Knowledge Matter, Results Count.

Jacob Bernoulli discovered this constant in 1683 by studying a question about compound interest:^{[5]}

- An account 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 of the account at year-end will be $2.00. What happens if the interest is computed and credited more frequently during the year?

If the interest is credited twice in the year, the interest rate for each 6 months will be 50%, so the initial $1 is multiplied by 1.5 twice, yielding $1.00×1.5^{2} = $2.25 at the end of the year. Compounding quarterly yields $1.00×1.25^{4} = $2.4414…, and compounding monthly yields $1.00×(1+1/12)^{12} = $2.613035… If there are *n*compounding intervals, the interest for each interval will be 100%/*n* and the value at the end of the year will be $1.00×(1 + 1/*n*)^{n}.

Bernoulli noticed that this sequence approaches a limit (the force of interest) with larger *n* and, thus, smaller compounding intervals. Compounding weekly (*n* = 52) yields $2.692597…, while compounding daily (*n* = 365) yields $2.714567…, just two cents more. The limit as *n* grows large 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 offers an annual interest rate of *R* will, after *t* years, yield *e*^{Rt} dollars with continuous compounding. (Here *R* is the decimal equivalent of the rate of interest expressed as a percentage, so for 5% interest, *R* = 5/100 = 0.05)

https://en.wikipedia.org/wiki/E_(mathematical_constant)

Sitworld Analytics Team

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