# ACT Math : Complex Numbers

## Example Questions

### Example Question #1 : How To Add Complex Numbers

Complex numbers take the form , where  is the real term in the complex number and  is the nonreal (imaginary) term in the complex number.

Which of the following is incorrect?

Explanation:

Complex numbers take the form , where  is the real term in the complex number and  is the nonreal (imaginary) term in the complex number.

Thus, to balance the equation, add like terms on the left side.

### Example Question #1 : How To Divide Complex Numbers

Simplify:

Explanation:

Multiply both numberator and denominator by :

### Example Question #2 : How To Divide Complex Numbers

Evaluate:

Explanation:

First, divide 100 by  as follows:

Now dvide this result by :

### Example Question #3 : How To Divide Complex Numbers

Evaluate:

Explanation:

First, divide 100 by  as follows:

Now, divide this by :

### Example Question #31 : Squaring / Square Roots / Radicals

Evaluate:

Explanation:

First, evaluate :

Now divide this into :

### Example Question #5 : How To Divide Complex Numbers

Evaluate:

Explanation:

First, evaluate  using the square pattern:

Divide this into :

### Example Question #2 : How To Divide Complex Numbers

Complex numbers take the form , where  is the real term in the complex number and  is the nonreal (imaginary) term in the complex number.

Simplify:

Explanation:

This problem can be solved very similarly to a binomial such as . In this case, both the real and nonreal terms in the complex number are eligible to be divided by the real divisor.

, so

### Example Question #3 : How To Divide Complex Numbers

Complex numbers take the form , where  is the real term in the complex number and  is the nonreal (imaginary) term in the complex number.

Simplify by using conjugates:

Explanation:

Solving this problem using a conjugate is just like conjugating a binomial to simplify a denominator.

Multiply both terms by the denominator's conjugate.

Simplify. Note .

Combine and simplify.

Simplify the numerator.

The prime denominator prevents further simplifying.

Thus, .

### Example Question #11 : Complex Numbers

Simplify:

Explanation:

This problem can be solved in a way similar to other kinds of division problems (with binomials, for example). We need to get the imaginary number out of the denominator, so we will multiply the denominator by its conjugate and multiply the top by it as well to preserve the number's value.

Then, recall  by definition, so we can simplify this further:

This is as far as we can simplify, so it is our final answer.

### Example Question #1 : How To Multiply Complex Numbers

The solution of  is the set of all real numbers  such that:

Explanation:

Square both sides of the equation:

Then Solve for x:

Therefore,