It is understood in the field of mathematics that values in the continuous spectrum CANNOT be proved to be equal. In other words, five is NOT NECESSARILY equal to five!
Some people might ask: If no two values can be proved to be equal, how does mathematics handle values?! It's a good question... And the answer is simple: Thats why calculus was formed!! Mathematics is based on limit theorems. We can say that five sometimes equals five, yet five never equals six!! I say that five sometimes equals five, because this might not be true all the time!!
I have provided a FORMAL MATHEMATICAL PROOF of these claims. I have published the proof in PDF format (140 KB) and MS WORD format (56 KB).
I urge you to print the proof, and double check the validity of every statement and transformation. I have taken like three hours of my time to write this proof, in order to check and re-check every statement I put... Yet it is important to have peer reviews!!
I also urge you to look at the proof to learn the good way to prove a point... Always start with 1- definition of notations, then 2- definition of terms, followed by 3- axioms and assumptions. The three mentioned parts are crucial to a good proof! Finally, always follow logical inferencing, and then all should be good.
Now, apart from formalities, allow me to explain the claims informally:
First consider the value of ZERO/ZERO. It is well-known that the answer is ANY VALUE!! Some people think that the answer is "no value", but thats mathematically wrong statement!
Now, ZERO/ZERO is any value. Is One a value?! Yes, it is. Is Two a value?! Yes, it is. But we all know that any two equal values when divided the result is One... Right?! Well, in one case ZERO/ZERO in fact gave the value of One. But also, in a second case ZERO/ZERO had the value 2... Obviously this means that ZERO is not equal to ZERO in that case. Combine the above two phenomenas, we find that ZERO is sometimes equal to ZERO, and sometimes not so!!
Since we showed that ZERO is NOT NECESSARILY equal to ZERO, we can generalize this result to all values by adding assumed equal values to both sides of [0!=0]...
Another argument goes like this: ZERO*INFINITY is any value (just like ZERO/ZERO). We can think of multiplication as a magnification operator. Even the ZERO when magnified "enough" can be shown to deviate from ZERO!!! This also demonstrates the above claims about the continuous spectrum.
PS: For additional insight, read the "physical justification" in my comment here
PS: Download the formal mathematical proof: PDF DOC
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2 comments:
"Since we showed that ZERO is NOT NECESSARILY equal to ZERO, we can generalize this result to all values by adding assumed equal values to both sides of"
you are contradicting equality definition with this statement, if you assume 0!=0 then you cannot add 2 assumed equal values to both sides of the equation and reach cancellation that they are not equal, because you added same value to something that is not equal in the first place. yes there's this branch of math dealing with limits and series of functions and the question that is always raised, are we close enough? ( if we take taylor series of functions for example, till which order should we start ignoring the residue (Rn)), the answer would depend on how precise you wanna be (or what order they ask you to reach in the test) but there's also another whole side of math that deals with discrete values)
"because you added same value to something that is not equal in the first place." - We know that: if X does not equal Y, then adding equal values to both sides gives also an inequality, consider:
6 != 7
6+2 != 7+2
I admit that my use here of the equality sign is inconsistent, as I was explaining the idea "informally". If you want to read formally defined explanations, read the proof provided in PDF format.
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