help on basic properties of complex number

**rcl333** · August 31st 2010, 11:25 AM

I got confused on the operation and representation of complex numbers. It is
quite basic, and please help me understand. My question is as follows:

(1) Since 1=exp(2*pi*i), it seems that, for any real number a, we may have

exp(a*i)=exp((2*pi*i)*(a/(2*pi)))=(exp(2*pi*i))^(a/(2*pi))
= 1^(a/(2*pi))
=1

(2) For any complex number z, since 1+ exp(2*pi*i), it seems that

z^(1/4)=(1*z)^(1/4)=((exp(2*pi*i))^(1/4)*z^(1/4)
= (exp((pi/2)*i))*z^(1/4)
= i*z^(1/4)
by the same token, since 1=exp(-2*pi*i), it seems that

z*(1/4)=(-i)*z^(1/4)

All those are confusingly wrong, and I wish to understand why. Many thanks for you kindness to help.

**Defunkt** · August 31st 2010, 01:35 PM

$a^{b c} = (a^b)^c$ is not generally true for complex numbers.

**awkward** · August 31st 2010, 01:36 PM

Originally Posted by rcl333

I got confused on the operation and representation of complex numbers. It is
quite basic, and please help me understand. My question is as follows:

(1) Since 1=exp(2*pi*i), it seems that, for any real number a, we may have

exp(a*i)=exp((2*pi*i)*(a/(2*pi)))=(exp(2*pi*i))^(a/(2*pi))
= 1^(a/(2*pi))
=1

(2) For any complex number z, since 1+ exp(2*pi*i), it seems that

z^(1/4)=(1*z)^(1/4)=((exp(2*pi*i))^(1/4)*z^(1/4)
= (exp((pi/2)*i))*z^(1/4)
= i*z^(1/4)
by the same token, since 1=exp(-2*pi*i), it seems that

z*(1/4)=(-i)*z^(1/4)

All those are confusingly wrong, and I wish to understand why. Many thanks for you kindness to help.

Hi rcl333,

You confusion is due to the fact that some of the identities we take for granted about exponentiation of real numbers do not hold for complex numbers. In particular, it is not true that $(e^a)^b = e^{ab}$ . Try reading the section on powers of complex numbers on this web page,

Exponentiation - Wikipedia, the free encyclopedia

especially the sub-section titled "Failure of Power and Logarithm Identities". Then if you are still confused, let us know.

**rcl333** · August 31st 2010, 06:42 PM

I am extremely grateful for your kind and clear explanation with the Wikipedia reference. Now I understand that this rule does not always work for complex numbers. Please help me understand also when can we use this rule? and when can not?, because in Wikipedia site with the subsection "roots of the unity", it used the expression that (exp(2*pi*i*(1/n)))^k=exp(2*pi*i*k/n) in terms of the rule (e^a)^b=e^(ab).

Many thanks again.

Originally Posted by awkward

Hi rcl333,

You confusion is due to the fact that some of the identities we take for granted about exponentiation of real numbers do not hold for complex numbers. In particular, it is not true that $(e^a)^b = e^{ab}$ . Try reading the section on powers of complex numbers on this web page,

Exponentiation - Wikipedia, the free encyclopedia

especially the sub-section titled "Failure of Power and Logarithm Identities". Then if you are still confused, let us know.

**awkward** · September 1st 2010, 02:15 PM

The reason that

$(e^{2 \pi i (1/n)})^k = e^{2 \pi i k / n}$

is that, in this case, it doesn't matter which branch of the complex logarithm of $e^{2 \pi i (1/n)}$ is chosen. The "obvious" choice is $2 \pi i (1/n)$ , but $2 \pi i (1/n) + 2 m \pi i$ is also a valid choice for any integer m. (This is because $e^{2 m \pi i} = 1$ .)

But for any choice of the branch of the logarithm, we have
$e^{(2 \pi i (1/n) + 2 m \pi i) k} = e^{(2 \pi i k/n + 2 k m \pi i)} = e^{2 \pi i k/n} \cdot e^{2 k m \pi i} = e^{2 \pi i k/n} \cdot 1 = e^{2 \pi i k/n}$
because km is an integer.

The reason this trick does not work in general for $(e^a)^b$ is that $bm$ is not, in general, an integer.

**rcl333** · September 1st 2010, 03:06 PM

Thanks, it is very helpful and clear.

**rcl333** · September 2nd 2010, 08:42 AM

I appreciate the help earlier on the exponent rules on complex numbers. Please forgive my naiveness to the following followup questions:

Since for complex numbers, there are many branches of solutions for fractional power, how would we define which branch to choice, for example, if we have a function F(z)=a^z where a is a positive number (or even positive integer, this kind of function is basic in complex analysis), it shouldn't take multiple values, and which branch defines this function?

In particular, even in the real number syatem, 1 and -1 all serve as the squared root of 1. What is the meaning of 1^(1/2)? Which value and when should be adopted?

Thanks.

**Defunkt** · September 2nd 2010, 08:50 AM

Principal value - Wikipedia, the free encyclopedia

The meaning of $\sqrt{x}$ for $x \in \mathbb{R}$ is the positive number y such that $y^2 = x$ .

**awkward** · September 2nd 2010, 02:09 PM

Originally Posted by rcl333

I appreciate the help earlier on the exponent rules on complex numbers. Please forgive my naiveness to the following followup questions:

Since for complex numbers, there are many branches of solutions for fractional power, how would we define which branch to choice, for example, if we have a function F(z)=a^z where a is a positive number (or even positive integer, this kind of function is basic in complex analysis), it shouldn't take multiple values, and which branch defines this function?

In particular, even in the real number syatem, 1 and -1 all serve as the squared root of 1. What is the meaning of 1^(1/2)? Which value and when should be adopted?

Thanks.

In the real numbers, the convention is that $\sqrt{x}$ always denotes the positive root. The familiar identities like $\sqrt{xy} = \sqrt{x} \cdot \sqrt{y}$ depend on this convention.

In the complex numbers, no such convention works well in all cases, so if you need a single value of one of the multi-valued "functions", you must pick a branch of the function for that purpose. You may need to pick different branches for different problems. Fundamentally, however, one must simply be always aware of the fact that there are multiple possibilities for values of functions like the exponential and logarithmic functions.

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