Compound interest and the logarithms
Saturday, April 14, 2012 2:30:58 AM
Compound is not a new notion. It is used when we Aggregate fine material to form a solid mass. The Binomial Theorem is also not new. Indian vedic siddhantas and temple songs were developed using the Binomial pattern to work out intricate and fine rhythms. This knowledge went to The Chinese through cultural exchange then by the Arabs to Europe. Pascal Picked up the triangle version of the binomial theorem and used it in various ideas.
Descartes and De Fermat's algebraic approach to Geometry leads to the development of equations for curves and a new set of puzzles for the "clever" to solve. Particularly tangent to and area under curves. Curves were treated of by the Greeks and areas were found by mechanical means in certain cases. The tangent problem s were gradually being broadened, and methods of solution becoming more general. Gradually solutions stsrt you involve aspects of the Binomial theorem.
Richard Witt's book Arithmeticall Questions, published in 1613, was a landmark in the history of compound interest. It was wholly devoted to the subject (previously called anatocism), whereas previous writers had usually treated compound interest briefly in just one chapter in a mathematical textbook. Witt's book gave tables based on 10% (the then maximum rate of interest allowable on loans) and on other rates for different purposes, such as the valuation of property leases. Witt was a London mathematical practitioner and his book is notable for its clarity of expression, depth of insight and accuracy of calculation, with 124 worked examples.
Big interest was generated in binomial theorem by merchant userers. The development of compound interest explained by using the Binomial Theorem , especially by Richard Witt was of great impotance in financing many trade and exploration ventures. The promised returns and the opportunities offered by the Banks, and the Coffee houses, encouraged the speculation needed to venture in the development of trade and the empire. The down side was that those investors were promised fabulous returns which meant that the trade was exploitative, and deliberately so. Compound interest has that effect on those that generate wealth.
The increased interest Led to lots of tables of binomial expansion, and more interest in the Binomial Theorem. Newton was the first to recognise through the connection to the tangent area problem how to generalise it to the binomia;; series.
There were many tables , gradually increasing the number of years or the number of compounding intervals. Newton realised that if n was just allowed to be very large, the formula for each term could be worked out. He found the formula for each term for any n.
Newton's work on the binomial theorem is nothing short of remarkable. He begins, as did Wallis, by making area computations of the curves , and tabulating the results. He noticed the Pascal triangle and reconstructed the formula
for positive integers n.
Now to get to compute , i.e. n=1/2, he simply applied this relation with n=1/2. This of course generated an infinite series because the terms do not terminate.
Next he generalized to function of the form for any n. This gave him the general binomial theorem - but not a proof.
He was able to determine the power series for by integrating the series for , written according as the binomial series. In modern notation, we have
Now integrate to get the series
From this Newton developed his Fluxions, and from the methods of compounding he develops his method of compounding tangents. Along the way he develops logarithmic series for e and sine.