Why are some math questions undefined?

Questions like 0 X infinity and 1 to the power of infinity. But why are these math questions undefined? Like anything multiplied by 0 is always 0, and 1 to the power of anything is always 1.

Update:

Please try and make your explanations easier, I really don't get any of them...

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• Pascal
Lv 7

Actually, these expressions are indeterminate, not undefined. There is a difference.

An indeterminate form is one where knowing the limits of the two subexpressions doesn't tell you anything about the limit of the expression taken as a whole. To say that 0*∞ is indeterminate means that, if there are two functions f(x) and g(x), and [x→c]lim f(x) = 0, and [x→c]lim g(x) = ∞, it is not possible to find [x→c]lim f(x)*g(x) from these two limits alone. Similarly, 1^∞ is indeterminate because if you know that [x→c]lim f(x) = 1, and [x→c]lim g(x) = ∞, it is not possible to determine what [x→c]lim f(x)^g(x) from these two limits. Contrast that with determinate forms: if you know (for instance) that [x→c]lim f(x)=2, and that [x→c]lim g(x) = 3, then it can be established from these two facts alone that [x→c]lim f(x)*g(x) = 2*3 = 6.

Note that the fact that these expressions are indeterminate does not make them undefined - if they appear as atoms they may evaluate to a specific number depending on the number system used (although it would have to be larger than the real numbers, since ∞ is not a real number). In the extended real line, 1^∞ is indeed 1 and 0*∞ is sometimes defined as zero (although often it is left undefined, mostly in contexts where the extended real line is used to represent limiting processes). Note that this is number-system dependent, however. On the real projective line (which also contains an element denoted ∞), 0*∞ cannot be consistently defined (but you get in return that 1/0 IS defined on the real projective line, and has the value ∞, whereas 1/0 is not defined on the extended real line).

When you say "anything multiplied by 0 is always 0", that is not to be taken literally. For example "McDonald's multiplied by 0" means nothing. Likewise "diplomacy multiplied by 0".

Your sentence is correct if the word anything stands for one of these (among others):

* any natural number

* any integer

* any rational number

* any real number

* any complex number

* etc.

"Infinity" is none of those. So until you put "infinity" into a formal context and then define multiplication within that context, your statement about infinity is left undefined...

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EDIT 1:

Pascal, I maintain that the term "undefined" is exactly correct. A binary operation, such as multiplication, is defined on a specific domain, for example N x N. Within that scope, the specific multiplication (say) "2 * pi" would be "undefined" simply because here one of the operands is not in the domain.

You are of course right to point out that, IN THE VERY SPECIFIC CONTEXT OF ANALYSIS, it would be unsatisfactory to extend the operation from the domain R x R to the domain (R union {infinity}) x (R union {infinity}).

However, the question as asked never suggested that real analysis was the underlying context. In algebra, for instance, your line of reasoning is irrelevant, and we would rather wonder if there was a way of respecting the ring-structure of the real numbers through a given extension by the element "infinity". (I don't think there is.)

My point essentially had to do with formalism. "Infinity" is NOT a real number, under ANY circumstance. The concept of "infinity" may be given specific and formal meanings in various contexts, but that is a matter of DEFINITION, and the various contexts where we define it must not be mixed with one another. The term "infinity" in the context "limit towards infinity" does not mean the same thing as in the context of "cardinality of an infinite set", nor as in the context of "compactification of the complex plane by adding the infinity point", etc.

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EDIT 2:

Pascal: 0^0 is usually undefined. Of course, you may define it as you stated, and indeed on two occasions have I seen authors using that definition as a shorthand to avoid complicating expressions that involved summations, but in both cases, the definition was stated loudly and explicit---a sign that this is by no means an agreed definition across the board...

some math questions just can't be calculated.

For example 0 to the power of 0.

Rule 1: Anything to the power of 0 it 1

Rule 2: 0 times anything is 0

You can't tell if 0^0 = 1 or 0 so it is undefined.

Jon and ChristineF are both wrong.

For one thing, 0^0 = 1.

Also, anything * 0 = 0 because you don't have any of them. Even if anything is McDonald's or infinity. For example, in infinite cardinal arithmetic, aleph0 * 0 = 0. In infinite ordinal arithmetic, omega * 0 = 0. Christine, it is true that "infinity" and "McDonald's" have no meaning but if we were to give them any, it would have to respect what we already believe about 0.