# SSAT Upper Level Math : How to graph complex numbers

## Example Questions

### Example Question #11 : How To Graph Complex Numbers

Give the number which, when multiplied by , yields the same result as if it were increased by 6.

No such number exists.

Explanation:

Let  be the number in question. The statement "[a number] multiplied by  yields the same result as if it were increased by 6" can be written as

We can solve this for  as follows:

Rationalize the denominator by multiplying both numerator and denominator  by the conjugate of the denominator, which is :

### Example Question #12 : How To Graph Complex Numbers

Define an operation  as follows:

For all complex numbers ,

.

If , evaluate .

None of the other choices gives the correct answer.

Explanation:

,

by our definition, can be rewritten as

or

Taking the reciprocal of both sides, then multiplying:

### Example Question #401 : Geometry

Raise  to the power of 4.

The expression is undefined.

Explanation:

### Example Question #14 : How To Graph Complex Numbers

Define an operation  as follows:

For all complex numbers ,

Evaluate .

None of the other choices gives the correct answer.

Explanation:

### Example Question #15 : How To Graph Complex Numbers

Give the number which, when added to 20, yields the same result as if it were subtracted from .

No such number exists.

Explanation:

Let  be the number in question. The statement "[a number] added to 20 yields the same result as if it were aubtracted from " can be written as

Solve for :

### Example Question #16 : How To Graph Complex Numbers

Define an operation  as follows:

For all complex numbers ,

Evaluate

Explanation:

Multiply both numerator and denominator by the conjugate of the denominator, , to rationalize the denominator:

### Example Question #17 : How To Graph Complex Numbers

Subtract  from its complex conjugate. What is the result?

Explanation:

The complex conjugate of a complex number  is , so the complex conjugate of  is . Subtract the former from the latter:

### Example Question #18 : How To Graph Complex Numbers

Give the product of  and its complex conjugate.

The correct answer is not given among the other responses.

The correct answer is not given among the other responses.

Explanation:

The product of a complex number  and its conjugate  is

which will always be a real number. Therefore, all four given choices, all of which are imaginary, can be immediately eliminated. The correct response is that the correct answer is not given among the other responses.

### Example Question #19 : How To Graph Complex Numbers

Add  to its complex conjugate. What is the result?

Explanation:

The complex conjugate of a complex number  is , so  has  as its complex conjugate; the sum of the two numbers is

### Example Question #20 : How To Graph Complex Numbers

Multiply the complex conjugate of  by . What is the result?