# QUICK HELP complex numbers

Let Z1 = 3-i
Z2=7+2i express (1/Z1)-(1/Z2) in form a+bi
SOMEONE plzzzzzzz HELP ME!!! I dont have a clue as to how to do this
What do I do?
Where do I start?

Have you ever rationalized the denominator of a fraction before? This is pretty much the same thing. For instance, when you have a fraction such as $$\frac{5}{\sqrt{3}+2}$$, you multiply the top and bottom by the conjugate, because you know a difference of squares will result in a rational number, because the square root of a rational squared is a rational. The same idea applies here: just keep in mind that $$i=\sqrt{-1}$$, and apply the same concept.

Hope that helps! :)

Do you know what a "complex conjugate" is?

Parth Dave said:
Do you know what a "complex conjugate" is?

Yes I do it is the opposite sign well when dividing you take the denominator and divide by the conjugate I know that but in this question I dont know what to do or what order to do it in plz help me plzzz

BobG
Homework Helper
nolachrymose said:
Have you ever rationalized the denominator of a fraction before? This is pretty much the same thing. For instance, when you have a fraction such as $$\frac{5}{\sqrt{3}+2}$$, you multiply the top and bottom by the conjugate, because you know a difference of squares will result in a rational number, because the square root of a rational squared is a rational. The same idea applies here: just keep in mind that $$i=\sqrt{-1}$$, and apply the same concept.

Hope that helps! :)
nolachrymose described it pretty well.

If you substituted i for the square root of 3,

$$\frac{5}{\sqrt{-1}+2}$$,

and multiplied by the conjugate, you'd get:

$$\frac{5(\sqrt{-1}-2)}{-1 - 4} = 2 - \sqrt{-1}$$

The i is square root of negative one. If you have 3i and square it, you get -9. Other than keeping the negative signs straight, it's just like working with a square root.

Ive never rationalized the denominator of a fraction so im a little lost I tried multiplying 1/3-i first by the conjugate and got the answer of 3+i/10 and then i did the same for z2 and got 7-2i/53 but if i subtract the two i dont get the right answer plz show me how do i divide first subtract use conjugates or what?

As far as I can tell you rationalized the two properly. Unless you made a subtraction error, your answer should be right. What did you get as the final answer?

a)89/24-73/24i

b)89/24+73/24i

c)-89/24+73/24i

d)89/24i-73/24

I dont get anything close to these when i take those two answers and subtract them. Please help me Ive been doing this one forever

BobG
Homework Helper
All of their choices are wrong.

So what did you get? Something close to

$$\frac{89+73i}{530}$$

BobG
Homework Helper
I think I know where they made their mistake. You can do this two ways.

$$\frac{1}{3-i}-\frac{1}{7+2i}$$

$$\frac{3+i}{9-(-1)}-\frac{7-2i}{49-(-4)}$$

$$\frac{3+i}{10}-\frac{7-2i}{53}$$

$$\frac{(159+53i)-(70-20i)}{530}$$

$$\frac{89+73i}{530}$$

Or:

$$\frac{1}{3-i}-\frac{1}{7+2i}$$

$$\frac{7+2i}{(3-i)(7+2i)}-\frac{3-i}{(3-i)(7+2i)}$$

$$\frac{7+2i-3+i}{(21-(-2))+(6i-7i)}$$

$$\frac{4+3i}{23-i}$$

$$\frac{(4+3i)(23+i)}{(23-i)(23+i)}$$

$$\frac{(92+(-3))+(69i+4i)}{529-(-1)}$$ Here's where they made their mistake

$$\frac{89+73i}{530}$$

You can break this up into two separate fractions, if you want:

$$\frac{89}{530}+\frac{73}{530}i$$

Instead of multiplying (23-i)(23+i) and getting 529+1, they got 23+1. So, the answer they most likely picked is (b)

arildno
Homework Helper
Gold Member
Dearly Missed
A great example of why multiple choice exams are (IMO) inferior to some other exam types which are less sensitive to designer flaws.

BobG
Homework Helper
arildno said:
A great example of why multiple choice exams are (IMO) inferior to some other exam types which are less sensitive to designer flaws.
A flaw?! Or is it really a hidden 'feature'? (how come we don't have a 'shifty eyed' smilie?)

"B's for everyone who answers all the questions right. A's for everyone who catches my error and figures out exactly where I made my mistake!"

BobG said:
All of their choices are wrong.

So what did you get? Something close to

$$\frac{89+73i}{530}$$