General Discussion > Math and Science

Seemingly Complex Numbers Paradox

(1/1)

Sorunome:
So, in uni with a few other students we had a seemingly paradox with complex numbers, but I think we solved it, wanted to share it with you guys anyways:
##i = e^{i\frac{\pi}{2}} = e^{\frac{1}{4}(2\pi i)} = (e^{2\pi i})^{\frac{1}{4}} = 1^{\frac{1}{4}} = 1##MathJax.Hub.Queue(["Typeset", MathJax.Hub, document.getElementById("bbclatex6630b77eba0c8")]);console.log("Queued!");
On first and second glance this seems to be math-defying as ##i##MathJax.Hub.Queue(["Typeset", MathJax.Hub, document.getElementById("bbclatex6630b77eba0db")]);console.log("Queued!"); is clearly not equal to ##1##MathJax.Hub.Queue(["Typeset", MathJax.Hub, document.getElementById("bbclatex6630b77eba0e8")]);console.log("Queued!");. Below is the solution to this paradox we came up with.
Spoiler For Solution we came up with: As ##i##MathJax.Hub.Queue(["Typeset", MathJax.Hub, document.getElementById("bbclatex6630b77eba109")]);console.log("Queued!"); clearly does not equal to ##1##MathJax.Hub.Queue(["Typeset", MathJax.Hub, document.getElementById("bbclatex6630b77eba117")]);console.log("Queued!"); there must be an issue here. We believe the issue to lie within the ##1^{\frac{1}{4}}##MathJax.Hub.Queue(["Typeset", MathJax.Hub, document.getElementById("bbclatex6630b77eba125")]);console.log("Queued!");. As this is taking a root the answer is not unique, but there are actually multiple answers to ##1^{\frac{1}{4}}##MathJax.Hub.Queue(["Typeset", MathJax.Hub, document.getElementById("bbclatex6630b77eba132")]);console.log("Queued!");.
Let me re-format that a bit to make it clear:
##
1^{\frac{1}{4}} = x \\
\sqrt[4]{1} = x \\
1 = x^{4}
##MathJax.Hub.Queue(["Typeset", MathJax.Hub, document.getElementById("bbclatex6630b77eba13f")]);console.log("Queued!");

From this point on it is quite obvious, there are multiple solutions to ##x##MathJax.Hub.Queue(["Typeset", MathJax.Hub, document.getElementById("bbclatex6630b77eba14c")]);console.log("Queued!");, that is ##1##MathJax.Hub.Queue(["Typeset", MathJax.Hub, document.getElementById("bbclatex6630b77eba158")]);console.log("Queued!");, ##-1##MathJax.Hub.Queue(["Typeset", MathJax.Hub, document.getElementById("bbclatex6630b77eba165")]);console.log("Queued!"); and also ##i##MathJax.Hub.Queue(["Typeset", MathJax.Hub, document.getElementById("bbclatex6630b77eba174")]);console.log("Queued!"); and ##-i##MathJax.Hub.Queue(["Typeset", MathJax.Hub, document.getElementById("bbclatex6630b77eba182")]);console.log("Queued!");.
So we have to say:
##
1^{\frac{1}{4}} = \{-1, 1, -i, i\}
##MathJax.Hub.Queue(["Typeset", MathJax.Hub, document.getElementById("bbclatex6630b77eba190")]);console.log("Queued!");
So now clearly our real solution, ##i##MathJax.Hub.Queue(["Typeset", MathJax.Hub, document.getElementById("bbclatex6630b77eba1a0")]);console.log("Queued!");, is present!
So yeah, just thought it would be fun to share :P

I am nowhere near xedas math skills, please forgive me

Matrefeytontias:
Nice, it got me :P

Legimet:
Actually, the reason it doesn't work is because 1/4 isn't an integer. If a, b are real and c is an integer, then

##e^{c(a+bi)}=e^{ac}\cdot e^{bci}=e^{ac}(\cos cb + i\sin cb)=(e^a)^c(\cos b + i\sin b)^c=(e^{a+bi})^c##MathJax.Hub.Queue(["Typeset", MathJax.Hub, document.getElementById("bbclatex6630b77ebab45")]);console.log("Queued!");

But note that we used De Moivre's law, which only holds for integers.

Navigation

[0] Message Index