(-1)^(2/3) = 1 ?
Most web pages about secondary school algebra that I have seen are logically inconsistent when it comes to exponentiation. On the one hand, they claim (x [sup] a [/sup]) [sup] b [/sup] = x [sup] (ab) [/sup] whenever the exponentiations are defined. Then they allow operations like (-1)[sup] (2/3) [/sup] to be defined and usually give examples of how to work such things out (integer powers of odd roots of negative numbers).. So if you let x =- 1, a = 2/3 and b = 3/2 in the above law, you get the contradiction that 1 = -1.
Before anyone decides to wax eloquent on the properties of the complex numbers, the cube roots of unity etc. , let me say that I'm talking about whether these web pages use a consistent set of assumptions for the real numbers. The complex numbers are interesting, but they are no excuse for presenting an inconsistent set of axioms for the real numbers.
And I'm talking about exponentiation as an operation that is supposed to produce at most one result, not as a notation for the set of several roots to an equation.
It's particularly interesting that many web pages state that x [sup] a [/sup] is only defined when x is a positive number. But later, they cannot resist showing students examples of how to compute the odd roots of negative numbers. As to whether x [sup] (6/10) [/sup] = x [sup] (3/5) [/sup] when x < 0, most pages avoid that issue.
Mathematical software and calculators are also inconsistent. Try computing (-2)[sup] (0.6) [/sup] with various calculators. There is some excuse for that. They are trying to read the users mind to determine if he wants to stay in the real number system.
I think the correct treatment of the above issues has to do with correctly distinguishing between the operation of exponentiation (which obeys "the laws of exponents") and a common extension of that operation, which doesn't.
Before anyone decides to wax eloquent on the properties of the complex numbers, the cube roots of unity etc. , let me say that I'm talking about whether these web pages use a consistent set of assumptions for the real numbers. The complex numbers are interesting, but they are no excuse for presenting an inconsistent set of axioms for the real numbers.
And I'm talking about exponentiation as an operation that is supposed to produce at most one result, not as a notation for the set of several roots to an equation.
It's particularly interesting that many web pages state that x [sup] a [/sup] is only defined when x is a positive number. But later, they cannot resist showing students examples of how to compute the odd roots of negative numbers. As to whether x [sup] (6/10) [/sup] = x [sup] (3/5) [/sup] when x < 0, most pages avoid that issue.
Mathematical software and calculators are also inconsistent. Try computing (-2)[sup] (0.6) [/sup] with various calculators. There is some excuse for that. They are trying to read the users mind to determine if he wants to stay in the real number system.
I think the correct treatment of the above issues has to do with correctly distinguishing between the operation of exponentiation (which obeys "the laws of exponents") and a common extension of that operation, which doesn't.
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