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# 𝑒 as a limit

Sal continues the discussion on e, this time digging deeper into the mathematical definition of 𝑒. Created by Sal Khan.

## Want to join the conversation?

• so if the interest was lets say 7% instead of 100% would you put .07/n or instead of 1/n?
• Exactly! The general compound interest formula is (1 + r/n)^n, where r is the rate. Obviously 100% = 1 and 7% = 0.07, so you did a good job.
• What other applications are there for e?
• also if you ever see ln or ln() that is a reference to e as the base for log. Log base e of a number is called the natural log
• I don't understand what limits are when Sal talks about it at . Can someone explain?
• Let's consider the function:
𝑓(𝑥) = [1 + (1/𝑥)]ˣ
If we plug in larger and larger values of 𝑥, we see that the value of 𝑓(𝑥) seems to be getting closer and closer to a certain value around 2.71.... It turns out, when we use an infinitely large value for 𝑥, we get the exact value of 𝑒. More succinctly, we can say that the limit of 𝑓(𝑥) as 𝑥 tends to ∞ is 𝑒. Essentially, the limit helps us find the value of a function 𝑓(𝑥) as 𝑥 gets closer and closer to some value. You will learn more about limits and a more rigorous definition later in Precalculus and Calculus.
• Will this work if n is equal to a very large number i.e 999,999,999,999,999 because i tested it and the results were more than 3 so is there like a limit for n?
• Interestingly Google's "hidden" calculator does the same thing. I'm pretty sure this is a result of the limited precision with which floating point numbers are stored.

In other words, since computer chips can't store an infinite number of digits (that would take an infinite amount of of memory) numbers for intermediate steps get approximated and this can lead to inaccuracies like the one you noticed.

So, yes this works better for larger numbers, but only if you keep all of the information!

http://stackoverflow.com/questions/2100490/floating-point-inaccuracy-examples
• Sal references another video or videos at . What are those videos?
• It might be covered in some video later in this sequence of logarithm-related videos, though I doubt it, but it is mentioned in Vi Hart's pi videos under Recreational Math -> Doodling in math class.
• Instead of using 100% in (1+100%/n)^n can we take any other interest rates? Will that reach e as well?
• If you use a number other than 100% for the infinite series you won't get `e`, no.
• are there any other magical numbers other than pi e and i
• Is e irrational? What to do with e in logarithms?
• So what actually IS e? He says what pi represents but never explains what e represents. All he said was that is was some cool number. plz help
• Most of the more fundamental properties of e are rooted in calculus.

The main one is this: consider the set of functions a^x, where a is some positive constant. So 2^x, 5^x, π^x, etc.

If you take a point on one of these functions (x, a^x) and draw the tangent line to the function there (that is, the line that touches the curve at the chosen point and nowhere else), the slope of the line will be ln(a)·a^x.

If we take the function e^x, then, the slope of the tangent line is ln(e)·e^x, or just e^x. The value of the function is equal to the slope of its tangent line. And e is the only choice of number that will cause this.
• How do I calculate the natural log of a number using a scientific calculator that doesn't have an "ln" button? E has convinced my brain to retire at the ripe old age of 17. Thanks, e.
• It's tough doing natural log without a ln button, but is definitely possible. I'm assuming that your calculator has a normal "log" and "e" button.

lets say we want to calculate the natural log of 6.
We could rewrite the natural log as an exponential equation.
In this case the log turns into e^x = 6.
We take the normal log (base 10) of both sides:
In doing this, we strip x off of being an exponent and instead multiply it with e.

e^x = 6
x*(log e) = log (6)
We divide by log e on both sides, and our x is whatever's left on the right side after doing some calculations.
We could also apply base of change and our answer becomes log_e 6. This is not surprising, since this is exactly equal to ln 6.
Hope this helps, it's my first time posting on Khan!