Discovery of The Mathematical Constant E
Bernoulli discovered the constant e by studying a question about compound interest which required him to find the value of the following expression (which is in fact e):
One example is an account that starts with $1.00 and pays 100 percent interest per year. If the interest is credited once, at the end of the year, the value is $2.00; but if the interest is computed and added twice in the year, the $1 is multiplied by 1.5 twice, yielding $1.00×1.5² = $2.25. Compounding quarterly yields $1.00×1.254 = $2.4414..., and compounding monthly yields $1.00×(1.0833...)12 = $2.613035....
Bernoulli noticed that this sequence approaches a limit (the force of interest) for more and smaller compounding intervals. Compounding weekly yields $2.692597..., while compounding daily yields $2.714567..., just two cents more. Using n as the number of compounding intervals, with interest of 100%/n in each interval, the limit for large n is the number that came to be known as e; with continuous compounding, the account value will reach $2.7182818.... More generally, an account that starts at $1, and yields (1+R) dollars at simple interest, will yield eR dollars with continuous compounding.
Read more about this topic: Jacob Bernoulli
Famous quotes containing the words discovery of the, discovery of, discovery, mathematical and/or constant:
“Your discovery of the contradiction caused me the greatest surprise and, I would almost say, consternation, since it has shaken the basis on which I intended to build my arithmetic.... It is all the more serious since, with the loss of my rule V, not only the foundations of my arithmetic, but also the sole possible foundations of arithmetic seem to vanish.”
—Gottlob Frege (1848–1925)
“The discovery of the North Pole is one of those realities which could not be avoided. It is the wages which human perseverance pays itself when it thinks that something is taking too long. The world needed a discoverer of the North Pole, and in all areas of social activity, merit was less important here than opportunity.”
—Karl Kraus (1874–1936)
“There is a great discovery still to be made in literature, that of paying literary men by the quantity they do not write.”
—Thomas Carlyle (1795–1881)
“What he loved so much in the plant morphological structure of the tree was that given a fixed mathematical basis, the final evolution was so incalculable.”
—D.H. (David Herbert)
“Sigh no more, ladies, sigh no more,
Men were deceivers ever,
One foot in sea and one on shore,
To one thing constant never:
Then sigh not so, but let them go,
And be you blithe and bonny,
Converting all your sounds of woe
Into Hey nonny, nonny.”
—William Shakespeare (1564–1616)