@Rob: Awesome, very glad it helped!
Hi Kalid: thank for doing this. I’m a 73 year old geezer but still interested in learning. I took 3 semesters of calculus (didn’t get good, or even fair, grades) and I don’t think I ever learned where ‘e’ came from! Fortunately I made my living working for the telephone company, mostly finding the broken wire or bad connection or low electron tube or all the other things that went wrong with the system until we made it all bits and fiber and they didn’t need me anymore. how the technical world has changed!. Just love your articles, wish I had been able to learn a lot of this stuff when I was a lot younger but now it’s mostly “entertainment” but I will try to “inflict” the knowledge on my grandchildren. thanks! elliot
great
what is e^ infinity, e^ - infinity, same with -e
pl
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Hi Kalid!
I loved the article. I am a biologist but I am interested in maths. I have a question about the e number. Why this expression, e^-rt, can be simplified to this other (1-r)^t ? Thanks for this so comprehensive article!
Hi Marc, I really appreciate the comment. I’m with you – math can be so boring and dry unless presented properly, I’m happy if this was able to help :).
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Hi Javier, check out http://betterexplained.com/articles/developing-your-intuition-for-math/ for some more on e :).
Hi Peter, thanks for the note. Great point – I think the key is “your change depends on how much you have”. With a barrel and pressure, the pressure depends on the remaining water. Or with a population of deer, the more there are, the higher the (absolute) number of deer each year (even if it’s just 10% more or whatnot – the actual number each year keeps increasing). Appreciate the note about what works, I’ll have to think about how to work it in.
Thank you sooooo much!! You are such a great teacher!
Excellent work Kalid to explain the concepts in simple pedagogic way. We need articles like this to make all the concepts simpler and appealing to everyone.
Hi kHlaid,
In your statement,
“Mathematically, if we have x splits then we get 2^x times more “stuff” than when we started. With 1 split we have 2^1 or 2 times more. With 4 splits we have 2^4 = 16 times more.”
I understand that you say 2^x because we are doubling ?
Assuming we are doubling,
1 becomes 2 i.e 1 split gives 2^1 = 2 (2 times more or 100% growth)
2 becomes 4 i.e 2 split gives 2^2 = 4 (4 times more or 300% growth)
this ‘x’ times more … is it computed with respect to the original value of 1?
because when 4 becomes 8 i.e 4 splits gives 8 => 2^4 = 16 (16 times more or 1500% growth)
Assuming even this is computed w.r.t the original value of 1.
When 1 becomes 8, its just 700% growth and doesn’t tally ! 
Am I missing something ?
Regards,
Joe
Thanks Victor, that’s a typo, fixing it up now.
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Hi Harmony, really glad it helped!
Comment from Dr. J which was eaten by the spam filter:
Posting again to get LaTeX to display the equation:
I like this website, but I’m questioning your use of the word “growth.” Doesn’t growth represent a change or gain, i.e., the final amount less the initial amount?
It seems that when you use the word “growth” in your equations, you really mean “total amount” or “balance.”
Consider the example of semiannual compounding at 100% annual interest in the “Money Changes Everything” section. After one year, a $1 investment will be worth $2.25. The balance is $2.25, but the growth is $1.25 = $2.25 (final amount) – $1.00 (initial investment). So
$$\text{total amount} = (1 + 100%/2)^2 = 2.25$$
Please tell me if I am misunderstanding your terminology. Thanks.
Great point, I need to clarify. e is indeed the total, not the change. I was thinking of starting at 1 and multiplying in the “growth factor” of e (so going from 1.0 to 2.718) but that may not be clear. I’ll have to revisit the terminology used to clarify.
Thanks Keerthi!