Thiago P.

What is e

Posted in Math

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Basic Compound Growth

To model growh in a system that doubles every time unit and that starts with a value of 1, one could use:

growth = 2^t where t is how many time units to continue growth.

 

The above function can be transformed in to a more generic function

growth = initialValue * (1 + returnRate)^t

 

If we have 2 marbles, and for 3 days, we double the amount of marbles we have

If we have 2 marbles, and for 3 days, we gain 50% more marbles

 

Compounded n timers per time unit

In the above model, interest/growth is compounded at fixed timer intervals. It is possible to compound interest multiple times within a time interval.

For example, if we start with $100 and gain 100% interest every year, we could instead gain 50% interest every half year, or 25% interest every 4th of a year and so on. This will result in larger gains than just simple compound growth.

The equation for this is

growth = initialValue * (1 + returnRate/n)^(n*t) where n is how many times to split the time interval to.

 

Continuous growth e

Given the fallowing:

As you increase the number of times to siplit the time unit (i.e. increase n), the output values will get closer and closer to the constant e.

In other words, e is the result of compounding 100% on smaller and smaller time intervals.

If we start with $1 and gain 100% interest every year. for 1 year We could divide the year into 1000 parts (n = 1000) and the final result would be about e.

growth = 1 * (1+1/1000)^(1000 * 1)

growth = (1/1000)^1000

If n were 1 million, or 1 trillion, the final result would still be about e.

 

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