Editing E (mathematical constant) (section)

=== Compound interest ===
[[File:Compound Interest with Varying Frequencies.svg|right|thumb|The effect of earning 20% annual interest on an {{nowrap|initial $1,000}} investment at various compounding frequencies. The limiting curve on top is the graph <math>y=1000e^{0.2t}</math>, where {{mvar|y}} is in dollars, {{mvar|t}} in years, and 0.2 = 20%.]]

Jacob Bernoulli discovered this constant in 1683, while studying a question about [[compound interest]]:<ref name="OConnor" />
{{Blockquote|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 {{nowrap|1=$1.00 × 1.5<sup>2</sup> = $2.25}} at the end of the year. Compounding quarterly yields {{nowrap|1=$1.00 × 1.25<sup>4</sup> = $2.44140625}}, and compounding monthly yields {{nowrap|1=$1.00 × (1 + 1/12)<sup>12</sup> = $2.613035...}}. If there are {{mvar|n}} compounding intervals, the interest for each interval will be {{math|100%/''n''}} and the value at the end of the year will be $1.00&nbsp;×&nbsp;{{math|(1 + 1/''n'')<sup>''n''</sup>}}.<ref name="Gonick"/><ref name=":0" />

Bernoulli noticed that this sequence approaches a limit (the [[force of interest]]) with larger {{mvar|n}} and, thus, smaller compounding intervals.<ref name="OConnor" /> Compounding weekly ({{math|1=''n'' = 52}}) yields $2.692596..., while compounding daily ({{math|1=''n'' = 365}}) yields $2.714567... (approximately two cents more). The limit as {{mvar|n}} grows large is the number that came to be known as {{mvar|e}}. That is, with ''continuous'' compounding, the account value will reach $2.718281828... More generally, an account that starts at $1 and offers an annual interest rate of {{mvar|R}} will, after {{mvar|t}} years, yield {{math|''e''<sup>''Rt''</sup>}} dollars with continuous compounding. Here, {{mvar|R}} is the decimal equivalent of the rate of interest expressed as a ''percentage'', so for 5% interest, {{math|1=''R'' = 5/100 = 0.05}}.<ref name="Gonick">{{cite book
 | last = Gonick
 | first = Larry
 | author-link = Larry Gonick
 | year = 2012
 | title = The Cartoon Guide to Calculus
 | publisher = William Morrow
 | url = https://www.larrygonick.com/titles/science/cartoon-guide-to-calculus-2/
 | isbn = 978-0-06-168909-3
 | pages = 29–32
}}</ref><ref name=":0" />