If someone asked you what the number "e" is, and they didn't know anything about calculus, how would you explain the meaning and significance of this marvelous number? Here was how I approached the challenge in my book The Joy of x.
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In co-ordinates you need dx/dt = -y and dy/dt = x (a little picture shows this - it's just the tangent to the unit circle). And from there everything else follows pretty quickly...
The way I’ve done it (to econ students with no prior calculus) is in graphical steps:
1. geometric progression b^n, with various bases b
2. exponential b^x as interpolation of GP, graphing various simple bases b(>0)
3. observe there’s a unique b: 2
4. given which …
As usual, so beautiful explanations! I love the composition of the two last examples, involving distributions in space and time! I am looking forward to get the Joy of x in 2025!
I once had police pull up to examine my suspicious group of ballroom dancing hooligans. Upon searching my car, they found my Bible at the time: "The Joy of C" (a programming book). The officer angrily asked "What is C?", like it's some drug or something! The title of your book could cause mayhem😂
I read your book when I was in the 8th grade and it really changed the way I approached my math classes and now in college I’m studying statistics and data visualizations.
I'm not sure if it will be in-depth enough for you. The chapters are all short and light (like the one you read) and some readers feel that they want more. If so, you might prefer my book Infinite Powers.
The continuously compounded interest example is both standard and convincing enough.
The sequential dating example gives some "nerds have weird ideas about real life" vibes, though. You could illustrate the same underlying mathematical problem with a different real-life scenario.
The king will interview the N best painters in the country, and select one to commission a portrait of his beloved wife, the queen, to be delivered on her birthday.
Painters are very proud, though, and they will never offer their services to anyone who's rejected them before. Therefore...
I had success once telling someone that, in the same way as zero is a handhold for addition and subtraction, and one is for multiplication and division, e serves the same purpose for differentiation and integration. It's unhelpful technically, but informative for intuition.
I was just going to respond with my own favorite way of explaining it, and then I realized that it’s actually something that I learned from one of your books.
My attempt: It's an irrational number between 2 and 3 that was discovered and comes up naturally when studying different important scenarios about how the world works.
Just finished reading your book because of this post. I enjoyed it. It was relatable and digestible. I can't say I fully got all the concepts but that didn't diminish my enjoyment.
Oof. I struggled to understand algebra, never mind calculus. In trying to help me, my Dad started by saying, "OK. X is a number....." I interrupted him with, "Oh, no you don't! X is a letter and you can't change it!"
I dare not ask about 'e'.
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I once decided to explain e (and Euler's formula) at a party of non-mathematicians. We were asked to do 10 mins show and tell
I certainly didn't say it this way, but the underlying idea was to ask what generates translations around the unit circle
1. geometric progression b^n, with various bases b
2. exponential b^x as interpolation of GP, graphing various simple bases b(>0)
3. observe there’s a unique b: 2
The excerpt was just advanced enough to feel real and hold my interest, but just dumbed down enough for me to not get utterly left behind.
https://www.goodreads.com/book/show/714380.The_Discoverers
Long ago, I thought math was my future but pure math kicked my butt.
I still love watching things like 3Blue1Brown. Brought the kids to the NY Museum of Math years ago.
Math is beautiful :)
https://youtu.be/m2MIpDrF7Es?si=ZrZJKAEtQ1IYZTJ7
The sequential dating example gives some "nerds have weird ideas about real life" vibes, though. You could illustrate the same underlying mathematical problem with a different real-life scenario.
Painters are very proud, though, and they will never offer their services to anyone who's rejected them before. Therefore...
It’s like Bueller, but SOMEONE is there…
That’s what I would go with
I dare not ask about 'e'.