Playing around with imag. numbers.

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In summary, the conversation discusses the use of i as a tool in complex exponentials and analysis. It is also mentioned that i can be represented in polar form and its roots can be obtained by dividing the exponent and using the fact that ei(2npi)=1. The conversation then delves into the mathematical and physical interest of calculating powers of i, with the conclusion that it is extremely interesting and has various applications. The conversation also explores different ways to write i^(1/2) and its physical interpretations, such as using the geometric interpretation of complex number multiplication. Finally, the conversation discusses the fundamental theorem of algebra and how it relates to finding the solutions for i^(1/2).
  • #1
flatmaster
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We define i=Squrt(-1). However, from what math I know, we just use i as a tool in complex exponentials, complex analysis, etc. I curious what happens if you actually, take powers of i. Does this actually mean anything? Is it mathematically interesting. What is i^(1/2) other than just i^(1/2)?
 
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  • #2
i (like all complex numbers) has a polar representation. i=ei(pi/2). Roots of i can be obtained by dividing the exponent. For square root divide by 2. To get all the roots use the fact that ei(2npi)=1, for all n. So to get the other square root, use n=1.
 
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  • #3
Ok. I see how to calculate powers of i now, but is this at all mathematically or physically ineteresting?
 
  • #4
flatmaster said:
Ok. I see how to calculate powers of i now, but is this at all mathematically or physically ineteresting?

My answer is yes, but to explain it would require descriptions of lots of physics theory as well as mathematics.
 
  • #5
flatmaster said:
Ok. I see how to calculate powers of i now, but is this at all mathematically or physically ineteresting?

Extremely interesting. kth roots of unity (of which powers i^(1/n) are a special case) have a wide variety of applications.
 
  • #6
i^(1/2)= sqrt(2)/2+ i sqrt(2)/2 or sqrt(2)/2- i sqrt(2)/2.
 
  • #7
"What is i^(1/2) other than just i^(1/2)"

Do you mean what are physical interpretations of it, or what are other ways to write it? If it's the latter, then knowing the geometric interpretation of complex number multiplication helps a lot. Multiplying two complex numbers corresponds to adding their angles (with the positive x-axis) and multiplying their absolute values.

Now it's easy to see what two numbers squared give i. They must have magnitude 1, and since i makes an angle of 90 degrees, one root must make an angle of 45 degrees. Another solution would be the number on the unit circle that makes an angle of 225 degrees. According to the fundamental theorem of algebra, those are the only two solutions.
 
  • #8
Tobias Funke said:
"What is i^(1/2) other than just i^(1/2)"

Do you mean what are physical interpretations of it, or what are other ways to write it? If it's the latter, then knowing the geometric interpretation of complex number multiplication helps a lot. Multiplying two complex numbers corresponds to adding their angles (with the positive x-axis) and multiplying their absolute values.

Now it's easy to see what two numbers squared give i. They must have magnitude 1, and since i makes an angle of 90 degrees, one root must make an angle of 45 degrees. Another solution would be the number on the unit circle that makes an angle of 225 degrees. According to the fundamental theorem of algebra, those are the only two solutions.

This is what I was looking for. A geometrical way to express the analytical expression. I guess I didn't explain that very well.
 

What are imaginary numbers?

Imaginary numbers are numbers that when squared give a negative result. They are typically denoted by the letter "i" and are used in complex numbers to represent the square root of a negative number.

How do you add and subtract imaginary numbers?

To add and subtract imaginary numbers, simply combine the real parts and the imaginary parts separately. For example, (3+2i) + (1+4i) = (3+1) + (2i+4i) = 4 + 6i.

Can you multiply and divide imaginary numbers?

Yes, you can multiply and divide imaginary numbers just like real numbers. When multiplying, use the fact that i² = -1. When dividing, multiply the numerator and denominator by the complex conjugate of the denominator to eliminate the imaginary part.

What are the applications of imaginary numbers?

Imaginary numbers have many applications in mathematics and science. They are used in electrical engineering, quantum mechanics, signal processing, and many other fields to model and solve problems involving complex numbers.

How do you graph imaginary numbers?

Imaginary numbers can be graphed on the complex plane, with the real part represented on the x-axis and the imaginary part on the y-axis. The point (a+bi) on the plane would be plotted at the coordinates (a,b).

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