of the denominator which is resulting in

This is equal to

Since you can make that substitution of in place of in both numerator and denominator, leaving:

When you then cancel the negatives in both numerator and denominator (remember that , simplifying each term), you're left with a denominator of and a numerator of , which equals .

The absolute value is a measure of the distance of a point from the origin. Using the pythagorean distance formula to calculate this distance.

Combine like terms. Treat as if it were any other variable.

Substitute to eliminate .

Simplify.

Recall that the definition of imaginary numbers gives that and thus that . Therefore, we can use Exponent Rules to write

When dealing with imaginary numbers, we multiply by foiling as we do with binomials. When we do this we get the expression below:

Since we know that we get which gives us .

Use the FOIL method to simplify. FOIL means to mulitply the first terms together, then multiply the outer terms together, then multiply the inner terms togethers, and lastly, mulitply the last terms together.

The imaginary is equal to:

Write the terms for .

Replace with the appropiate values and simplify.

We know that . Therefore, . Thus, every exponent of that is a multiple of 4 will yield the value of . This makes . Since , we know that .

Combine like terms:

Distribute:

Combine like terms:

Use FOIL to multiply the two binomials.

Recall that FOIL stands for Firsts, Outers, Inners, and Lasts.

Remember that

To rationalize a complex fraction, multiply numerator and denominator by the conjugate of the denominator.