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You can find many texts and web pages that teach elementary algebra and say they are only dealing with the real numbers. They state general laws of exponents such as:
[tex] (x^a)^b = x^{ab} [/tex] "whenever the exponential operations are defined".
In discussing square roots they are careful to point out that the square root of a negative number is not defined. However, most cannot resist claiming that the cube root of a negative number can be taken. Even "achievement test" material may expect students to compute [tex] \sqrt[3]{-8} = -2 [/tex].
Using the above mentioned law of exponents, this will lead to disaster when [tex] X = -8, a = 2/3, b= 3/2 [/tex].
I think most math students have the old fashioned view that mathematics is not required to be logically consistent. They think it refers to a Platonic reality rather than collections of assumptions and definitions. ( I gather than some of the participants in the .9999.. .thread are of this school.) So, perhaps logical contradictions don't upset the social order.
However, from the modern point of view, it would be best if the elementary textbooks could get their story straight on the case of [tex] \sqrt[3]{-8}[/tex] and similar arithmetic. Something has to give. Either you have to put some more complicated restrictions on when [tex] x^a x^b = x^{ab} [/tex] or you have to say that [tex] \sqrt[3]{-8}[/tex] is undefined in the algebra of the real numbers. When a student brings up the above contradiction, what is liable to happen is that the instructor will start getting into some digression about the complex numbers, numbers have 3 cube roots etc. Fine, but if you are going to teach the complex numbers then don't say you are teaching a course restricted to the real numbers.
[tex] (x^a)^b = x^{ab} [/tex] "whenever the exponential operations are defined".
In discussing square roots they are careful to point out that the square root of a negative number is not defined. However, most cannot resist claiming that the cube root of a negative number can be taken. Even "achievement test" material may expect students to compute [tex] \sqrt[3]{-8} = -2 [/tex].
Using the above mentioned law of exponents, this will lead to disaster when [tex] X = -8, a = 2/3, b= 3/2 [/tex].
I think most math students have the old fashioned view that mathematics is not required to be logically consistent. They think it refers to a Platonic reality rather than collections of assumptions and definitions. ( I gather than some of the participants in the .9999.. .thread are of this school.) So, perhaps logical contradictions don't upset the social order.
However, from the modern point of view, it would be best if the elementary textbooks could get their story straight on the case of [tex] \sqrt[3]{-8}[/tex] and similar arithmetic. Something has to give. Either you have to put some more complicated restrictions on when [tex] x^a x^b = x^{ab} [/tex] or you have to say that [tex] \sqrt[3]{-8}[/tex] is undefined in the algebra of the real numbers. When a student brings up the above contradiction, what is liable to happen is that the instructor will start getting into some digression about the complex numbers, numbers have 3 cube roots etc. Fine, but if you are going to teach the complex numbers then don't say you are teaching a course restricted to the real numbers.
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