At the threshold, our psyche changes, getting prepared for entry into a venerated new realm. It is the first step into the realm. It must not be high. Mathematics is as old as the human race, it attracted and was attended by the best brains and served the humblest through the ages. It keeps growing and expanding in new directions; new concepts giving birth to new definitions, more theorems and their proofs following. There are simple (expected) results not necessarily having easy proofs − like the four colour problem and Fermat’s last theorem. There are bewildering results, which likes of Ramanujan alone know how it occurred to them (like approximations to π). In the textbooks on Mathematics, and also in the classrooms, big space-time is spent on the proofs of the theorems. Proofs are like a defence of the theorem to the (ignorant) judges. But someone will prepare the defence (proof) only if he or she is able to see the theorem (the truth).
The observations which help a person in seeing and arriving at the truth (theorem) convincingly; in the absence of which the drudgery of building of the proof will not be taken up; deserve greater importance in the learning of Mathematics. The observations (called examples or counter-examples in Mathematics) precede the theorems, definitions and even the concept itself. At the threshold of Mathematics, shall we wait a while, enjoying our first observations!
The real encounter with great things will take time and effort. If the teaching of Mathematics follows this sequence, beginning with dis- cussions on examples leading to the concept, it would provide the student with the joy of discovery at every level of learning and train him/her better in Mathematical reasoning. The common practice however has been that the definitions are presented to the learner as if obtained from the blue and theorems obtained through a magic wand. And then examples are presented to illustrate the definitions and theorems. However, the scientific learning should involve exposure to a variety of examples and situations that take a student to such a height that the concept and related theorems twinkle to him simultaneously. The granting of the incubation period is essential before the natural birth of a concept or definition and their properties (theorems) to a learner. I do believe that Mathematics Laboratory work must not be only the Physical verification of the results but include experiments that lead you to the results.
At the threshold of Mathematics, in the following few activities, an attempt is made to illustrate the proposal made in the preceding paragraphs. Activities suggested are not meant to be read as a part of text by the students. It is a sharing of one teacher to other fellow-teachers; hoping them to develop and share activities for myriad of concepts and theorems they are coming across. In this attempt, few topics have been taken up at each level; early school, secondary school and college.
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AN INTERESTING PRIME NUMBER PROBLEM
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What are both primes p > 0 for which 1/p has a purely periodic decimal expansion with a period 5 digits long?
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Of course one must know what a prime number is... and I'm sure all of you know that a prime number is an integer greater than 1 (by definition, the number 1 is not a prime) that is evenly divisible only by itself and 1.
The first few primes are 2,3,5,7,11,13,17,19,23,29, ...
Let P = a prime satisfying the stated conditions.
Then 1/P = 0.abcdeabcdeabcde.....
Note: The letters a,b,c,d,e represent digits. The expression abcde is not a product.
OK, 1/P = 0.abcdeabcdeabcde...
==> 100000(1/P) = abcde.abcdeabcdeabcde.....
Hence, 100000(1/P) - (1/P) = 99999(1/P)=(abcde) ==> 1/P = (abcde)/99999.
Now factoring 99999 yields 99999 = (3)(3(41)(271).
Hence 1/P =(abcde)/[(3)(3)(41)(271)].
This means that the only possibilities for P (remember, it is a prime) are 3, 41, and 271, BECAUSE....
the only way abcde can be "reduced" to 1 will be if
(i) it is divisible by (3)(41)(271), yielding 1/3;
(ii) it is divisible by (3)(3)(271), yielding 1/41;
(iii) it is divisible by (3)(3)(41), yielding 1/271.
Remember now... these are only possibilities. They have to be checked. Since 1/3 = 0.333333... (period = 1), it doesn't satisfy required condition. As indicated above
1/41 = .0243902439 .... (period = 5)
and
1/271 = .0036900369.... (period = 5)
do satisfy the stated conditions. Hence, the prime numbers requested are 41 and 271.
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Reference : http://www.herkimershideaway.org/writings/primp.htm
similarily one can find for the n-periods.
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