Math Is Fun Forum

  Discussion about math, puzzles, games and fun.   Useful symbols: ÷ × ½ √ ∞ ≠ ≤ ≥ ≈ ⇒ ± ∈ Δ θ ∴ ∑ ∫ • π ƒ -¹ ² ³ °

You are not logged in.

#1 2009-08-16 09:22:10

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

Offline

#2 2009-08-16 09:33:18

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)


In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

Offline

#3 2009-08-16 09:37:01

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.


Why did the vector cross the road?
It wanted to be normal.

Offline

#4 2009-08-16 10:04:31

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.

Offline

#5 2009-08-18 00:32:53

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".


"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..."

Offline

Board footer

Powered by FluxBB