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### Exponential Decay

```Date: 04/28/2003 at 20:07:43
From: Sylvia
Subject: Percentage

The English language evolves naturally in such a way that 77% of all
words disappear (or are replaced) every 1000 years.  Of the basic
list of words used by Chaucer in 1400 A.D., what percentage should we
expect to find still in use today?

```

```
Date: 05/01/2003 at 09:55:27
From: Doctor Ian
Subject: Re: Percentage

Hi Sylvia,

This isn't really a percentage problem.  It's an exponential decay
problem.

So now you know less than you did before, right? :^D

Suppose there were N words in English back in 1400 AD. Then if the
theory is correct, we'd expect to find those words disappearing as
follows:

Year   Remaining
----   -------------
1400   N
2400   (0.23) * N      Because if 77% disappear, 23% remain
3400   (0.23)^2 * N    Now there are just 23% of the previous 23%.
4400   (0.23)^3 * N    And so on.

So this corresponds to an equation:

[(year - 1400)/1000]
fraction remaining = 0.23

For a quick check, substitute 2400, 3400, and 4400 for 'year' and make
sure you get what's in the table. The nice thing about the equation is
that we can use it for _any_ year, not just years that happen to be a
multiple of 1000 years from 1400.

Now we're ready to find out how many of those words are around today.
The year is about 2000, so

[(2000 - 1400)/1000]
fraction remaining = 0.23

[0.6]
= 0.23

[6/10]
= 0.23

6 (1/10)
= (0.23 )

To evaluate this, raise 0.23 to the 6th power, and then take the 10th
root of that.

Does this help?

- Doctor Ian, The Math Forum
http://mathforum.org/dr.math/
```
Associated Topics:
High School Exponents

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