Fractional Exponent of Negative Numbers

[Zalajbeg]

I have been thinking on this topic for a while.

As I have seen on various sites:
1) a^(b/c) means "c"th root of "a^b".
2) Also an even root of a negative number does not exist in real numbers.

Then I want to investigate this formula:

a=(-4)^(1/2)

Mustn't it be equal to b=(-4)^(2/4)

However if we calculate the power first;

a=square root of [(-4)^1]= Not a real number
b=4th root of[(-4)^2]= 2

What do I miss? I have some confussion about decimal exponents but I hope the answer of this question will solve it.

 

[Millennial]

That specific law of exponents does not work in negative numbers.
A way to understand this is that your method entails $\sqrt{-a}=(-a)^{2/4}=a^{1/4}$, but clearly $a^{1/4}\cdot a^{1/4}=\sqrt{a}\neq \sqrt{-a}$.
Your mistake is that you are applying the law $(ab)^{c}=a^c\cdot b^c$, which isn't valid for negative a or negative b.

 

[Zalajbeg]

Thanks for your answer.

But then, how to handle fractional exponent problems:

For example how to handle this one;

(-2)^(4/5)

 

[Millennial]

The number -2 can be represented in polar form as $2e^{i\pi}$, and hence the problem of exponentiation reduces to evaluating $2^{4/5}e^{4i\pi/5}=2^{4/5}(\cos(4\pi/5)+i\sin(4\pi/5))$.

Euler's identity is often employed to give the definition of complex exponentiation.

 

[Diffy]

@Millennial

I really like your posts, they are succinct (reducing the chance of confusion) and very accurate.

Thanks,

 

[Zalajbeg]

The number -2 can be represented in polar form as $2e^{i\pi}$, and hence the problem of exponentiation reduces to evaluating $2^{4/5}e^{4i\pi/5}=2^{4/5}(\cos(4\pi/5)+i\sin(4\pi/5))$.

Euler's identity is often employed to give the definition of complex exponentiation.

Thanks for your answer.

So, If I want to determine if an negative base has a real root or not, then should I approach it by expressing it as an complex number?

Couldn't I say if, for example, -2^1.0954 has a real root or not easily?

 

[Diffy]

Thanks for your answer.

So, If I want to determine if an negative base has a real root or not, then should I approach it by expressing it as an complex number?

Couldn't I say if, for example, -2^1.0954 has a real root or not easily?

Well sure you could. But you are going to be able to say that based on Millennial's answer.

Based on his formula, n^(x) where n < 0 will have a real root iff sin(x*pi) = 0. Otherwise it will be a complex number.

 

[uart]

Couldn't I say if, for example, -2^1.0954 has a real root or not easily?

Before answering that, let me just add some more info to the previous question regarding (-2)^(4/5).

Using the complex definitions of exp and log we can write it as,
$$e^{0.8 \log(-2)} = e^{0.8 (\log(2) + i (1 + 2k) \pi))}$$

So we see that if take this expression over the complex numbers, then it's multi-valued. Basically we get a bunch of different answers, for k = 0, 1, 2, etc. It's easy to see however, that in this case they repeat after k=4, so there are 5 unique values. We also note that for k=2 we get a purely real solution (since 0.8 * i 5 pi = i 4 pi).

In general, say we have (-x)^(p/q) where "x" is a positive real and fraction p/q is reduced to it's simplest form. Then we can write it as

$$e^{\frac{p}{q} \log(-x)} = e^{\frac{p}{q} (\log(x) + i (1 + 2k) \pi))}$$

Here we can see that if "q" is odd then there must be some value "k" for which $1+ 2k = q$, and the corresponding value will be real. Conversely we can show that for "q" even there will be no real valued solutions.

 

[Millennial]

Complex exponentiation is not multi-valued, complex logarithm is. It is just that complex exponentiation gives the same answer for different numbers, to be precise, $p^q=e^{p\log(q)}$ is equal to $\left(p+\frac{2\pi in}{\log(q)}\right)^{q}=e^{p\log(q)+2\pi in}$ for $n\in \mathbb{Z}$ through the periodicity of trigonometric functions. However, since complex exponention is not a bijection, complex logarithm is multivalued, if $\log(z)=w$ holds, then for $n\in \mathbb{Z}$, $\log(z)=w+2\pi in$ also holds.

Just wanted to fix that error.

Also, if you don't know it, complex logarithm on the principal branch is defined by $\log(z)=\log|z|+i\text{arg}(z)$. It is easily derivable from the polar form of a complex number. Of course, through my last argument above, a general form that covers not just the principal branch but all branches is given by $\log(z)=\log|z|+i\text{arg}(z)+2\pi in$ for $n\in \mathbb{Z}$.

@Millennial

I really like your posts, they are succinct (reducing the chance of confusion) and very accurate.

Thanks,

I'd like to thank you for the complement :)

 

[Zalajbeg]

Firstly thanks all for your wise answers.

It is clearer to me now.

In general, say we have (-x)^(p/q) where "x" is a positive real and fraction p/q is reduced to it's simplest form. Then we can write it as

$$e^{\frac{p}{q} \log(-x)} = e^{\frac{p}{q} (\log(x) + i (1 + 2k) \pi))}$$

Here we can see that if "q" is odd then there must be some value "k" for which $1+ 2k = q$, and the corresponding value will be real. Conversely we can show that for "q" even there will be no real valued solutions.

I was thinking about "the simplest form". I have got the logic now. But as I am not good at complex numbers it will take a while to understand why it is required to be simpliest form. I think I can find it after some study.

Thank you all again for your great help.

 

[Dickfore]

In complex numbers, the rule:
$$(z_1 \, z_2)^{\alpha} \stackrel{?}{=} z^\alpha_1 \, z^\alpha_2$$
does not hold in general when α is not an integer. The reason behind this is because non-integer powers in complex numbers are defined through the principal branch of the complex logarithm (Log):
$$z^\alpha \equiv \exp \left( \alpha \, \mathrm{Log} (z) \right)$$
and, for this branch, the product rule:
$$\mathrm{Log}(z_1 \, z_2) \stackrel{?}{=} \mathrm{Log}(z_1) + \mathrm{Log}(z_2)$$
does not hold in general. This is because:
$$\mathrm{Log} (z) \equiv \ln \vert z \vert + i \, \mathrm{Arg} (z)\,, \ -\pi < \mathrm{Arg} (z) \le \pi$$
and the principal arguments (Arg) do not add:
$$\mathrm{Arg}(z_1 \, z_2) \neq \mathrm{Arg}(z_1) + \mathrm{Arg}(z_2)\,,$$
since they are restricted to the above stated interval.

For example:
$$\vert -4 \vert = 4\,, \ \mathrm{Arg}(-4) = \pi$$
$$(-4)^{\frac{1}{2}} = \exp \left(\frac{1}{2} \, ( \ln 4 + i \, \pi ) \right) = e^{\ln 2} \, e^{i \, \frac{\pi}{2}} = 2 \, i$$
On the other hand, $(-4)^2 = 16$, and:
$$\vert 16 \vert = 16\,, \ \mathrm{Arg}(16) = 0$$
$$(16)^{\frac{1}{4}} = \exp \left(\frac{1}{4} \, ( \ln 16 ) \right)= e^{\ln 2} = 2$$

 

[uart]

Firstly thanks all for your wise answers.

It is clearer to me now.
I was thinking about "the simplest form". I have got the logic now. But as I am not good at complex numbers it will take a while to understand why it is required to be simpliest form. I think I can find it after some study.

Thank you all again for your great help.

Looking at Millenial's and Dickfore's reply I realize I was wrong to say is was multi-valued, since the complex power is defined using only the principle branch of log. That is, using only k=0 in the equation I wrote.

Think of this as being a bit like the distinction between the (multiple) solutions to the equation $x^2 = 4$ and and the (single) value of $\sqrt{4}$. To keep sqrt as a single valued function it is usually defined as only the positive value and we use an explicit plus or minus ($\pm$) to write all the solutions to the given equation.

In the same manner we can see a close relation to the value of $(-2)^\frac{p}{q}$ and the solutions to the equation $x^q - (-2)^p = 0$. The equation does have "q" solutions, but $(-2)^\frac{p}{q}$ is defined as just the one of these values corresponding to the logarithm principle branch.

Just wanted to fix that error.

No problems, thanks for that. I hope that the above also helps to clarify the situation to the OP.