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Advanced Math/Complex numbers - DeMoivre's Thm

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Expert: - 5/4/2008


Question

Complex Number
Please see the image attached. I am currently studying A level Maths. Thank you for your time.

Answer
Questioner:   jenny
Category:  Advanced Math
Private:  No
 
Subject:  complex numbers

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Hi, Jenny,

Your question: Please see the image attached.

>> OK, I have it, but it really is better if you try typing it in.  You might find it helps your own understanding.
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I am currently studying A level Maths. Thank you for your time.  

>> Alas, the educational system here in West Berzerkistan is different from yours and I have no idea what A level means.  Try sending the chapter-section heading from the current page of your text next time.

It looks as if you are doing 'Basic operations on complex numbers in polar form and DeMoivre's Theorem'.
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So you have to change numbers in 'standard' or 'rectangular' form, z = x + iy, to polar form, or  z = r cis(t).

[cis t is an abbreviation for  cos t + i sin t]

[The t is theta, which the crude computers here in West Berzerkistan cannot make.]

The scheme is to:

1. Make a CLEAR (and I mean clear) diagram.
2. Deduce the r (distance or modulus) and t (angle or amplitude) from your diagram.

Of course, you have:

1. r = sqrt(x^2 + y^2) and  
2. t = arctan(y/x), sort of. [You have to deduce the proper quadrant of t from that CLEAR diagram.]
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Note: Since my recent difficulties I have been reducing my typing.  I use a lot of abbreviations:

s  = sin x
c  = cos x
s2 = sqrt(2)
s3 = sqrt(3)
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The complex numbers z1 and z2 are given by

A. z1 = -2 - 2i

OK, the point (-2,-2) is in Q4 and you should get the angle as either 315 degrees or 7pi/4 (or -45 degrees or - pi/4), and
r = 2s2.  [s2 means sqrt(2)]
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B. z2 = s3 + i  [s3 means sqrt(3)]

OK, (s3,1) is in Q1 and you should get the angle as 30 deg or pi/6,
and  r = 2

(i)Find the modulus and argument of z1 and z2, and write each number in polar form.

We have, from the above,

r1 = 2 s2,  t1 = - 45 deg (I'll use that one.)
r2 = 2,     t2 = 30 deg

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(ii)Hence find the modulus and argument of z1z2 and z1^2 z2^2.

Basic operations:

z1z2 = r1r2 cis(t1 + t2)

z1z2 = (2 s2)(2) cis(-45 + 30)

z1z2 = 4 s2 cis(-15 deg)
z1z2 = 4 s2 cis(345 deg), if you must have 0 <= t <= 360

z1^2 z2^2 = (z1z2)(z1z2) = (4 s2 cis(-15 deg))(4 s2 cis(-15 deg))

= 16(2) cis (-30 deg)
= 32 cis (330 deg )
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(iii)Using the polar form of z1 and z2, find the modulus and argument of z2^7/z1^4   and hence write this quotient in the form a + bi.

DeMoivre's Thm: (remember, t means theta)

z^n = (r cis t)^n = r^n cis (nt)

z2^7 = (2 cis 30)^7 = 2^7 cis (210)

z1^4 = (2s2 cis(-45))^4 = 16(4) cis(-180) = 2^6 cis (+180)

Basic operation:

z3/z4 = (r3/r4) cis (t3 - t4)

Divide:
z2^7    2^7 cis (210)
----- = -------------- = 2 cis 30
z1^4    2^6 cis (+180)

Now you want to make that x + iy, right?

2 cis 30 = 2(cos 30 + i sin 30)

2(s3/2 + i (1/2)) = s3 + i.

That's it.

Paul Klarreich

Expertise

I can answer questions in basic to advanced algebra (theory of equations, complex numbers), precalculus (functions, graphs, exponential, logarithmic, and trigonometric functions and identities), basic probability, and finite mathematics, including mathematical induction. I can also try (but not guarantee) to answer questions on Abstract Algebra -- groups, rings, etc. and Analysis -- sequences, limits, continuity. I won't understand specialized engineering or business jargon.

Experience

I taught at a two-year college for 25 years, including all subjects from algebra to third-semester calculus.

Education/Credentials
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