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
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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
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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)
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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...
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the above example is 100 * 1.083 ^ 12 which is 260 instead of 100 -> 200 you get 260 with daily interest the daily rate is 100/365 = 0.2739 so: 100 * 1.002739 ^ 365 = 271.38... which is pretty close to the 2.718.... multiplier
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u can calculate to infinite precision because technically your dollar is made up of pennies and those pennies can be subdivided into milli-pennies that can start compounding themselves and those into micro pennies etc... but in practice no one does this
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natural things however DO do this a plant w/ 1 square centimeter of surface area "1 dollar" actually contains millions of cells. it's like 1 dollar divided into millions of pieces, each reproducing itself this is why e growth is "natural"
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the "natural" inclination is self replication and that replicant contains the ability to self replicate as well. so natural growth is e^x at a systems level even though at the micro level it is 2^x
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also, while the ratio of two fibonnaci numbers converges towards the golden ratio 1.618, the SQUARE (surface area) of the numbers converges towards 2.618 which is prettty close to 2.718 the difference is like cell death or the monthly interest roundinghttps://youtu.be/ahXIMUkSXX0?t=83 …
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the above is just speculation though going back to the 2.718 as *THE* natural growth rate, it also explains why the derivative of e^x is also e^x a derivative basically says "what is the rate of change along this function"
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when you ask the natural growth function "at what rate are you growing at?" it'll reply "i'm just being natural ME!" that is... at any given snapshot of a natural growth situation that snapshot itself wholly contains enough information to calculate the growth rate
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e.g. a snapshot of a plant can tell how fast it'll grow because growth is correlated with volume of cells and the snapshot has that info vs with a snapshot of a car u need a before/after snapshot as well in order to calculate the limit b/c otherwise it's just a static car
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