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intuition on e, euler's number 2.718 if u make a copy of urself you start from 1 and end up with 2. if all the copies do this again u end up with 4, 8, 16, and so on. this is exponential growth 2^x

the basis of exponential growth is that the reproduced stuff embeds the FULL growth capacity of that which is being reproduced. there's a no-loss fractal embedding. it's creating a "perfect" copy that is capable of making further perfect copies

every other form of exponential growth is just a multiplier on the base 2 example above e.g. if instead of making one copy of urself u can make 2 copies then you start with 1, then 3, then 9 and so on. 3^x

where the magical 2.718 number comes in is when you leave the discrete world when you make a copy you are creating an instant copy of yourself. u go from from 1 -> 2

however if you have 1000 dollars in a bank account with 100% annual interest given monthly, it doesn't go from 1000 -> 2000 instantly. as soon as u get each month's payment, it can immediately start to grow as well. e.g. (100/12 = 8.3% monthly)

intuitively you will sense that with this compounded growth, you will get MORE than x2 (obviously) but probably less than x3 and that's correct. the more you break down the payments to allow compounding to happen ASAP the more u approach the limit of 2.718...