In this video, we build Euler’s number **e** from first principles, starting with compound interest and continuous growth. Then we see why the exponential function (e^x) is unique, why its derivative equals itself, and why the same mathematical structure keeps appearing across seemingly unrelated problems.
By the end, you’ll see why **π naturally belongs to circles, while e naturally belongs to change itself.**
### Chapters
00:00 — Why Does e Appear Everywhere?
00:27 — Building e From Compound Interest
01:20 — Continuous Compounding
01:35 — The Limit That Creates e
01:44 — Why e Represents Continuous Growth
01:55 — Population Growth & Calculus
02:18 — Why the Derivative of eˣ Is eˣ
02:57 — Exponential Decay
03:29 — e in Probability
04:03 — e in Statistics & the Bell Curve
04:29 — Euler’s Identity: eⁱπ + 1 = 0
04:49 — Why e Really Appears Everywhere
05:10 — e Is the Number of Change
#Mathematics #EulerNumber #Calculus #MathExplained #ExponentialGrowth #Probability
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I hope that mathematical of continuous change always lead to the Mathematics Normal Naturale!
Now you know Influencial Matters previously given with Gigantic Space Motherships, you can easily be vanished in time.
good information, I liked it, thank you
How do u know
What i wrote
E=p=i=1=0
The simple explanation for non-mathematical people is that any process which grows continuously at a steady rate in proportion to the total amount (not the starting amount) will have an exponential element to it. It doesn't mean fast or huge, it means steady and continuous! So even if the rate is small, say a population growing at a rate of 1% per year, the growth will be exponential.
Well said !
If Pi is the number behind area/volume (3 dimensional space) and e is the number behind growth/decay/change then isnt Pi multiplied by e (8.539734….) the number behind the reality of universe through time?
What bank pays 100%?
In the real world it is multiplicative growth and decay that appears everywhere, not the number e, which is just one choice of "base", one that appeals to mathematicians. In radioactivity and drug decay the base chosen is 0.5, it is half-life that is quoted, not 1/e life. Sadly, no bank has ever multiplied my savings by e to any power, the base used is (1+r), where r is the interest rate. [e may appear in banking, but only in the software that does numerical computation of (1+r)^N].
e does feature heavily in mathematics (and even in electronic engineering), so if you live in the world of mathematics it may be fair to say that e is everywhere.
Awesome explanation! This excelent video unlocked the exact perspective I was looking for. TKS!
It doesn’t. You are too well trained to see it everywhere.
Thanks for posting this video. A nice summary of how e appears across Differentiation, Integration and Probability.
Thanks. I found that fascinating even though I didn't fully understand it all. 👍
Let me know what bank is this that pays 100% interest. All of the ones I know of pay close to nothing. Second, e is not a number. It is a letter. They teach you that in elementary school. And thirdly, it appears everywhere because people keep writing it everywhere. Tell them to stop if it bothers you.
I hear that pi also keeps popping up, as well as the Fibonacci sequence 😊
I changed schools often during adolecsence and teen years. Your videos are refreshing lost knowledge for me, and filling gaps. Thank you for your clear, concise visual reminders.
I've watched several videos trying to understand where e comes from and why it exists in the first place. Inevitably, some of the explanations were familiar, but you also offered fresh perspectives and connections I hadn't come across before. I can't say I fully understand it yet, but I definitely feel I've taken another step forward. Thanks for that!