Science Forums 2

[Don Blazys]

To: Qfwfq.

When you say that the "indeterminate form":
(0/0) is just as "undefined" as the "undefined operation":
(6/0), then you are dismissing the expression:
(0/0) as "meaningless", when in fact, there are plenty of cases, contexts and circumstances in which equations such as:
(0/0)=1 can be viewed as "meaningfull". (Especially in the context of "limits".)

There are, however, no conditions under which equations such as:
(6/0)= N can be construed as meaningfull, so the expressions:
(0/0), and:
(6/0) must be viewed as "fundamentally different" from each other.

(0/0) "exists" as a logical construct.
(6/0) "does not exist" as a logical construct.

Also, (0/0)= (infinity) if and only if (infinity)*0=0. (Again, the meaning/value of (0/0) depends entirely on the context in which it occured.)

I suppose that different mathematicians have different criteria for what constitutes a "valid result". For me (and most mathematicians my age), those criteria are "consistency" and "beauty".

My garage is the same (consistent) regardless of whether I enter it from the front door, or the back door.

In the same way, at x=1, the equation:

(T/T)a^x=T(a/T)^((xln(a)/(ln(T))-1)/(ln(a)/(ln(T))-1))=

(T/T)a=T(a/T)^((ln(a)/(ln(T))-1)/(ln(a)/(ln(T))-1))=T(a/T),

so that letting T=a results in:

(a/a)a=a(a/a)^(0/0)=a(a/a)=

(1)a=a(1)^(0/0)=a(1),

where, for the sake of consistency, we must conclude that the expressions:

(1) and (1)^(0/0)

are both equal, and "equally trivial" and may therefore be disregarded so as to result in the identity:

a=a=a.

Look at it this way. If, in the above case, we were somehow able to demonstrate that:

(1) and (1)^(0/0)

do not have the exact same value, then we would have to make the further "assumption" that logarithms are inherently "flawed", and must therefore be eliminated from mathematics!

I would then be forever remembered as the "mathematician who demonstrated that the properties of logarithms are bogus".

Like I said before, the proof is merely an unavoidable consequence of "cohesive terms", which are, in turn, merely an extention of the properties of logarithms,... and those properties are definitely not bogus.

Don.