On Mathematics Education: Algebra vs. Calculus

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This time I am going to take a break from heavy use of $ \mathrm\LaTeX$ like I used to do in my earlier posts. I will focus on the education of mathematics as is being provided to students in India. My sources of information in this regard are:
  • memories from my school/college days (which are still relevant in current time)
  • interaction with current school/college students
  • mathematics books (both Indian and foreign, old and new) available on the market
The subtopic of the post should not be taken too literally, rather it reflects two approaches in the teaching of mathematics: 1) algebraic approach which I don't like and find unsuitable for any serious teaching of mathematics, and 2) the calculus approach which is the way mathematics teaching should be, but is currently not being practiced anywhere as far as the books show.

Irrationality of π(PI): Lambert’s Proof Contd.

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Irrationality of Continued Fractions

We have the following results about the irrationality of some continued fractions:
If $ a_{n}, b_{n}$ are positive integers then:
1) The infinite continued fraction $$\frac{b_{2}}{a_{2} +}\,\frac{b_{3}}{a_{3} +}\,\frac{b_{4}}{a_{4} +}\,\frac{b_{5}}{a_{5} +}\,\cdots $$ converges to an irrational value, provided that $ a_{n} \geq b_{n}$ for all values of $ n$ starting from a certain value $ n = n_{0}$.

2) The infinite continued fraction $$\frac{b_{2}}{a_{2} -}\,\frac{b_{3}}{a_{3} -}\,\frac{b_{4}}{a_{4} -}\,\frac{b_{5}}{a_{5} -}\,\cdots $$ converges to an irrational value, provided that $ a_{n} \geq b_{n} + 1$ for all values of $ n$ starting from a certain value $ n = n_{0}$ and the condition $ a_{n} > b_{n} + 1$ must hold for an infinite number of values of $ n$.

Irrationality of π(PI): Lambert's Proof

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Introduction

After mentioning about the Lambert's famous proof of irrationality of $ \pi$ in an earlier post, it is now time to give it to the readers in its entirety. I need to reiterate the fact that being a far more direct proof than the modern proofs of Ivan Niven, it is still highly neglected by modern authors and educators. The idea of the proof is really elementary but based on the concept of continued fractions which are now deleted from the high school mathematical syllabus. Why this topic is now left out is still unclear to me. One reason which I can guess of is that the manipulations of continued fractions are not so simple (compared to those of an infinite product or a series). The visible form of the continued fraction does not give any idea about its value unless we do the calculations.