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I’ve been extraordinarily fortunate to have had two math teachers in my high school (I’m currently a senior in AP Calc BC) who teach exactly the same way, since for high school students our ability to conceptualize hasn’t completely developed yet. I have to say, that I’ve never learned more in my life on any subject in so few words and so little time. I think that your approach on intuition is the only way that anything (that mathematicians think they completely understand) should be taught when introducing new/abstract topics.
One of my math teachers has a mantra that sums it up perfectly:
“My job,” he says, “is to make the new stuff look like the old stuff.”
Before he blows our minds with some new info that leaves jaws on the floor, he states his motto, and he follows up with “So let’s start with what we already know/have defined…”
That’s teaching. Thank you for blowing my mind.
@Colby: Wow, thank you for the comment! I completely agree with that mantra of “making the new stuff look like the old stuff”. You really need to start with the previous ideas (since that’s how most new ideas get developed – variations / combinations of existing ideas) and work your way up. If done correctly, the new ideas almost seem “obvious” or “inevitable” – which is great, because it means they make sense! Thanks again for the note.
[…] brainful – take a break if you need it. Hopefully, sine is emerging as its own pattern. Now let's develop our intuition by seeing how common definitions of sine […]
[…] exploring Circle images on the World Wide Web, I found this webpage. “Our initial exposure to an idea shapes our intuition”. The article goes on to […]
I am a woman and I love the way you explained e. But I am confused on the derivative definition. I understand that e represents the fastest continuous 100% growth (at the end of an interest period). The derivative definition appears to be an “snapshot” which says that my current amount is equal to my growth rate (100%) at this instance in time. My problem is that I don’t know how one would arrive at “e” (the value at the end of an interest period) from an instaneous snapshot of a single point in time. In other words, how did this define “e”? Could you help me understand or correct my misperceptions?
Marisha
@Marisha: Great question. The tricky thing about the derivative definition is it describes properties of e without saying what number it is! It’s like telling someone “The number I’m thinking of is 3 more than half its value”. Technically, you’ve described what number you’re thinking of (x = 3 + x/2, so x = 6) but you didn’t make it easy!
The derivative definition (d/dx e^x = e^x) is saying “Your rate of change at any time x (d/dx e^x), is exactly equal to your current amount at time x (e^x).” That is, your instantaneous growth rate (per unit time) is 100% of your current amount. If you’re at 10, you expect to grow 10 in the next unit of time (but as you hit 11, you expect to grow 11 in a unit of time… and as you hit 12, you expect to grow 12 in a unit of time…).
This isn’t much to work with – we need to find some exponential function that equals its own growth curve. Trial and error it is.
We can say “Hrm, maybe e is 2” and see – does d/dx 2^x = 2^x? It doesn’t… d/dx 2^x, the rate of change, grows too slowly (see graph). d/dx 2^x is only about 70% of 2^x, not equal.
Ok, how about 3^x? Whoops, that grows a smidgen too fast! d/dx 3^x is about 110% of 3^x (graph). But now we can try 2.5 (too slow), 2.7 (really close, 99%)… 2.71 (even closer!)… until we get a guess for “e” is satisfyingly close to 100%. (By the way, e goes on forever without repeating, so we don’t really know the exact value… we just have a really good guess).
There are other ways of calculating e, even formulas for it (that have an infinite number of steps, but the more steps you take the closer you get). The derivative definition sort of forces you to use trial and error to guess what number e might be. Hope this helps!
i always thought how Euler ,Newton got their formula …is that their intuition or their hard work …they have solved such problem that we can see only through modern days computers…
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Hi, I totally agree with you kalid. In school they just go straight to the hard complicated bit without mentioning why e.g. x is x and what happens when there is 2 in front of it. Thank you for these posts and also can you please tell me how to do/find the discriminant, completing the square, solving and sketching graphs. Thanks.
I LOVE THIS
@Siddartha: Thanks :).
Hey Kalid,
Great website!! Just a quick question…with regard to your interesting explanation of e’s Taylor series, could you maybe explain to me the math as to why the interest of the interest is 1/2! and the interest of the interest of the interest is 1/3! and so on and so forth? I see the principal value and the direct interest in the series, but after that, I’m a little lost as to why those next terms are the mathematical expressions of the subsequent interests…
Thanks so much!!
Sincerely,
Stephen
This is so sexy! Beautiful!
Thanks, Kalid! Makes perfect sense…please keep up the great work
@Stephen: Great question. The reason is we’re constantly computing integrals to find the interest, interest that the interest earned, etc. Breaking it down:
- Original value: 1 (this is our starting amount, and can be written 1/0! = 1)
- First-level interest: 1 (this is our basic 100% return, and can be written 1/1! = 1)
- Second-level interest: 1/2! (this is the interest on our 100% return, which is the integral of “x”: 1/2 * x^2 where x = 1)
- Third-level interest: 1/3! (interest on our second-level interest (1/2 x^2) is 1/3 * 1/2 x^3 = 1/6 x^3 = 1/6)
and so on. Basically, e^x can be seen as
1 + x + x^2/2! + x^3/3! + x^4/4! + …
and each term is computing the interest on the term before it by taking its integral. We plug in x = 1 to get e^1 = e. Hope this helps! (I should write a follow up on this, it’s a good point). Also, see the article on sine to see another example of this “interest on interest” pattern: http://betterexplained.com/articles/intuitive-understanding-of-sine-waves/
Thanks Mehdi – appreciate the suggestions. I’ll put them on my list :).
So we start in a corner of a circle? You’ve lost me already 
[…] which means, in English, “e changes by 100% of its current amount” (read more). […]