Nice complex numbers problem (1 Viewer)

[maniacguy]

Prove |Arg(z)| > | | |z| -1| - |z-1| | for z a complex number.

This is a very nice idea (IMO) and one worthy of consideration, although I doubt it'll crop up in an HSC exam.

Hint : Algebraically bashing this will more than likely get you into an evil mess! Not least because you'll end up with a trig inequality that you then have to differentiate, and absolute value signs running around here, there and everywhere! Think about what complex numbers mean.

 

[ND]

Shouldn't it be >=, cos when z is purely real and +ve, arg(z) = 0 and so does RHS.

 

[maniacguy]

Yes, you're right, sorry
And I was thinking about the equality as well just before posting...

 

[tonberry_kun]

i dont even understand the question...
i'm fuct for hsc arent i?

 

[enak]

So many | lines

 

[Affinity]

bashed it , can't find an elegant solution

consider the points Z, A=(Z-1), and a point B on the line OZ where ZB = 1 and Z is not between O an B.

If OB < OA let C be a point on OA such that OC = OB (case 1)
else let C be a point on OB such that OC = OA (case 2)

now, OA represents |z-1| and OB ||z|-1|

the RHS of the inequality = CA (case 1)or CB (case 2)

AB^2 = 1 + 1 - 2cos@ by the cosine rule note angle AZB =@ the argument of Z.

for case 1,
AB + OB > OA = OC +AC (triangle inequality)
AB > CA

for case 2,
AB + OA > OB = OC + CB (triangle inequality)
AB > CB

so in either case, AB > RHS

the question reduces to prove:
@^2 > AB^2

RTP @^2 > 2(1 - cos@)

RTP @^2 + 2cos@ -2 >0

this holds for @ = 0

d(LHS)/d@ = 2@ - 2sin@, and since @ > sin@ for @>0

LHS increases for positive @

the inequality holds for positive @, and notice that LHS is even, so the inequality holds for all values.

 

[deyveed]

Affinity and ND, what are you both ranked for Maths Ext 2?
How did you both get so good at Ext2?

 

[maniacguy]

Grrr... (I'm having some difficulty logging on. Possibly need to adjust security settings)

Apologies to affinity - I posted the wrong problem first.

The original was meant to have LHS = |z||Arg(z)|

(The problem becomes considerably easier, and the ideas are introduced earlier)

On the positive side, Affinity has demonstrated the validity of using a geometrical arguement based on the Argand diagram in proving the result that was given. (Great work on that, Affinity!)

(It's still a nice proof, though, I think. Does anyone know any of the hsc question setters? (for future years, of course. This year's was done back in February) )

 

[enak]

maniac: perhaps you would like to edit your first post to make it correct, then people attempting it wouldn't have so much trouble

 

[freaking_out]

Originally posted by deyveed
Affinity and ND, what are you both ranked for Maths Ext 2?
How did you both get so good at Ext2?

yeah, did u guys go to tutors?

 

[ND]

Originally posted by freaking_out
yeah, did u guys go to tutors?

I'd be interested to hear Affinity's reply to this, as he seems to have an "Affinity" for maths. I however am only doing it for my UAI.

The only tutoring i've ever had for maths is 1 hour with spice girl and 2 with turtle.

 

[Affinity]

Originally posted by ND
I'd be interested to hear Affinity's reply to this, as he seems to have an "Affinity" for maths. I however am only doing it for my UAI.

The only tutoring i've ever had for maths is 1 hour with spice girl and 2 with turtle.

haven't had any, unless dad counts.. and that was fews years ago.

never knew turtle tutored, must be very interesting.

 

[freaking_out]

Originally posted by ND
I'd be interested to hear Affinity's reply to this, as he seems to have an "Affinity" for maths. I however am only doing it for my UAI.

The only tutoring i've ever had for maths is 1 hour with spice girl and 2 with turtle.

how much did they charge u.

 

[ND]

Originally posted by freaking_out
how much did they charge u.

They made me realise how bad at maths i really am.

edit: just realised that could be taken the wrong way, what i mean is, watching them work makes me feel so inferior.

 

[freaking_out]

Originally posted by ND
They made me realise how bad at maths i really am.

edit: just realised that could be taken the wrong way, what i mean is, watching them work makes me feel so inferior.

hmm...yeah, i'm suprised that they didn't get 100...so there must b ppl smarter than them. *shrugs*

 

[ND]

Originally posted by freaking_out
hmm...yeah, i'm suprised that they didn't get 100...so there must b ppl smarter than them. *shrugs*

I don't know about smarter, maybe just better at english.

 

[McLake]

Originally posted by ND
I don't know about smarter, maybe just better at english.

2 people from my high school got 100 last year, no maths (all humanities)

 

[Affinity]

think I got what maniacguy has in mind

scrap the calculus part
@ = Arc AB centered at Z > AB

 

[deyveed]

What are you both ranked?

 

[ND]

In my class? 1/1. Heh.