Math Is Fun Forum

random_fruit

Member

Registered: 2008-12-25

Posts: 39

What do we get from log (-1)?

So, the square root of minus 1 is called i and gives us imaginary then complex numbers.  Another function that is not defined for negative numbers (or zero, for that matter) is logarithm.  Has anyone ever used log(-1) or perhaps ln(-1) to extend maths?  If so, does anyone have a URL so I can read about it? (and then get confused!)

Thanks,

random_fruit

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

Re: What do we get from log (-1)?

Hi random_fruit

Ln(z) = ln(|z|) + i*Arg(z)

I thought it was equal to π i

A friend says this is a better answer.

But I don't agree.

Last edited by bobbym (2009-08-16 11:00:14)

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In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
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mathsyperson

Moderator

Registered: 2005-06-22

Posts: 4,900

Re: What do we get from log (-1)?

Funny you should say that, because you get logs of negative numbers by involving complex numbers.
Within the reals, log is the inverse of e.
ie. log(a) = b <--> b = e^a.

Complex logs work in much the same way - you find the complex log of something by saying what power of e gives that number.
Fairly famously, e^iπ = -1, so your example is fairly easy: log(-1) = iπ.
In general, you'd put the number in modulus-argument form and use that:

e^(x+iy) = e^x * (cos y + isin y), where x and y are real.

Since e^z = e^z+2πi = e^z+4πi = ..., all complex logarithms are multi-valued.
To get around this, people sometimes refer to the principal logarithm, which is defined to have imaginary part between 0 ≤ y < 2π.

Every complex number has a complex logarithm, except for 0.

This Wikipedia article goes into more detail about it, although it might be a bit confusing.

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random_fruit

Member

Registered: 2008-12-25

Posts: 39

Re: What do we get from log (-1)?

Thanks for the answers - so it seems if one tried to define some "new maths" involving ln(-1) it would turn out that sqrt(-1) could be defined using the "new" value ln(-1). 

Yes, I did know that e^iπ = -1

Thanks for both of the replies, and I'll go away and have a ponder on this.

Ricky

Moderator

Registered: 2005-12-04

Posts: 3,791

Re: What do we get from log (-1)?

Thanks for the answers - so it seems if one tried to define some "new maths" involving ln(-1) it would turn out that sqrt(-1) could be defined using the "new" value ln(-1).

That is precisely correct.  In fact, in complex analysis we can define a square root if and only if we can define a log.

Where l(z) is a branch of the log.  Intuitively

But there are problems with just saying "z^1/2".

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"In the real world, this would be a problem.  But in mathematics, we can just define a place where this problem doesn't exist.  So we'll go ahead and do that now..."