Raising things to i power, I came across 4.8104773809655


by SteveRives
Tags: power, raising, things
SteveRives
SteveRives is offline
#1
Jun10-05, 10:27 AM
P: 57
I was working on the general meaning of taking things to the i power -- I was pondering the meaning of

e^pi*i = -1

The proof for this thing is well established, I was musing about meaning. An obvious question was:

What to the i power goes to zero?

As I was hunting for that number (on my TI-83 while driving to work -- and it is very hard to press [2nd] i on that thing with one hand), I came across this number:

4.8104773809655

Namely:

4.8104773809655^i => i

...at least on my TI-83. 4.81047738096535^i on the google calculator.

What is this number, and who has done work on it?

Regards,

Steve Rives
Phys.Org News Partner Science news on Phys.org
Simplicity is key to co-operative robots
Chemical vapor deposition used to grow atomic layer materials on top of each other
Earliest ancestor of land herbivores discovered
Zurtex
Zurtex is offline
#2
Jun10-05, 10:51 AM
Sci Advisor
HW Helper
P: 1,123
For:

[tex]x^i = i[/tex]

The solutions for x are:

[tex]e^{\frac{\pi}{2} + n\pi} \quad \forall n \in \mathbb{Z}[/tex]

So if you hadn't already guessed:

[tex]e^{\frac{\pi}{2}} \approx 4.8104773809653516554730356667 \ldots[/tex]

As for the equation:

[tex]i^x = 0[/tex]

Well there exists no complex solution for x. In fact the only time:

[tex]a^b = 0[/tex]

For a and b in complex numbers is when [itex]a=0[/itex] and [itex]b \neq 0[/itex].
SteveRives
SteveRives is offline
#3
Jun10-05, 10:54 AM
P: 57
Of coruse! Only, I found it the hard way just now:

ln(4.81047738...) is 1.5707963

And that's pi/2

mathwonk
mathwonk is offline
#4
Jun10-05, 07:12 PM
Sci Advisor
HW Helper
mathwonk's Avatar
P: 9,421

Raising things to i power, I came across 4.8104773809655


he did not ask when i^x = 0, he asked (what)^i -->0?

so we might look at a^i = e^(i ln(a)) = cos(ln(a)) + i sin(ln(a)).

but again it is clear that this number has absolute value 1, so cannot approach zero, at least not for real a.

now for complex a, we just need to solve for when e^z goes to 0.

but e^(-n)-->0 for example, so therefore e^i(in) -->0 too. so x = in satisfies

e^(ix) goes to zero as n goes to infinity, i.e. as x goes to i.infinity.


Register to reply

Related Discussions
fundamental things, emergent things General Discussion 47
Raising a Bucket out of a Well Introductory Physics Homework 6
Raising Air Pressure? Classical Physics 5
Are all things in the universe caused by things that cause things like themselves? General Discussion 10
raising a ladder Introductory Physics Homework 13