Values In The Continuous Spectrum

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

A Mathematical Challenge: Integrating A Sinusoidal Signal

Introduction:
For the past two days my head has been processing at maximum load!! Thanks goes to an anonymous poster. Mr. Anonymous has brought my attention to an older post, titled "Inconsistent Circuits Formula". His comments triggered my thoughts, and as some do notice, our thoughts aren't clear all the time. At the time I posted my old entry, I had a limited vision - Today, I come with new understanding: An understanding of the nature of the phenomena of resonance.

I will not post my newly-reached understanding in this post. I have a clear vision of the phenomena of resonance, but I want to support it with some mathematical justification, and more importantly, numerical analysis data. The results should be ready within two weeks depending on my free time - No promises though! You can read the results here.

Mr. Anonymous has argued about my bold statement "We cannot prove two physical (even mathematical) quantities to be equal". This post should indirectly justify my position. I am still considering to post a more direct argument can be found here. Just keep in mind, my position is strictly regarding inclusion of equalities for continuous quantity spectrum.

Finally, this mathematical challenge question was tailored by me in order to clarify a point I want to make, so try to solve it, and tell me your results.

The Mathematical Challenge:

The Final Answer:
ZERO (assuming that no noise exists in the signal)
ANY VALUE is the general solution
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Explained Answer:
The interval [0, 2*PI/a] represents one period of the sinusoidal signal cos(ax). When integrating a sinusoidal signal over one period the answer is ZERO. In short, the answer is ZERO regardless of the value of a.

A source of confusion might arise due to the use of the value ZERO for 'a'. The easiest approach is to devise an answer which is independent of 'a', as has been explained above. Note that the sinusoidal signal has a constant value of ONE over the finite region. The interval of the integration extends to ( 2*PI/a ), which also extends to the infinity. For this reason, the transformation of cos(ax) to ONE is NOT valid, because the sinusoidal signal is equal to 1 ONLY in the finite region, but the integration extends to the infinite region which implies that this transformation leads to incorrect results.

This solution applies only if we consider that there is NO NOISE in the sinusoidal signal. If we consider the possibility of existence of noise, then the integration will take the value of the integral of noise over the whole interval.
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Update 1: The answer to the challenge has been posted.
Update 2: The paper about resonance has been posted here.