My Math Forum 1=2 or 0/x NOT = 0

 Algebra Pre-Algebra and Basic Algebra Math Forum

 May 23rd, 2018, 03:00 AM #1 Newbie   Joined: May 2018 From: United Kingdom Posts: 1 Thanks: 1 1=2 or 0/x NOT = 0 I have either proved that 1=2, 0/x for a value of x not equal to 0... Or I have made a mistake. I started with e^i*pi=-1 I then squared both sides to get: e^2*i*pi=1 Taking the log of both sides allows me to remove the power and multiply by the log for: 2*i*pi*log(e)=log(1) S1) 2*i*pi*log(e)=0 To extract the value of i, sqrt(-1), I divide both sides by 2*pi*log(e) i=0/2*i*pi*log(e) 0/x=0 i=0 sqrt(-1)=0 Square both sides for: -1=0 +2 both sides: 1=2 Help... Thanks from v8archie
 May 23rd, 2018, 03:17 AM #2 Global Moderator     Joined: Oct 2008 From: London, Ontario, Canada - The Forest City Posts: 7,879 Thanks: 1087 Math Focus: Elementary mathematics and beyond Your mistake lies in applying logarithmic identities which apply to real numbers to a complex number. Not all of these identities are necessarily true for complex numbers. See here for more information.
 May 23rd, 2018, 03:21 AM #3 Math Team   Joined: Jan 2015 From: Alabama Posts: 3,261 Thanks: 894 Your error is when you take the logarithm of a complex number. A complex number can be written in polar form as $\displaystyle re^{i\theta}= re^{i(\theta+ 2\pi n)}$ since "$\displaystyle e^{i\theta}= \cos(\theta)+ i\sin(\theta)$" is periodic with period $\displaystyle 2\pi$. Taking the logarithm, $\displaystyle \ln(re^{i(\theta+ 2n\pi)}= \ln(r)+ i(\theta+ 2n\pi)$ for n any integer. That is, ln is multivalued. Last edited by skipjack; May 23rd, 2018 at 05:45 AM.
May 23rd, 2018, 03:31 AM   #4
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Quote:
 Originally Posted by keylewer I have either proved that 1=2, 0/x for a value of x not equal to 0... Or I have made a mistake.
This is great, thank you. Too many are unwilling to admit that they might have erred.

 May 23rd, 2018, 01:37 PM #5 Global Moderator   Joined: May 2007 Posts: 6,607 Thanks: 616 A simple way to look at it is $e^{2\pi i}=1=e^0$, but $2\pi i\ne 0$. Thanks from JeffM1

 Tags 0 or x, broken, e^i*pi, oops

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