# ACT Math : Complex Numbers

## Example Questions

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### Example Question #1 : Complex Numbers

Subtract  from , given:

Explanation:

A complex number is a combination of a real and imaginary number. To subtract complex numbers, subtract each element separately.

In equation  is the real component and  is the imaginary component (designated by ). In equation  is the real component and  is the imaginary component. Solving for ,

### Example Question #1 : Complex Numbers

Simplify the exponent,

.

Explanation:

When you have an exponent on the outside of parentheses while another is on the inside of the parentheses, such as in , multiply the exponents together to get the answer: .

This is different than when you have two numbers with the same base multiplied together, such as in . In that case, you add the exponents together.

### Example Question #3 : 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:

Solving this equation is very similar to solving a linear binomial like . To solve, just combine like terms, being careful to watch for double negatives.

### Example Question #4 : 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:

A problem like this can be solved similarly to a linear binomial like /

### Example Question #5 : 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 equations simplifies into ?

Explanation:

This equation can be solved very similarly to a binomial like .

### Example Question #6 : Complex Numbers

Suppose  and

Evaluate the following expression:

Explanation:

Substituting for  and , we have

This simplifies to

which equals

### Example Question #1 : Complex Numbers

What is the solution of the following equation?

Explanation:

A complex number is a combination of a real and imaginary number. To add complex numbers, add each element separately.

First, distribute:

Then, group the real and imaginary components:

Solve to get:

### Example Question #8 : Complex Numbers

What is the sum of  and  given

and

?

Explanation:

A complex number is a combination of a real and imaginary number. To add complex numbers, add each element separately.

In equation  is the real component and  is the imaginary component (designated by ).

In equation  is the real component and  is the imaginary component.

### Example Question #9 : Complex Numbers

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

Simplify:

Explanation:

When adding or subtracting complex numbers, the real terms are additive/subtractive, and so are the nonreal terms.

### Example Question #10 : Complex Numbers

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

Can you add the following two numbers: ? If so, what is their sum?