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kev
14th December 2007, 10:35 AM
Ok, my mouth wrote a check my math skills can't cash, and I am looking for some help.

Background:
In my HS Biology class, we were discussing the idea of genetic variation and its impact on the ultimate diversity of individuals within a species as well as its impact on the overall diversity of life. The next units we will be covering are on evolution, and the one major point I wanted to illustrate with my students is the power of time, coupled with tiny genetic variations, in the overall process of life.

In my experience, the number one problem students (and others) seem to have with comprehending evolution is the idea of tiny change over long periods of time. Too often, students think of genetic mutation as if it is the "X-Men" movie - a small genetic mutation = a super power. Or, some simple life form suddenly gives birth to a vastly different offspring, which is now better suited to its environment.

In my attempt to illustrate the power of time on relatively insignificant amounts of change I used a standard example of investing $. $2000 invested in the stock market, returning an average of 10% per year, over the course of 65 years can turn into more than a million dollars. I use this example because kids like money, and it shows how something relatively small can turn into something big in what SEEMS like a long time (to us.)

I then pointed out how insignificant 60-70 years really is. I posed the question: What if, instead of 70 years, we looked at the change over a million years. If your money doubled every 7 years, how much would you have in a million years?

I know this is not an exact correlation of how genetic variation/evolution works. Rather, my goal was simply to illustrate how something rather insignificant can turn into something incomprehensible when the element of LARGE amounts of time are introduced.

So - What I am looking for:

I don't really care if I even get the exact answer to the problem. But, I am curious if anyone out there could provide the answer in scientific notation or, better yet, come up with how many numbers would be in the answer. $2000, doubling in value every 7 years, for 1 million years.

I jokingly told the kids that if anyone gave me the answer, written out, I would give them 1 extra credit point, for every 7 digits, on their semester test. Well, quite a few of them have spent the last 24 hours trying to come up with the answer. I don't think that any of them can actually get the real number, but many are trying to use scientific notation, or come up with how many numbers are in the answer. I would like to reward them for their efforts for these attempts.

Thanks for any help that might be out there.

Professor Yaffle
14th December 2007, 10:44 AM
Deleted, misread the OP

Madalch
14th December 2007, 11:00 AM
I don't really care if I even get the exact answer to the problem. But, I am curious if anyone out there could provide the answer in scientific notation or, better yet, come up with how many numbers would be in the answer. $2000, doubling in value every 7 years, for 1 million years.
If it doubles every 7 years, it will double 1000000/7 times = 142857 times (with the repeating decimal).

$2000 * 2^142857 will be quite a bit.

142857 multiplied by the log of 2 gives 43004 (and change), so we'll call it 2e+43004 (the two coming from the change).

Multiply by 2000, and we get 4e+43007.

Jimbo07
14th December 2007, 11:07 AM
$2000 invested in the stock market, returning an average of 10% per year, over the course of 65 years can turn into more than a million dollars.

...

I then pointed out how insignificant 60-70 years really is. I posed the question: What if, instead of 70 years, we looked at the change over a million years. If your money doubled every 7 years, how much would you have in a million years?

...

I don't really care if I even get the exact answer to the problem. But, I am curious if anyone out there could provide the answer in scientific notation or, better yet, come up with how many numbers would be in the answer. $2000, doubling in value every 7 years, for 1 million years.



Two approaches:

i) Doubling -> 1000000/7 = 142857.142857... periods. Now consider, 2000 * 2= 4000 and 4000 * 2 = 8000, etc. So, raising 2 to the power of 2 (22) (for 2 periods) * 2000 gives 8000. We already calculated (let's say) 142857 periods, giving 2142857 * 2000! Depending on your calculator, you may have trouble with this calculation ;) = about 3.492e+43007. I don't know the name for that number!

ii) Compound Interest -> P(1+i)n, where P is your principle, i is your rate in decimal form and n is your number of periods. At 10%, I get 2000(1+0.1)1000000 = about 9.686e+41395. Impress them with a really small interest rate: one hundredth of one percent (about what I get on my savings account :D). 2000(1+0.0001)1000000 = about 5.349e+46 (for your reference, 1 billion is 1.0e+9!!)

I think the discrepancy lies in the fact that the doubling thing contains assumptions that I don't remember right now. For those willing to embarrass me, I'm at work right now, and will have to go back and do some checking later.

Madalch
14th December 2007, 11:15 AM
I think the discrepancy lies in the fact that the doubling thing contains assumptions that I don't remember right now. For those willing to embarrass me, I'm at work right now, and will have to go back and do some checking later.

The discrepancy lies in the fact that "doubling every 7 years" and "10% compound interest" aren't exactly equal. Doubling every 7 years exactly would require interest of 10.40895136blahblahblah...

(ETA: Using 10.4089 % interest in your formula gives 2.42e+43007, which is close enough for government work.)

Jimbo07
14th December 2007, 11:18 AM
The discrepancy lies in the fact that "doubling every 7 years" and "10% compound interest" aren't exactly equal. Doubling every 7 years exactly would require interest of 10.40895136blahblahblah...

Thought so, thanks!

Yllanes
14th December 2007, 11:27 AM
The complete result with 10% compound interest, rounded to the nearest cent, is in the attachment (a text file, zipped to meet the size requirements of the forum).

Jimbo07
14th December 2007, 11:44 AM
I found (as an estimate of pi) 3141 in there... anyone else wanna try some numerology?

:D

Madalch
14th December 2007, 11:47 AM
I like this thread- without it, I never would have realized that the calculator function on my computer can handle numbers up to 10^43007. I'm used to calculators that conk out at 10^100.

2000*2^(1000000/7) = 3.8558919e+43007

Garrette
14th December 2007, 12:43 PM
I am neither a biologist nor a mathematician, and I realize the OP is meant to illustrate a principle and not an exactitude. However, I think the 10% per year is too high. Not only is it an excellent financial return, it is a phenomenally fast rate of evolution.

I think your point will be gotten across just as well and more honestly with a much smaller compound interest rate.

shadron
14th December 2007, 12:46 PM
The Excel spreadsheet, among others, has the formula in #4 ii built in. The CRC math tables books have tables of amounts compounded at x interest for y time. Now, all you have to do is impress them with the meanings of small changes to that exponent on the right of the "e".

kev
14th December 2007, 12:58 PM
Garrette - I agree with you. I will certainly use a different example in the future. I used the 10% because it is in the ball park of historically average returns of the market. I did it rather off-hand as simply a quick example - then, well, it kind of took on a life of its own.

I also agree about the 10%/doubling as an evolutionary comparison - definitely not a good choice, because that is not how it works. Anyone have any suggestions/analogies that would be a more accurate representation of genetic change in an evolutionary sense over time? Something that HS kids could relate to. (that is why I used money)

Thanks for the help everyone (especially yllandes for the number) - the kids love it.

On a side not. I just had about 30-40 16 year old kids spend the better part of their free time for the last 24 hours trying to figure this out - using calculators, internet, calling older friends in college as engineers, badgering the math teachers etc, attempting to come up with patterns, etc. Many figured out the "formula" for trying to solve it - just did not have the computing power. Some came up with answers through 1000 yrs and the like, 2 figured out the approximate numer of "numbers" in the answer. Kind of fun to see them trying to grasp the enormity of the answer. I will try to come up with a better example in the future though, before I open my mouth.

Thanks again - this site is better than google when an unknown answer is needed.

Ziggurat
14th December 2007, 01:16 PM
I also agree about the 10%/doubling as an evolutionary comparison - definitely not a good choice, because that is not how it works. Anyone have any suggestions/analogies that would be a more accurate representation of genetic change in an evolutionary sense over time? Something that HS kids could relate to. (that is why I used money)


Well, one thing to consider is renormalizing your time scale. It isn't strictly years which govern evolutionary iterations, it's generations. Humans have generations much longer than a year. Single-celled organisms, however, have generations much shorter than a year - on the order of an hour or so for bacteria, or roughly 104 generations per year. So now if you allow for even 0.001% variation per generation, you can get about 10% variation over a year.