The MOST FAMOUS number in MATH explained WITHOUT Calculus! | Math for ALL
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- Опубліковано 5 лют 2025
- The number e is often taught in advanced math to students who are studying calculus, since its technical definition involves limits. However, its technical definition is uninspiring and it's not clear what the motivation behind it is. In fact, many people who work with e (including mathematicians) may not be aware of why its original founder, Bernoulli, discovered this number. In this video, we will explain e in a way that is accessible to a fifth grader, or indeed anyone who is familiar with the concept of percentages and compound interest. I will phrase it as a problem of how to maximize our revenue by choosing "how many times" to compound our interest over a year - watch the video to understand this simply but in depth! We will also briefly talk about and explain the most famous equation in math (Euler's equation) at the end of the video: e^{π√(-1)} + 1 = 0.
The concept in this video is extremely fundamental throughout all levels of math including high school math, college math, precalculus, calculus, and math beyond calculus. After watching this short video, you will have an intuitive as well as a rigorous grasp of what the number e really represents. The foundation here will prepare you to do well on standardized tests for USA College Admissions like the SAT and ACT, AP Calculus, and homework, assignments, tests and exams in middle school, high school and college math!
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Happy Christmas and Holidays to you and your family!!! 🥳🎊🎉 I hope you have a wonderful time as the year comes to a close and I am wishing you all the best! 😊
Loved the explanation. I never knew the explanation before. You explained the number e so beautifully
The way you give the proof or give the explanation is so beautiful . I am sure in a few months millions of students will start watching your videos .
Merry Christmas to you . Wish you will have millions of subscribers soon
Thank you so much!!! 😊 I am so happy to read your comments! 😊 Happy Christmas!!! 🥳🎊🎉
It's a pleasure to see the clarity with which you illustrate the matter. This gave me a wonderful insight. To dive even deeper into the topic, I would find the actual proof interesting.
Hi @alex_be9698 thank you so much for your kind comment and feedback! 😊 I am so happy to hear that you enjoyed the video! 🥳 Yes, I would love to do a sequel video that you would find interesting to watch! Which statement are you referring to that you would like to see an actual proof of? (I think I mentioned at least a couple of statements at the end of the video about e. However, the video itself can be taken as the definition of e, so doesn't require proof, but there are other definitions of e equivalent to this one, and that equivalence can be proven (if you have another definition of e in mind).) I hope you have an amazing day/evening/night and Happy New Year!!! 🥳🎉🎊
@@MathMasterywithAmitesh First of all: The derivation of e was already mindblowing for me as an uninformed amateur 😄And yes, in addition, after your short "teaser" at the end of the video, I'm looking forward to a derivation or proof of the euler identity. You see: Your good explanations make us viewers just want even more of it. 😃
Hi @@alex_be9698 ! Thank you so much for your very kind comment! 😊 I see, I wanted to make sure that I understood what proof you were referring to :) I will definitely do a followup video on Euler's identity! 😊 I am so happy you enjoyed this video and that you are interested to learn more about e! I wish you an amazing weekend! 😊
@@MathMasterywithAmitesh Great. I'm really looking forward to your next videos. 👍 It's amazing that you as an excellent mathematician share inspiring insights with all of us. Thanks a lot. Have a great weekend, too. 🙂
hey amitesh, if you had to pick one specific number, which one would you say is the most important (i.e, which one's discovery advanced the field of mathematics the most)? You do mention e being at least one of the most important but personally which would you say holds #1? I've seen some people say 0, others e and some even the imaginary number i (which is rare, but still), I'd be curious to see your answer. Merry Christmas!
Hi @justinferland6129 yes, that is a great question and one that I thought about recently! 😊 The most logical (and boring answer) is they are all equally important, but here is a more interesting one: 😅
If you are evaluating importance by "which number advanced the field of math the most", then I think 0 and 1 are ruled out (because they were known for a very long time before calculus was invented). I think π is very famous, but you can develop the foundations of precalculus/calculus without it I think (although, of course, it does appear a lot and is closely related to many concepts in calculus). You just need to do everything in terms of degrees, rather than radians, and avoid taking perimeters of circles/circular arcs (which comes up relatively infrequently).
The question becomes more tricky when considering e vs i = √(-1). If you combine them, then you can do everything, since you can define e^z for complex numbers z, and then you can derive sin(x) and cos(x) using Euler's formula e^{ix} = cos(x) + isin(x). Of course, now π arises as the smallest positive number x such that e^{ix} + 1 = 0 etc. In this sense, e^x is the most important function in math.
However, it's a tricky comparison of e vs i = √(-1). With e, we get introduced to fundamental ideas of sequences and series (e also has a famous series definition e = 1 + 1/2! + 1/3! + 1/4! + ...), and e^x, which is a fundamental function in calculus and the theory of differential equations. With i, we get introduced to the complex numbers and the concept of being able to solve *any* polynomial equation, which is really crucial (even early like making sense of the quadratic formula, or later in linear algebra, for example, with finding eigenvalues etc.).
I think I would have to say i is more important because it already plays a central role in precalculus before e, and polynomials are extremely fundamental in math. I think math really started to advance as well in terms of the development of abstract ideas, once mathematicians became comfortable with idea of the square root of a negative number (and complex numbers in general). However, it's close and I was leaning toward e when I first read your comment! 😊 What do you think? I am curious to know your thoughts too! 😊
I hope you have an amazing day/evening/night! I wish you a very happy Christmas and happy holiday season!!! 🥳🎉🎊