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# 𝑒 and compound interest

Sal introduces a very special number in the world of math (and beyond!), the constant 𝑒. Created by Sal Khan.

## Want to join the conversation?

• Why did Sal choose to use 100% interest? Couldn't this same principle be used/proven with different interest, for example 80% or 500%?
• If you pick different numbers for x, it doesn't approach e anymore. Instead it'll approach e^r, where r is the interest rate. Example: if the interest rate is 200%, then it'll approach e^2.

You can plug this right into google and see:
(1+2/365)^365 = 7.3488
e^2 = 7.389
• At , shouldn't the yearly computation be 365.25 to account for leap years?
• Technically. It was most likely simplified for simplicity. You will usually see 365 days per year in textbooks for these kinds of problems.

Hope that helped!

Jonathan Myung
• How do you find n in the compound interest equation?
• Brilliant question! To find n, you need to use natural logarithm function.

Suppose you have a future value formula PV * (1+r)^n = FV where:
PV stands for present value;
FV stands for future value;
r stands for interest rate; and
n stands for a number of periods

So PV * (1+r)^n = FV can be rearranged to
(1+r)^n = FV/PV
Then we take natural logarithm ln
ln(1+r)n = ln(FV/PV)
Then we divide both sides by ln(1+r) and we get
n=(ln(FV/PV))/ln(1+r)

If you haven't learned about natural logarithms go to Logarithms playlist in the Algebra section.
If you are not very familiar with present value and future value formulas then the next playlist will cover Time Value of Money which is a very important concept.
• what's mean e?
• So if you borrow \$1 at 100% annual interest compounded monthly for 2yrs. Would the answer be 1 ( 1 + 100%/24 ) ^ 24 or 1 ( 1 + 100%/12 ) ^ 24? I'm leaning towards the first but not sure.
• Yes, it's the first one.
(1 vote)
• Why do 12 months and 1 year have different percentages?
• That depends on interest calculation frequency.
• How do you solve a problem backwards?
• You take the same values but the inverse calculating sign.
For example,
X-3+6=13
You would solve backwards as shown,
13-6+3=X
Hence, X=10

Hope this helps!
(1 vote)
• Why is e such a small number and how did they calculate e if it is infinite?
• e is indeed infinite; although we have calculated some of the digits does not mean that we calculated all infinite digits.

e is also a small number since if we keep putting on compound interest, your interest money will be more smaller every increment.
As there is an infinite amount of increments, the interest money will be increasingly minuscule and approach a certain sum of money: 2.71828... !