Interesting question: why is ln(-1) in polar form....?

In summary, ln(-1) is in polar form because it allows us to represent all possible values of the multi-valued function, ln(x). It cannot be simplified further and is closely related to the single-valued function, Log(z). Ln(-1) also has interesting properties, such as being a square root and cube root of unity. It has many applications in mathematics and science, particularly in complex analysis, number theory, and physics.
  • #1
rabualeez
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Hi all,
I was doing some math and I stumbled upon a very interesting thing. When I do ln(-1), I get πi, and when I turn that into polar coordinates on the calculator, it gives me πeiπ/2 . Why is that? I'm very curious to know, because they are so intertwined!

Thank you
 
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  • #2
Hello Rabu, :welcome:

Apparently, ##i## and ##e^{i\pi/2}## are one and the same. Can you see why ?
The calculator responds in a form ##|\alpha| \,e^{\operatorname{\, Arg}(\alpha)}## which I think is reasonable for complex numbers.
 
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1. Why is ln(-1) in polar form?

The natural logarithm function, ln(x), is defined for positive real numbers. However, when we extend it to complex numbers, it becomes a multi-valued function. This means that there are infinite possible values for ln(-1), depending on the branch of the complex plane we choose. One common way to represent complex numbers is in polar form, using their magnitude and argument. In this form, ln(-1) can be expressed as 0 + (2n+1)iπ, where n is any integer. Therefore, we say that ln(-1) is in polar form because it allows us to represent all possible values of this multi-valued function.

2. Can ln(-1) be simplified further?

No, ln(-1) cannot be simplified any further. In fact, it is considered an "unsimplifiable" expression because it involves the imaginary unit, i.

3. How is ln(-1) related to the complex logarithm function?

The complex logarithm function, denoted as Log(z), is a single-valued function that is defined for all non-zero complex numbers. It is closely related to ln(x) and is equal to ln(x) + iArg(x), where Arg(x) is the principal argument of the complex number x. Therefore, ln(-1) can also be expressed as Log(1) = 0 + (2n+1)iπ, where n is any integer.

4. What other properties does ln(-1) have?

In addition to being a multi-valued function and having infinite possible values, ln(-1) also has some interesting properties. For example, it is the only complex number whose square is equal to 1, making it a square root of unity. It is also the only complex number whose cube is equal to -1, making it a cube root of -1.

5. How is ln(-1) used in mathematics and science?

Ln(-1) has many applications in mathematics and science, particularly in complex analysis, number theory, and physics. It is used to solve equations involving exponential and logarithmic functions, and it plays a crucial role in understanding the behavior of complex numbers. In physics, ln(-1) is used in the study of oscillations and waves, as well as in quantum mechanics. It is also used in signal processing and electrical engineering.

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