# What is e

##### Posted in Math

## 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

- initalValue = 2
- returnRate = 1
- t = 3
- growth = 16

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

- Everything same as above except returnRate is now 0.5

## 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:

- An inital value of 1
- 100% return rate (return rate = 1)
- per ever 1 time unit (t = 1)
- Compounded n timers per time unit

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**.

## Definitions

- Compound Growth: Applying an amount if interest, based on the current amount, at
**set time intervals** - Continuous Growth: Applying an amount of interest, based on the current amount,
**at the smallest time intervals possible**i.e. continuously.