Why is Complex Exponentiation Not Commutative?

When dealing with complex numbers, the order in which you apply exponents matters. This article explores the non-commutative nature of the power sequence, specifically focusing on expressions involving negative real numbers and the implications of raising powers to other powers.

The Commutativity of Exponents

For positive real numbers, the commutativity of raising to a power is generally accepted. However, the situation changes for negative real numbers, which should be treated as complex with an argument of 180 degrees. This means that expressions like ((-1)^{1/2}^2) and ((-1)^2^{1/2}) yield different results, despite the apparent similarities.

Example Exploration

Let's compare the two expressions:

((-1)^{1/2}^2) ((-1)^2^{1/2})

For the first expression, ((-1)^{1/2}^2), we can apply two methods:

((-1)^{1/2}^2 (i^2) -1) ((-1)^{1/2}^2 (-1^{1/2}) cdot (-1^{1/2}) -1)

Both methods yield (-1).

For the second expression, ((-1)^2^{1/2}), we derive:

[(-1)^2^{1/2} (1^{1/2}) 1]As a result, ((-1)^{1/2}^2 -1 eq 1 (-1)^2^{1/2}).

Complex Exponents and the Non-Commutativity Issue

The issue arises because raising a power to another power is not commutative in complex exponents. The expressions ((-1)^{n/2}^2) and ((-1)^2^{n/2}) are not equal for all integer values of (n).

Consider (n 2), comparing ((-1)^{2/2}^2) and ((-1)^2^{2/2}):

[(-1)^{2/2}^2 (-1)^2 1][(-1)^2^{2/2} 1^{1} 1]Both expressions yield 1, which corrects the misunderstanding about the non-commutative nature of exponents.

Complex Exponential Function and Multi-Valued Results

The complex exponential function (e^z^w eq e^{wz}) can have multiple solutions. For ((-1)^{n/2}), the expression can be rewritten as:

[(-1)^{n/2} e^{ipi}^{n/2} e^{ipi n/2}]

Here, (x) is an arbitrary integer. Since (n) is an odd integer, (cos(npi/2) 0). The value of (sin(npi/2)) depends on whether (n/2x1) leaves a remainder of 1 or 3 when divided by 4. If the remainder is 1: (sin(npi/2) i). If the remainder is 3: (sin(npi/2) -i).

In both cases, squaring these results yields (-1), leading to the conclusion that ((-1)^{n/2}^2 -1) when (n) is odd.

For ((-1)^2 e^{ipi}^2 e^{i2pi} 1), raising to the power of (n/2) will also give 1. This shows that the results can be dissimilar despite the initial similarity in structure.

Conclusion

The non-commutativity of complex exponents highlights the importance of careful handling of complex numbers in exponentiation. While there are no strict rules about obtaining the same result when swapping exponents, understanding the underlying principles helps in accurately interpreting and solving complex expressions.

Further Reading

For more detailed information, see:

Failure of Power and Logarithm Identities