Question

Can someone explain this question to me, and how to go about answering it? (question posted below)

I think the reason it's in quotations is that it's a complex number . . . but I'm not completely sure. From what I understand, this number can be a square root. Ex. √24/25 + 7/25i times √24/25 + 7/25i = 24/25 + 7/25i.

Can someone help me understand?

Answer 1

is this from a funny math book ? :D

Answer 2

Hah, noo, it's a math problem my professor assigned me for precalculus 2.

Answer 3

He made it up on his own.

Answer 4

i liked this funny way of writeing questions :D

Answer 5

ok start by this
find:-
\(\large( \frac{24}{25}+\frac{7}{25} i)^{\frac{1}{2}}\)

Answer 6

u know that
z=x+yi
x=r cos theta
y= r sin theta

Answer 7

Yes, I know that. So, would (24/25+72/5i)^1/2 be √24/25 + 7/25i ?

Answer 8

r=1
\(\large \theta =\cos ^{-1} x\)

\(\large \theta = \sin^{-1} y \)

so find theta first

Answer 9

cos^-1 = 25/24 and sin^-1 = 25/7 ?

Answer 10

cos ^-1(24/25)=sin^-1 (7/25) = 16,26... (from calculator)
then write the equation

\(z=\cos (\theta+2\pi) +\sin (\theta+2\pi) i\)
and use this formula :-
\((z)^n=(\cos \theta +\sin \theta i)^n=\cos n \theta +\sin n \theta i\)

Answer 11

So 16= cos^-1 and 26= sin^-1? (I just want to make sure I understand before I proceed with the rest)

Answer 12

no lol i made a typo its 16.26

Answer 13

Ooohhhh ok

So z= cos(16.26 + 2π) + sin(16.26 + 2π)i ?

Answer 14

yesss !

Answer 15

Ok, so then (z)^n = (cos (16.26) + sin(16.26)i )^n = cos n(16.26) + sin n(16.26)i ?

Answer 16

well u have n=1/2 since its sqrt right ?
just to make it clear there would be two solution
z= cos(16.26 /2) + sin(16.26 /2)i
and
z= cos(16.26/2 + π) + sin(16.26/2 + π)i

Answer 17

Oohhh I see. So n=1/2. Just to be sure, can you clarify where pi came from?

Answer 18

sure :)
z= cos(θ ) + sin(θ )i
is a perodic function of period 2π , which means
cos(θ )=cos(θ + 2nπ)
sin(θ )=sin(θ+ 2nπ)
z= cos(θ + 2nπ) + sin(θ+ 2nπ)i

mmm ok ?

Answer 19

so give n two values 0,1

Answer 20

Ohhh I see. And then you get two different solutions, right?

Answer 21

yep ^^

Answer 22

Since if n=0 then 2nπ would equal 0, and if n=1, then 2nπ will equal 2π

Answer 23

Ok!

Answer 24

Ok, so I'm at this point where z= cos(16.26 /2) + sin(16.26 /2)i and
z= cos(16.26/2 + π) + sin(16.26/2 + π)i. Is there anything else I need to do?

Answer 25

nope ^^

Answer 26

Ok. So, the answer to the original question would be that there is no square root of 24/25 + 7/25i because of those multiple solutions? (or am I completely off the track here?)

Answer 27

no lol " " are sense of humar its not related to math :P
@ganeshie8 are they for grammer thing lol ?

Answer 28

Ok, I'm really sorry, but I'm confused . . . . So the quotations don't mean anything? It's just there to mess with me (to put it bluntly)?

Answer 29

i think so ^_^
at least they dint mean anything to me ... mmm

Answer 30

Hm . . . the question asks if the quotations mean that something is presumed that shouldn't be . . . . But I can't see anything like that. Unless I'm missing it . . . .

Answer 31

gtg nw :D
cya

Answer 32

Ok, thanks for your help!

Answer 33

np :)

Answer 34

Is anyone able to help with that last bit then? About the presuming part?

Answer 35

there is no "THE" square root for any complex number

Answer 36

there are always "TWO" square roots for any complex number

Answer 37

while asking the question in quotation marks, he presumed that there is an unique square root - which is not right.

Answer 38

Oohhh, I see. Dang, and I already knew that too. I can't believe I missed that . . . but thank you so much for explaining, I really appreciate it.

Answer 39

np :)

Answer 40

mmm i hate eng xD