Complex numbers question

In summary, the equation (z1z2)^w=(z1^w)(z2^w) is not valid for z1=z2=-1 and w=-i. In order for the equation to hold for all complex values of w, the complex values of z1 and z2 must be expressed in the length-angle representation of a complex number, where z=r e^(i φ), with 0 ≤ φ < 2π and r ≥ 0. Using logs on complex numbers is not recommended.
  • #1
Lucy Yeats
117
0

Homework Statement



Verify that the equation (z1z2)^w=(z1^w)(z2^w) is violated for z1=z2=-1 and w=-i.
Under what conditions on the complex values z1 and z2 does the equation hold for all
complex values of w?

Homework Equations





The Attempt at a Solution



((-1)x(-1))^(-i)=1^(-i)=e^ln(1^(-i))=e^(-iln1)=e^0=1

(-1)^(-i)x(-1)^(-i)=(-1)^(-2i)=((-1)^2)^(-i)=1^(-i) but this is the same as above.

Thanks in advance for helping. :-)
 
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  • #2
Hi Lucy Yeats! :smile:

I'm afraid you can't just use the regular rules for "ln" on complex numbers.
"ln" is not a well defined function for complex numbers.
It's a so called multi-valued function.

You have a similar problem with a^(bc).
It is not just equal to (a^b)^c.
It's more complex. ;)Try using the length-angle representation of a complex number.
That is, ##z=r e^{i \phi}##, where ##0 \le \phi \lt 2\pi## and ## r \ge 0##.
So try writing -1 as ##e^{i \pi}## and 1 as ##e^0##.
 
  • #3
I know that ln(z1z2)=lnz1+lnz2+2πin, where n is an integer.

I don't see why I need to use logs in this question. why can't I say:

(-1x-1)^(-i)=1^(-i)
(-1)^(-i)x(-1)^(-i)=((-1)^(-i))^2=(-1)^(-2i)=((-1)^2)^(-i)=1^(-i)

Thanks for helping!
 
  • #4
Sorry, I see what you mean about the a^bc thing. Ignore the last post and I'll try again.
:-)
 
  • #5
The step
(-1)^(-2i)=((-1)^2)^(-i)
does not hold.

Try the same step with e^(i pi), which is -1.

And no, you should not use logs in this question - at least not on complex numbers, but at most only on real numbers.
Lucy Yeats said:
Sorry, I see what you mean about the a^bc thing. Ignore the last post and I'll try again.
:-)
EDIT: Too late. ;-)
 
Last edited:

1. What are complex numbers?

Complex numbers are numbers that contain 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 with i being the imaginary unit equal to the square root of -1.

2. 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.

3. 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 + 8i^2 = 15 + 22i - 8 = 7 + 22i.

4. Can you divide complex numbers?

Yes, you can divide complex numbers. To divide complex numbers, you multiply the numerator and denominator by the complex conjugate of the denominator. The complex conjugate is found by changing the sign of the imaginary part. For example, (3 + 2i)/(5 + 4i) = (3 + 2i)(5 - 4i)/(5 + 4i)(5 - 4i) = (3*5 + 3*(-4i) + 2i*5 + 2i*(-4i))/(5^2 - (4i)^2) = (15 - 12i + 10i - 8i^2)/(25 - 16i^2) = (15 - 2i)/41 = 15/41 - 2i/41.

5. Why are complex numbers important?

Complex numbers are important because they allow us to solve problems in mathematics and physics that cannot be solved with real numbers alone. They are also used in applications such as electrical engineering and signal processing. Complex numbers have many real-world applications and are essential in understanding and describing certain phenomena in science and mathematics.

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