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## Integrated math 3

### Course: Integrated math 3>Unit 5

Lesson 2: The constant e and the natural logarithm

# 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%?
• I noticed the same thing, so I tried plugging in different numbers (x) for the numerator in the expression (1+x/n)^n, and found that this equals e^x as n->infinity. Kind of neat!
• At , shouldn't the yearly computation be 365.25 to account for leap years?
• Definitely it should not be! Even if it sounds clever from mathematical point of view, in real world it doesn't work this way. In fact, there are several methods how to measure a year in days. https://en.wikipedia.org/wiki/Day_count_convention#Actual_methods
• 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?
• `e` stands for Eulers's number which was named after Swiss mathematician Leonhard Euler who found this irrational constant.
https://en.wikipedia.org/wiki/E_(mathematical_constant)
• 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.
• Why do 12 months and 1 year have different percentages?
• That depends on interest calculation frequency.
• Why is e, so special and magical as Sal describes it? Why don't we just say 2.7 rounded?
• The same could be said about pi, which could just be called 3.14 rounded. It's just the way things are, I suppose.
• 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... !