Solving (-1)^i Using MATLAB: A Step-by-Step Guide

In summary, the conversation is about trying to calculate (-1)^i using the formula x^ni = cos (ln (x)^n) + i sin (ln (x)^n). The person tried to use MATLAB and got an answer, but is looking for help in solving it with steps. They also mention using polar form and complex exponential, but are not sure how to use them in this case. The conversation ends with someone suggesting to use Euler's formula and the other person trying to figure out how to apply it.
  • #1
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Calculate (-1) ^ i

I tried using the formula x^ni = cos (ln (x)^n) + i sin (ln (x)^n)

but i cannot solve it. i used MATLAB to get this answer 0.0432139182637723 + 0i

but i don't know how to solve it with steps.. can i get some assistance please.

thank you.
 
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  • #2
aks_sky said:
Calculate (-1) ^ i

I tried using the formula x^ni = cos (ln (x)^n) + i sin (ln (x)^n)

but i cannot solve it. i used MATLAB to get this answer 0.0432139182637723 + 0i

but i don't know how to solve it with steps.. can i get some assistance please.

thank you.

Do you know how to write -1 in polar form?
 
  • #3
yup the polar form will just be cos (theta) + i sin (theta) and the modulus here is 1.. correct?
 
  • #4
aks_sky said:
yup the polar form will just be cos (theta) + i sin (theta) and the modulus here is 1.. correct?

Well, that's a particular complex number, but it's actually expressed in rectangular form x + iy, where x = cos(theta) and y = sin(theta).

Do you know how to write -1 in terms of "e", i.e., do you know what a complex exponential is? It would help to know what background can be assumed for this exercise.
 
  • #5
well what i did was... x = ln (-1)^i
which is.. i ln (-1)
then in terms of "e" i will get... e ^ i ln (-1)

which gives me cos (ln (-1)) + i sin (ln (-1))
but i can't go any further to get the answer
 
  • #6
aks_sky said:
well what i did was... x = ln (-1)^i
which is.. i ln (-1)
then in terms of "e" i will get... e ^ i ln (-1)

which gives me cos (ln (-1)) + i sin (ln (-1))
but i can't go any further to get the answer

What I was trying to get at is, have you been exposed to Euler's famous formula:

[tex]e^{i\pi} = -1[/tex]

If so, then you can easily use this to get the answer you want.
 
  • #7
yup i know that formula.. but how do i use it here?.. i tried to use that formula too but dint work.. maybe i did something wrong?
 
  • #8
aks_sky said:
yup i know that formula.. but how do i use it here?.. i tried to use that formula too but dint work.. maybe i did something wrong?

Well, you're trying to find (-1)^i, right? So what is the natural thing do to both sides of Euler's formula?
 
  • #9
um not sure exactly.
 
  • #10
aks_sky said:
um not sure exactly.

Oh, come on!

What operation do you do to -1 to obtain (-1)^i? (This isn't a trick question!) Just do that operation to both sides of Euler!
 
  • #11
What jbunnii is trying to say is:

[tex](-1) = (e^{i\pi})[/tex]
[tex](-1)^i = ...[/tex]

Use basic algebra here.
 
  • #12
we take logs of both sides
 
  • #13
ohhh yup i get what you asking
 

1. What are complex numbers?

Complex numbers are numbers that are expressed in the form of a + bi, where a and b are real numbers and i is the imaginary unit (√-1). They are used to represent quantities that involve both real and imaginary components.

2. How do complex numbers differ from real numbers?

Unlike real numbers, which can be represented on a number line, complex numbers cannot be visualized in this way. Additionally, while real numbers have only one solution for equations such as x^2 = 9, complex numbers have two solutions in the form of ±3i.

3. What is the purpose of using complex numbers in problem-solving?

Complex numbers are used in various fields of science and mathematics, such as engineering, physics, and signal processing. They allow for more accurate and efficient calculations and can represent physical quantities that cannot be described using only real numbers.

4. How are complex numbers represented on a graph?

Complex numbers can be represented on a graph using the complex plane, with the real component plotted on the x-axis and the imaginary component on the y-axis. The point where the two axes intersect represents the origin, or 0 + 0i.

5. What are some common operations performed with complex numbers?

Some common operations with complex numbers include addition, subtraction, multiplication, and division. Other operations such as finding the modulus (absolute value) and conjugate of a complex number are also frequently used.

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