Euler's Number

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e



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Basic Description

Bacteria (or humans) multiplying, radioactive uranium decaying, or bank accounts swelling as a result of accruing interest, are all similar phenomena in a certain fundamental way. The rate at which a bacteria population is increasing at any moment is directly proportional to population at that moment, and totally analogous statements are true about the rate of uranium decay and the rate at which a bank account balance increases. The term exponential growth is used to describe scenarios like these.

There is, however, a large problem with the seemingly innocuous phrase "at any moment" from the previous description. If a bank account is accruing interest at every moment, how does the balance not become infinite fairly quickly? Also, the population of a bacteria colony is probably not, increasing at every moment, because there are almost certainly moments when none of the bacteria in the population are multiplying.

The number  e , which like  \pi has a specific value, but not one that is easily written using normal decimals, comes from taking a closer look at the apparent paradox in a bank account constantly accruing interest. The observations that come from studying this problem can also be applied to studying situations like a multiplying bacteria colony and radioactive decay, where change is not happening constantly, but pretending that it does gives a good approximation of reality.

A More Mathematical Explanation

Suppose a bank account starts with a balance of  A dollars and [...]

Suppose a bank account starts with a balance of  A dollars and that it earns an interest rate of a whopping 100%. This does not, counter-intuitively, necessarily mean that the account will accumulate exactly  A dollars in interest over the course of the year. Instead, the amount of interest will depend on how often that interest is compounded, as is illustrated in the examples below.

If interest is paid 1 time, at the end of the year

In this scenario, when the end of the year arrives, the account will receive 100% of its current balance, $A , in interest. This will give the account a balance of $2A at the end of the year.

If interest is paid 2 times throughout the year

In this scenario, the account will first receive interest after 6 months. Because only half a year has elapsed, the account only receives 50% of its current balance, A, in interest. This will give the account a balance of 1.5A.

Six months later, the account will receive interest again. Because only half a year has elapsed since the last interest payment, the account only receives 50% of its current balance in interest. The current balance, however, is not A, but 1.5A, so the balance will be 1.5(1.5A) = 2.25A at the end of a full year.

This is equivalent to saying that the balance at the end of the year is 2.25 times the balance at the end of the year.

If interest is paid 3 times throughout the year

In this scenario, the account will first receive interest after 4 months. The account will only receive 1/3 of its current balance, A, in interest, giving the account a balance of 4/3A.

The account will receive interest again after another 4 months. It will receive 1/3 of its current balance, which is not A, but 4/3A. This will give it a balance of 4/3(4/3A) = (4/3)^2A.

The account will receive interest one final time at the end of the year. It will again receive 1/3 of its current balance, which is now (4/3)^2A, giving the account a final balance at the end of the year of (4/3)^3A.

Because (4/3)^3 is roughly equal to 2.37, the balance at the end of the year is roughly 2.37 times the balance at the beginning of the year.

If interest is paid n times throughout the year =

The number  e can be defined in several different ways, all of which are equivalent, but none of which make




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