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For real x, ix is purely imaginary and hence it doesn't make sense for it to tend to a value / object which is an element of the extended real number system.
A more accepted / technically correct terminology would be,
Prasun here: Nope, i4π=(i4)π=1π=1 is not a valid step as we're dealing with complex numbers so we can't split the exponents as we please. You should do this instead: let x=i4π then ln(x)=4πln(i)=4π⋅ln(eiπ/2)=2π2i or x=ei⋅2π2=cos(2π2)+isin(2π2)≈0.6297+0.7769i.
@Pi Han Goh
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The main thing that I'd have said (I forgot to reply earlier) is that (ab)c=(ac)b=abc doesn't necessarily hold when some of a,b,c are non-reals.
I had seen a post on MSE regarding this once where an answer explained why the "general rules of exponentiation" doesn't hold when complex numbers are thrown into the mix. I was trying to find that post before I could reply him but it seems you already cleared it up. Thanks!
Yeah, I had given it much thought earlier. I was also thinking about complex infinity when I first posted that comment. But complex infinity is mostly used to denote a complex value with infinite magnitude but undefined argument.
But in this case, the argument is defined and is 2π (considering the limit is to positive infinity).
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This discussion board is a place to discuss our Daily Challenges and the math and science related to those challenges. Explanations are more than just a solution — they should explain the steps and thinking strategies that you used to obtain the solution. Comments should further the discussion of math and science.
When posting on Brilliant:
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[example link](https://brilliant.org)> This is a quote# I indented these lines # 4 spaces, and now they show # up as a code block. print "hello world"\(...\)or\[...\]to ensure proper formatting.2 \times 32^{34}a_{i-1}\frac{2}{3}\sqrt{2}\sum_{i=1}^3\sin \theta\boxed{123}Comments
For real x, ix is purely imaginary and hence it doesn't make sense for it to tend to a value / object which is an element of the extended real number system.
A more accepted / technically correct terminology would be,
x→∞lim(ix)=i∞
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hello prasun , i wanted ask u one more thing is i4π=1
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Prasun here: Nope, i4π=(i4)π=1π=1 is not a valid step as we're dealing with complex numbers so we can't split the exponents as we please. You should do this instead: let x=i4π then ln(x)=4πln(i)=4π⋅ln(eiπ/2)=2π2i or x=ei⋅2π2=cos(2π2)+isin(2π2)≈0.6297+0.7769i.
See this note to spot such pitfalls.
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The main thing that I'd have said (I forgot to reply earlier) is that (ab)c=(ac)b=abc doesn't necessarily hold when some of a,b,c are non-reals.
I had seen a post on MSE regarding this once where an answer explained why the "general rules of exponentiation" doesn't hold when complex numbers are thrown into the mix. I was trying to find that post before I could reply him but it seems you already cleared it up. Thanks!
Also, I'm wondering when did I get to Malaysia?!
No. Apply de-moivre's theorem you will get to know.
What about ∞~ ? reference .
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Yeah, I had given it much thought earlier. I was also thinking about complex infinity when I first posted that comment. But complex infinity is mostly used to denote a complex value with infinite magnitude but undefined argument.
But in this case, the argument is defined and is 2π (considering the limit is to positive infinity).
W|A verification.
Hence, the term "complex infinity" wouldn't be applicable here, in my opinion.