What's the deal with imaginary numbers?

  • Thread starter LHarriger
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Anyway, I thought I'd share this example of how the power rule for roots can be used to give incorrect results. This one is particularly dramatic, I think.In summary, the conversation discusses the ambiguity of the square root notation for complex numbers and how it differs from real numbers. The issue arises when using the power rule for roots, which can lead to incorrect results if both surds are negative square roots. The conversation also touches on the concept of branch cuts and the importance of being explicit about the ambiguity when using square roots in complex numbers.
  • #1
LHarriger
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Imaninary numbers, i=1 ??!

Ok, this is driving me crazy:
[itex]i^{2}=-1[/itex]
no problem here, but
[itex]i^{2}=\left(\sqrt{-1}\right)^{2}=\sqrt{(-1)^{2}}=\sqrt{1}=1[/itex]
oops...
I know that the error has to be in this step:
[itex]\left(\sqrt{-1}\right)^{2}=\sqrt{(-1)^{2}}[/itex]
(works fine for positive numbers)
and I am pretty sure it has something to do with principle roots.
However, I just can't figure out a clean argument for what exactly is wrong.
Please help!
 
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  • #2
The square root is ambiguous, ie, [itex]\sqrt{1}=\pm 1[/tex], and you need to be explicit about this ambiguity when using it or you'll make mistakes like that one.
 
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  • #3
StatusX said:
The square root is ambiguous, ie, [itex]\sqrt{1}=\pm 1[/tex], and you need to be explicit about this ambiguity when using it or you'll make mistakes like that one.
Actually, [itex]\sqrt{1}=1[/tex]. However the square roots of 1 are [itex]\pm 1[/tex].

The problem is that exponentiation for complex numbers differs than that for real numbers. In fact, if a and b are complex numbers, then we define:
[tex]a^b = e^{b (\log |a| + i\arg(a))}[/tex],
where [itex]\arg(a)[/itex] is chosen to lie in [itex](-\pi, \pi][/itex]. (Read up on branch cuts.)

From this it follows that [itex](a^b)^c = (a^c)^b[/itex] isn't necessarily true.
 
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  • #4
Thank you Morphism!
I recall learning complex exponentiation and branch cuts from my undergraduate complex analysis that I took 3 years ago. I was just too rusty to be able to put it too work on my own to solve this problem.
 
  • #5
morphism said:
Actually, [itex]\sqrt{1}=1[/tex]. However the square roots of 1 are [itex]\pm 1[/tex].

That notation isn't standard, and couldn't have been what the OP was using since he wrote [tex]\sqrt{-1}[/tex].

The problem is that exponentiation for complex numbers differs than that for real numbers. In fact, if a and b are complex numbers, then we define:
[tex]a^b = e^{b (\log |a| + i\arg(a))}[/tex],
where [itex]\arg(a)[/itex] is chosen to lie in [itex](-\pi, \pi][/itex]. (Read up on branch cuts.)

From this it follows that [itex](a^b)^c = (a^c)^b[/itex] isn't necessarily true.

Again, this is just one of many possible conventions. The important point is that for a rational number p/q (in lowest terms) and a complex number z, there are q complex numbers which all have equal claim to the name zp/q. And for irrational numbers, there are infinitely many. Complex exponentiaion is simply not a single valued function.

And [itex](a^b)^c = (a^c)^b[/itex] is still true, in the sense that for any [itex]a^b[/itex] and [itex]a^c[/itex] (remember, they're multivalued) there are choices of [itex](a^b)^c[/itex] and [itex](a^c)^b[/itex] so that equality holds, and so that both are equal to some [itex]a^{bc}[/itex]. The problem only comes up when you artificially try to make the function single valued.
 
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  • #6
StatusX said:
That notation isn't standard, and couldn't have been what the OP was using since he wrote [tex]\sqrt{-1}[/tex].

sqrt(-1) is a standard symbol: it means a primitive square root of -1, it does not mean both primitive square roots of -1, and of course there is no algebraic distinction between either.

Technically single valued function is a pleonasm - a function is single valued by definition. There is, perhaps, an inconsistent usage of the word function, here. However it would be better to stay away from these topics lest it confuse the reader. Incidentally, in my experience there are not q complex numbers with an equal claim to being z^{1/q}: that symbol is normally fixed at meaning the principal branch of the q'th root. This is not an artificial definition.
 
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  • #7
I'm not arguing that it's not convenient for many purposes to use a principal branch for multivalued functions, I'm just saying that the issue in the OPs question is more closely concerned with the multi-valuedness of the functions, not with the properties of a specific branch.
 
  • #8
L Harringer: I know that the error has to be in this step:
[itex]\left(\sqrt{-1}\right)^{2}=\sqrt{(-1)^{2}}[/itex]

(works fine for positive numbers)
and I am pretty sure it has something to do with principle roots.
However, I just can't figure out a clean argument for what exactly is wrong.

I heard one reason when taking algebra: the symbol "i" was invented to prevent such a mistake as:

[tex]\sqrt{-1}\sqrt{-1} =\sqrt{1}[/tex]

I can't vouch for that, but take it for what its worth.
 
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  • #9
robert Ihnot said:
L Harringer: I know that the error has to be in this step:
[itex]\left(\sqrt{-1}\right)^{2}=\sqrt{(-1)^{2}}[/itex]

(works fine for positive numbers)
That's where the error is. The product rule for surds does not work if both surds are negative square roots.
 
  • #10

1. What are imaginary numbers?

Imaginary numbers are numbers that can be written in the form of a+bi, where a and b are real numbers and i is the imaginary unit, defined as the square root of -1.

2. How is i=1 when it is defined as the square root of -1?

The notation i=1 is a common shorthand used in mathematics to represent the imaginary unit. It does not mean that i is equal to 1, but rather that i is a number that when squared, results in -1.

3. What is the purpose of imaginary numbers?

Imaginary numbers were initially thought to have no practical applications, but they have since been found to be incredibly useful in fields such as electrical engineering, quantum mechanics, and signal processing.

4. Can imaginary numbers be graphed on a number line?

No, imaginary numbers cannot be graphed on a traditional number line. Instead, they are represented on a complex plane, where the real numbers are plotted on the horizontal axis and the imaginary numbers are plotted on the vertical axis.

5. Are there any real-life examples of imaginary numbers?

Yes, imaginary numbers can be used to represent physical quantities such as alternating current in electrical circuits and the amplitude of a sound wave in audio engineering. They are also used in cryptography and in the study of fluid dynamics.

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