# Algebra in the complex domain

**URL:** <https://boards.straightdope.com/t/algebra-in-the-complex-domain/203523>\
**Category:** Factual Questions\
**Created:** [September 24, 2003, 12:19pm UTC](https://boards.straightdope.com/t/algebra-in-the-complex-domain/203523 "2003-09-24T12:19:32Z")\
**Posts on this page:** 20\
**Page:** 1

<div class="post-metadata">

**Author:** ![MC\_Master\_of\_Ceremonies](https://avatars.discourse-cdn.com/v4/letter/m/bb73d2/32.png) [@MC\_Master\_of\_Ceremonies](https://boards.straightdope.com/u/MC_Master_of_Ceremonies)\
**Post date:** [September 24, 2003, 12:19pm UTC](https://boards.straightdope.com/t/algebra-in-the-complex-domain/203523/1 "2003-09-24T12:19:32Z")

</div>

using x = π in Euler’s formula:

e[sup]xi[/sup] = cos(x) + isin(x)

e[sup]πi[/sup] = -1

square both sides:

e[sup]2πi[/sup] = 1

putting both sides to the power of ln(5)/2πi  
1[sup]ln(5)/2πi[/sup] = (e[sup]2πi[/sup])[sup]ln(5)/2πi[/sup] = e[sup]ln(5)[/sup] = 5

Now this looks incorrect as I have a power of 1 that is not equal to 1 (though it is a complex power).

Now I’ve been told that (a[sup]b[/sup])[sup]c[/sup] = a[sup]bc[/sup] doesn’t hold in the complex domain , but the other well-known identity:

i[sup]i[/sup] = e[sup]-π/2[/sup]

can seemingly be derived from the x = π case of Euler’s formula using this rule.

Of course using simlair reasoning you get this silly result:

e[sup]2πi[/sup] = 1 =\>

2πi = ln(1) = 0 =\>

-4π[sup]2[/sup] = 0

but all this suggests to me is that ln(x) is multi-valued.

Anyway basically I want to know whether:

1[sup]ln(5)/2πi[/sup] = 5

is a correct or incorrect equality.

---

<div class="post-metadata">

**Author:** ![The\_Great\_Unwashed](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/the_great_unwashed/32/4791_2.png) [@The\_Great\_Unwashed](https://boards.straightdope.com/u/The_Great_Unwashed)\
**Post date:** [September 24, 2003, 12:23pm UTC](https://boards.straightdope.com/t/algebra-in-the-complex-domain/203523/2 "2003-09-24T12:23:05Z")

</div>

“Now this looks incorrect as I have a power of 1 that is not equal to 1 (though it is a complex power).”

Why should that be surprising, (for instance) 1 has three cube roots, only one of them (1) is wholly real.

(PS: I didn’t check your calculations, just wanted to observe that I saw no surprises).

---

<div class="post-metadata">

**Author:** ![MC\_Master\_of\_Ceremonies](https://avatars.discourse-cdn.com/v4/letter/m/bb73d2/32.png) [@MC\_Master\_of\_Ceremonies](https://boards.straightdope.com/u/MC_Master_of_Ceremonies)\
**Post date:** [September 24, 2003, 12:27pm UTC](https://boards.straightdope.com/t/algebra-in-the-complex-domain/203523/3 "2003-09-24T12:27:41Z")

</div>

Yes, I suppose, but it’s a real answer, but then again 1 has two square-roots one of which isn’t one and is real.

I suppose I should of rephrased that into it looks incoorect as it has an answer that isn’t 1, -1, i or -i.

---

<div class="post-metadata">

**Author:** ![Angua](https://avatars.discourse-cdn.com/v4/letter/a/8baadc/32.png) [@Angua](https://boards.straightdope.com/u/Angua)\
**Post date:** [September 24, 2003, 12:44pm UTC](https://boards.straightdope.com/t/algebra-in-the-complex-domain/203523/4 "2003-09-24T12:44:20Z")

</div>

> [@](#):
>
> \*Originally posted by MC Master of Ceremonies \*
> 
> but all this suggests to me is that ln(x) is multi-valued.  
> \*\*

If x is complex, as in this case, then yes, ln(x) is a multi-valued function.

If one has e[sup]lnz[/sup]=z, then because of the periodicity of the function e[sup]lnz + 2n(pi)i[/sup], the logarithm function in the complex plane is multi-valued, which explains your strange results - ln is not uniquely defined.

---

<div class="post-metadata">

**Author:** ![The\_Great\_Unwashed](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/the_great_unwashed/32/4791_2.png) [@The\_Great\_Unwashed](https://boards.straightdope.com/u/The_Great_Unwashed)\
**Post date:** [September 24, 2003, 12:54pm UTC](https://boards.straightdope.com/t/algebra-in-the-complex-domain/203523/5 "2003-09-24T12:54:32Z")

</div>

> [@](#):
>
> \*Originally posted by MC Master of Ceremonies \*  
> \*\*Yes, I suppose, but it’s a real answer, but then again 1 has two square-roots one of which isn’t one and is real.
> 
> I suppose I should of rephrased that into it looks incoorect as it has an answer that isn’t 1, -1, i or -i. \*\*

1[sup]1/3[/sup] has three cubed root, namely,  
1  
-cos60 + _i_sin60  
and  
-cos60 - _i_sin60

Only one of which (1) is in the set {1, -1, _i_, -_i_}, I’m missing your point, obviously.

---

<div class="post-metadata">

**Author:** ![Orbifold](https://avatars.discourse-cdn.com/v4/letter/o/779978/32.png) [@Orbifold](https://boards.straightdope.com/u/Orbifold)\
**Post date:** [September 24, 2003, 2:00pm UTC](https://boards.straightdope.com/t/algebra-in-the-complex-domain/203523/6 "2003-09-24T14:00:34Z")

</div>

> [@](#):
>
> \*Originally posted by MC Master of Ceremonies \*  
> \*\*1[sup]ln(5)/2πi[/sup] = 5
> 
> is a correct or incorrect equality. \*\*

It is, believe it or not, a correct equality.

The reason is that exponentiation in the complex plane is defined as follows: a[sup]b[/sup] = exp(b ln(a)). But as you’ve noted, ln() is a multi-valued function when applied to complex numbers. Therefore, exponentiation is _also_ multi-valued in the complex domain.

Now consider 1[sup]ln(5)/2πi[/sup]. By definition, this must equal exp( (ln(5)/2πi) ln(1)). If we let ln(1)=0, then we get exp(0) which equals 1, which we expect. But since we’re considering complex numbers, ln(1) also equals 0+2πi, in which case 1[sup]ln(5)/2πi[/sup] = exp(ln(5)) = 5. In the same fashion, we can show that 1[sup]ln(5)/2πi[/sup] = exp(2 ln(5))=25, and so on.

If we restrict ourselves to the “primary branch” of the logarithm function…that is, if we arbitrarily require that the imaginary part of ln(z) always lies between -πi and πi…then some of these problems go away. But that in turn causes other problems, like breaking the rule that ln(ab) = ln(a)+ln(b). In short, complex numbers are weird.

---

<div class="post-metadata">

**Author:** ![MC\_Master\_of\_Ceremonies](https://avatars.discourse-cdn.com/v4/letter/m/bb73d2/32.png) [@MC\_Master\_of\_Ceremonies](https://boards.straightdope.com/u/MC_Master_of_Ceremonies)\
**Post date:** [September 24, 2003, 3:08pm UTC](https://boards.straightdope.com/t/algebra-in-the-complex-domain/203523/7 "2003-09-24T15:08:15Z")

</div>

Thanks **Orbifold** while your here can you solve another seemingly paradoxiacal result in a simlair vein:

e[sup]-π/2[/sup] = i[sup]i[/sup] =\>

e[sup]-πi/2[/sup] = i[sup]-1[/sup] =\>

e[sup]πi/4[/sup] = i[sup]1/2[/sup] =

(e[sup]πi[/sup])[sup]1/4[/sup] = i[sup]1/2[/sup] =

1[sup]1/4[/sup] = i[sup]1/2[/sup]

= 1, -1, i, -i

---

<div class="post-metadata">

**Author:** ![Achernar](https://avatars.discourse-cdn.com/v4/letter/a/e274bd/32.png) [@Achernar](https://boards.straightdope.com/u/Achernar)\
**Post date:** [September 24, 2003, 3:18pm UTC](https://boards.straightdope.com/t/algebra-in-the-complex-domain/203523/8 "2003-09-24T15:18:47Z")

</div>

You have a small error in your second-to-last step. exp(pi i) = -1, not 1.

---

<div class="post-metadata">

**Author:** ![MC\_Master\_of\_Ceremonies](https://avatars.discourse-cdn.com/v4/letter/m/bb73d2/32.png) [@MC\_Master\_of\_Ceremonies](https://boards.straightdope.com/u/MC_Master_of_Ceremonies)\
**Post date:** [September 24, 2003, 3:51pm UTC](https://boards.straightdope.com/t/algebra-in-the-complex-domain/203523/9 "2003-09-24T15:51:10Z")

</div>

Oops it should of been:

e[sup]-π/2[/sup] = i[sup]i[/sup] =\>

e[sup]-πi/2[/sup] = i[sup]-1[/sup] =\>

e[sup]πi/4[/sup] = i[sup]1/2[/sup] =

(e[sup]2πi[/sup])[sup]1/8[/sup] = i[sup]1/2[/sup] =

1[sup]1/8[/sup] = i[sup]1/2[/sup]

= 1, -1, i, -i

but I see what’s wrong with that anyway now, and all I’ve really proved is that the square root of i is equal to the sqaure root of i.

So th question that I really wanted to ask is: what is the square root of i?

---

<div class="post-metadata">

**Author:** ![The\_Great\_Unwashed](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/the_great_unwashed/32/4791_2.png) [@The\_Great\_Unwashed](https://boards.straightdope.com/u/The_Great_Unwashed)\
**Post date:** [September 24, 2003, 4:09pm UTC](https://boards.straightdope.com/t/algebra-in-the-complex-domain/203523/10 "2003-09-24T16:09:27Z")

</div>

> [@](#):
>
> \*Originally posted by MC Master of Ceremonies \*  
> \*\*So th question that I really wanted to ask is: what is the square root of i? \*\*

By DeMoivre’s Theorem they are  
cos45 + _i_sin45  
and  
-cos45 - _i_sin45

---

<div class="post-metadata">

**Author:** ![Orbifold](https://avatars.discourse-cdn.com/v4/letter/o/779978/32.png) [@Orbifold](https://boards.straightdope.com/u/Orbifold)\
**Post date:** [September 24, 2003, 4:11pm UTC](https://boards.straightdope.com/t/algebra-in-the-complex-domain/203523/11 "2003-09-24T16:11:43Z")

</div>

i has two square roots, (1+i)/sqrt(2) and (-1-i)/sqrt(2). Those are also the two possible numbers you get when evaluating i[sup]1/2[/sup] via the formula exp(ln(i)/2).

---

<div class="post-metadata">

**Author:** ![Enola\_Straight](https://avatars.discourse-cdn.com/v4/letter/e/dec6dc/32.png) [@Enola\_Straight](https://boards.straightdope.com/u/Enola_Straight)\
**Post date:** [September 24, 2003, 4:18pm UTC](https://boards.straightdope.com/t/algebra-in-the-complex-domain/203523/12 "2003-09-24T16:18:46Z")

</div>

If the square root of 1 = 1 and -1,  
maybe the square root of i = i and -i?

WAIT…

i^2=-1  
-i^2= (-1)(i)(-1)(i)=(-1)^2(i)^2=1\*-1=-1

HUMPH!

---

<div class="post-metadata">

**Author:** ![The\_Great\_Unwashed](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/the_great_unwashed/32/4791_2.png) [@The\_Great\_Unwashed](https://boards.straightdope.com/u/The_Great_Unwashed)\
**Post date:** [September 24, 2003, 4:40pm UTC](https://boards.straightdope.com/t/algebra-in-the-complex-domain/203523/13 "2003-09-24T16:40:59Z")

</div>

Generally if you express your complex number in polar terms (so r\<t is complex number of modulus r and angle t)

then the _n_th roots of r\<t are

(r[sup]1/_n_[/sup]) \<(t/_n_ + 360k_n_) where k = 1, 2,… …_n_-1

Which is pretty damned gorgeous (and one of the few things I can still remember about complex numbers).

---

<div class="post-metadata">

**Author:** ![MikeS](https://avatars.discourse-cdn.com/v4/letter/m/919ad9/32.png) [@MikeS](https://boards.straightdope.com/u/MikeS)\
**Post date:** [September 24, 2003, 6:12pm UTC](https://boards.straightdope.com/t/algebra-in-the-complex-domain/203523/14 "2003-09-24T18:12:09Z")

</div>

Psst, **The Great Unwashed**. That’s supposed to be 360k/n, right?

---

<div class="post-metadata">

**Author:** ![The\_Great\_Unwashed](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/the_great_unwashed/32/4791_2.png) [@The\_Great\_Unwashed](https://boards.straightdope.com/u/The_Great_Unwashed)\
**Post date:** [September 24, 2003, 6:49pm UTC](https://boards.straightdope.com/t/algebra-in-the-complex-domain/203523/15 "2003-09-24T18:49:50Z")

</div>

Might be. It could be a new theory of mine (all mine), and it works perfectly well if you don’t care for all the roots. So nyah!

Actually if one of the mods could correct that it’d be appreciated – okay it’ll make **MikeS** look pretty silly, but he’s been here two years and only posted forty-six times (pulls out calculator), that’s less than one post per fortnight! Why do have to pick on mine?

Snaffen-raffen **MikeS**.

---

<div class="post-metadata">

**Author:** ![Achernar](https://avatars.discourse-cdn.com/v4/letter/a/e274bd/32.png) [@Achernar](https://boards.straightdope.com/u/Achernar)\
**Post date:** [September 24, 2003, 10:06pm UTC](https://boards.straightdope.com/t/algebra-in-the-complex-domain/203523/16 "2003-09-24T22:06:14Z")

</div>

You also need k to go from 0 to n-1, or from 1 to n. If you only go from 1 to n-1, you miss one of the roots.

---

<div class="post-metadata">

**Author:** ![The\_Great\_Unwashed](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/the_great_unwashed/32/4791_2.png) [@The\_Great\_Unwashed](https://boards.straightdope.com/u/The_Great_Unwashed)\
**Post date:** [September 24, 2003, 11:37pm UTC](https://boards.straightdope.com/t/algebra-in-the-complex-domain/203523/17 "2003-09-24T23:37:30Z")

</div>

snaffen-raffen **Achernar**

Really, how could I be so right, and yet so wrong?

Right, except, that I multiplied where I should have divided, and that the range of values for k was “out by one”, how else did the Romans mess me up?

---

<div class="post-metadata">

**Author:** ![The\_Great\_Unwashed](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/the_great_unwashed/32/4791_2.png) [@The\_Great\_Unwashed](https://boards.straightdope.com/u/The_Great_Unwashed)\
**Post date:** [September 24, 2003, 11:42pm UTC](https://boards.straightdope.com/t/algebra-in-the-complex-domain/203523/18 "2003-09-24T23:42:55Z")

</div>

snaffen-raffen **Achernar**

Really, how could I be so right, and yet so wrong?

Right, except, that I multiplied where I should have divided, and that the range of values for k was “out by one”, how else did the Romans mess me up?

---

<div class="post-metadata">

**Author:** ![The\_Great\_Unwashed](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/the_great_unwashed/32/4791_2.png) [@The\_Great\_Unwashed](https://boards.straightdope.com/u/The_Great_Unwashed)\
**Post date:** [September 24, 2003, 11:49pm UTC](https://boards.straightdope.com/t/algebra-in-the-complex-domain/203523/19 "2003-09-24T23:49:31Z")

</div>

Oh yes, they made me double post, but besides that…

---

<div class="post-metadata">

**Author:** ![MC\_Master\_of\_Ceremonies](https://avatars.discourse-cdn.com/v4/letter/m/bb73d2/32.png) [@MC\_Master\_of\_Ceremonies](https://boards.straightdope.com/u/MC_Master_of_Ceremonies)\
**Post date:** [December 1, 2003, 9:59pm UTC](https://boards.straightdope.com/t/algebra-in-the-complex-domain/203523/20 "2003-12-01T21:59:01Z")

</div>

Sorry to resurect this thread, but something is bothering me about this, if

1[sup]ln(5)/2π[/sup] = 5

then

5[sup]2π/ln(5)[/sup] = 1

but surely if

x[sup]y[/sup] = 1

then

|x| = 1

I must of gone wrong somewhere, but where?

[Next page](https://boards.straightdope.com/t/algebra-in-the-complex-domain/203523.md?page=2)