Complex Numbers finding values

In summary, the problem is to find a value of k given that arg((z1z2)/2i) = ? and arg z2=2?=0, and arg z1=?/6. The solution involves expanding the arg function and using identities to simplify the expression, and knowing that arg(i) = ?/2.
  • #1
lunds002
21
0

Homework Statement



Given that arg((z1z2)/2i) = [itex]\pi[/itex], find a value of k.

Homework Equations



arg z2=2[itex]\pi[/itex]=0

arg z1=[itex]\pi[/itex]/6


The Attempt at a Solution



(([itex]\pi[/itex]/6)^k x (2[itex]\pi[/itex]))/ (2i) = [itex]\pi[/itex]

I'm not sure what to do with the imaginary number i..
 
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  • #2
lunds002 said:
Given that arg((z1z2)/2i) = [itex]\pi[/itex], find a value of k.

You're missing a k in the problem. Where is it missing?
 
  • #3
Oops, should be arg ( z1 x z2^k) / (2i)
 
  • #4
lunds002 said:
Oops, should be arg ( z1 x z2^k) / (2i)

You should be able to get it by expanding the arg function.

E.g. arg(z1z1)=arg(z1)+arg(z2).

(the rules are similar to log rules)
 
  • #5
Okay so then I get

arg(z1^k) + arg(z2) = pi
2i

(pi/6)^k + 2pi = pi
2i

(pi/6)^k + 2pi = 2i(pi)

Not sure what to do with the imaginary i
 
  • #6
lunds002 said:
Okay so then I get

arg(z1^k) + arg(z2) = pi
2i

(pi/6)^k + 2pi = pi
2i

(pi/6)^k + 2pi = 2i(pi)

Not sure what to do with the imaginary i

Check the wikipedia page on argument for how to deal with arg(z1/z2). It's similar to how you expand loga(x/y).

You should know what arg(i) is equal to.
 
  • #7
Okay so I know z1/z2 = r1/r2 cis (theta-[itex]\psi[/itex])

I don't really understand how that applies here though

And you're right, I do know that arg(i) = -1
 
  • #8
lunds002 said:
Okay so I know z1/z2 = r1/r2 cis (theta-[itex]\psi[/itex])

I don't really understand how that applies here though

And you're right, I do know that arg(i) = -1

http://en.wikipedia.org/wiki/Argument_(complex_analysis)#Identities

arg(i) is the angle formed by the z=i and the positive real axis. z=i is the line perpendicular to the positive real axis, so arg(i) is not -1 but ?
 

Related to Complex Numbers finding values

What are complex numbers?

Complex numbers are numbers that consist of both a real part and an imaginary part. They are written in the form a + bi, where a is the real part and bi is the imaginary part. For example, 3 + 2i is a complex number with a real part of 3 and an imaginary part of 2i.

How do you add or subtract complex numbers?

To add or subtract complex numbers, you simply add or subtract the real parts and the imaginary parts separately. For example, (3 + 2i) + (5 + 4i) = (3+5) + (2i+4i) = 8 + 6i. Similarly, (3 + 2i) - (5 + 4i) = (3-5) + (2i-4i) = -2 - 2i.

How do you multiply complex numbers?

To multiply complex numbers, you use the FOIL method, just like when multiplying binomials. For example, (3 + 2i)(5 + 4i) = 3(5) + 3(4i) + 2i(5) + 2i(4i) = 15 + 12i + 10i + 8i² = 15 + 22i - 8 = 7 + 22i.

How do you find the conjugate of a complex number?

The conjugate of a complex number is found by changing the sign of the imaginary part. For example, the conjugate of 3 + 2i is 3 - 2i. This is useful when dividing complex numbers, as it allows you to eliminate the imaginary part in the denominator.

How do you find the absolute value of a complex number?

The absolute value (or modulus) of a complex number is found by taking the distance of the number from the origin on the complex plane. It is calculated as the square root of the sum of the squares of the real and imaginary parts. For example, |3 + 2i| = √(3²+2²) = √13.

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