# SAT II Math I : Number Theory

## Example Questions

### Example Question #3 : Complex Imaginary Numbers

Simplify the following product:

Explanation:

Multiply these complex numbers out in the typical way:

and recall that  by definition. Then, grouping like terms we get

### Example Question #4 : Complex Imaginary Numbers

Identify the real part of

none of the above.

Explanation:

A complex number in its standard form is of the form: , where  stands for the real part and  stands for the imaginary part. The symbol  stands for .

The real part in this problem is 1.

### Example Question #5 : Complex Imaginary Numbers

Simplify:

Explanation:

To add complex numbers, find the sum of the real terms, then find the sum of the imaginary terms.

### Example Question #1 : Imaginary Numbers & Complex Functions

Simplify:

Explanation:

Start by using FOIL. Which means to multiply the first terms together then the outer terms followed by the inner terms and lastly, the last terms.

Remember that , so .

Substitute in  for

### Example Question #2 : Imaginary Numbers & Complex Functions

Simplify:

Explanation:

Start by using FOIL. Which means to multiply the first terms together then the outer terms followed by the inner terms and lastly, the last terms.

Remember that , so .

Substitute in  for

### Example Question #3 : Imaginary Numbers & Complex Functions

Simplify:

Explanation:

Start by using FOIL. Which means to multiply the first terms together then the outer terms followed by the inner terms and lastly, the last terms.

Remember that , so .

Substitute in  for

### Example Question #11 : Number Theory

Simplify:

Explanation:

Start by using FOIL. Which means to multiply the first terms together then the outer terms followed by the inner terms and lastly, the last terms.

Remember that , so .

Substitute in  for .

### Example Question #3 : Complex Conjugates

Simplify:

Explanation:

To get rid of the fraction, multiply the numerator and denominator by the conjugate of the denominator.

Now, multiply and simplify.

Remember that

### Example Question #12 : Number Theory

Write in standard form:

Explanation:

Multiply by the conjugate:

### Example Question #9 : Imaginary Numbers

Simplify the expression.