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## APยฎ๏ธ Calculus AB (2017 edition)

### Course: APยฎ๏ธ Calculus AB (2017 edition)ย >ย Unit 4

Lesson 6: Derivatives of exponential functions

# ๐ and compound interest

Sal introduces a very special number in the world of math (and beyond!), the constant ๐. Created by Sal Khan.

## Want to join the conversation?

• 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
• 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!
• 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... !
• how is 50% interest equal to multiplying the PV by 1.5?
could someone please explain the math behind it
• When you add 50% onto something, the resulting value is half of the original value, plus the original value. This simplifies to 3/2, or 1.5. We need to regard the entire value of the item when doing compound interest, because the interest value is generated based on the preceding iteration's total value.
• Can someone simply this
• Sure, let's simplify the concepts of "e" and "compound interest."

1. *e (Euler's Number)**:
"E" is a special mathematical constant known as Euler's number, denoted by the symbol "e." Its approximate value is approximately 2.71828. Euler's number is a fundamental constant that appears in various areas of mathematics, particularly in calculus, where it is the base for the natural logarithm.

The value of "e" is irrational, meaning it cannot be expressed as a simple fraction and its decimal representation goes on infinitely without repeating. It is an essential constant in various mathematical and scientific calculations and often arises in problems involving growth, decay, and rates of change.

One of the fundamental properties of "e" is the exponential function e^x, which has unique and useful mathematical properties, making it significant in many mathematical applications.

2. **Compound Interest
*:
Compound interest is a method of calculating interest on a principal amount that includes not only the initial principal but also the interest that has accumulated over previous periods. In contrast, simple interest is calculated only on the initial principal amount.

The formula for calculating compound interest is:
A = P(1 + r/n)^(nt)

Where:
A = the future value of the investment/loan, including interest
P = the principal amount (initial investment/loan)
r = the annual interest rate (expressed as a decimal)
n = the number of times that interest is compounded per year
t = the number of years the money is invested or borrowed for

In the formula, "A" represents the final amount after "t" years with compound interest, which includes both the original principal and the accumulated interest. Compound interest is a powerful concept that allows investments to grow exponentially over time, as interest is continuously added to the principal, leading to increasing returns.

It's essential to be aware of compound interest when saving or investing, as it can significantly impact the growth of your money over long periods. On the other hand, when borrowing, compound interest can cause the total amount owed to increase more rapidly than simple interest would.

it dont get simpler then this