Intuitively, we often think of real numbers as existing in one-to-one correspond
ID: 2970536 • Letter: I
Question
Intuitively, we often think of real numbers as existing in one-to-one correspondence with the points on a continuously-drawn line, the real number line. One way of expressing the completeness of the real numbers is to say that the real line has no holes. That is, the values that are
Intuitively, we often think of real numbers as existing in one-to-one correspondence with the points on a continuously-drawn line, the real number line. One way of expressing the completeness of the real numbers is to say that the real line has no holes. That is, the values that are "missing" from the rationals-such as, for example, sqrt(2)-are present in the real numbers. What do you see as the possible limitations of using this intuitive idea to prove the existence of certain limits in the real numbers? Do you think it is sufficient for the purposes of most students who study calculus to simply accept the existence of these limits without proof? From the perspective of teaching, how rigorous do you think the treatment of the real numbers has to be?Explanation / Answer
Intuition has always been a guide to humankind ever since the ever since the dawn of civilization.It is intuition that has restricted our ancestors to believe that the sun and the other 'planets' revolve around the earth and that the earth is the center of this "universe" for more than few hundred years but it happened out to be uttterly false and my point is that though intuition has in some sense a simplified look or understanding it cannot be a guide to the TRUTH and this is where I think it should be limited to just have a feeling of awe and beauty of the problems at stake but it should not be used as a guide to understand the comprehensiveness of the problem.And morevoer what is intuititve to one may not be intuitive to others so again there is a limitation in the sense of conveying the answer to others where as a definite proof can be understood logically leading to the correct solution but the logic and proof alone cannot give full comprehensive picture of the problem,they together with intuitive understnading gives the full meaning to the problem.
The idea of proof, which is understood in a rigid and absolute way by the mathematical community, seems to have been
considered the only valid conception. However, we consider it necessary to carry out a systematic study of the various meanings of proof, not just from the subjective point of view, but also in different institutional contexts.
So in this case of limitations in the real numbers we should notice that it is a continuos lineand that thinking of any two numbers in the real line you always find infinite nubers in between them no matter how close you consider the initial two numbers so the more you zoom into real line the more nubers you will find in other words the real line is of infinite density.
so by taking above points into consideration I would say it is not sufficient for the purposes of most students who study calculus to simply accept the existence of these limits without proof
From the perspective of teaching,not only in case of calcus or other topics in the entire subject of mathematics initutive understanding should always be followed with rigourous proof.
Related Questions
drjack9650@gmail.com
Navigate
Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.