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, discovery, mathematical and/or constant:
“Next to the striking of fire and the discovery of the wheel, the greatest triumph of what we call civilization was the domestication of the human male.”
—Max Lerner (b. 1902)
“The virtues of society are vices of the saint. The terror of reform is the discovery that we must cast away our virtues, or what we have always esteemed such, into the same pit that has consumed our grosser vices.”
—Ralph Waldo Emerson (1803–1882)
“All science requires mathematics. The knowledge of mathematical things is almost innate in us.... This is the easiest of sciences, a fact which is obvious in that no one’s brain rejects it; for laymen and people who are utterly illiterate know how to count and reckon.”
—Roger Bacon (c. 1214–c. 1294)
“The tears of the world are a constant quality. For each one who begins to weep, somewhere else another stops. The same is true of the laugh.”
—Samuel Beckett (1906–1989)