Monthly Archives: February 2017

It is hard to say, because the description of [math] pi [/math] predates surviving written records. However, my money is on taxes—ancient agrarian societies with central governments needed some way of fairly calculating taxes, and if you are taxed proportionally to how much land you have, it behooves you to have some reasonably accurate method of approximating the area of rectangles, triangles, and circles.

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This is a tardigrade:

Tardigrades came about during the cambrian explosion (a rapid diversification of species in the cambrian period), and have not only survived 5 (FIVE!!!!!) mass extinctions, but thrived during them. There are hundreds of distinct species of tardigrades, living on every continent and in every ocean, surviving in every climate.

The reason tardigrades are so tough is that they will dry out and exist in a desiccated state, it's metabolic rate dropping to 0.1% of normal. The creature's natural lifespan is less than a year, but species can survive decades under a rock in the desert or frozen in ice in the antarctic, get hit with a raindrop, and be up and copulating in the amount of time it takes you to put on socks. We slapped some on the outside of a spaceship for 10 minutes, and 68% of them survived (that's a lethal dose of cold and a lethal dose of radiation and a lethal dose of nothing to breath for pretty much everything else living).

The reason I'm droning on about this microscopic marvel is that the only advantage humans have over a tardigrade is a fairly impressive brain. We have done a lot with that single evolutionary trait, but it really isn't as unique as we like to make it out to be. Other species use tools, other species have communities, and other species have language. Being educated isn't evolution. Given enough time another species could very well be the next set of scribes and builders, and humans could be extinct. Tardigrades will still be thriving, though, and that's the goal of a species. They won evolution before the dinosaurs even came about.

Edit: It is wise to note that every species is equally evolved. It's not a process with an end goal in mind, it's ongoing and random. Traits can come and go, some are beneficial, some are not, some are specialized, others are pretty general. Species that survive something like a mass extinction tend to be highly flexible in the biomes they can live in, and it's a trait we see in many species across the world today. Those species that can survive the largest range of calamities have the best chances at survival, and that's probably the highest evolutionary trait.

Can you imagine how many species of single celled organisms died off before the first multicellular organism came into being? One of them simply had to have enough survivability to evolve into something vastly different…

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If a solar storm is powerful enough to damage devices in your house, then your devices are the least of your problems. Not being vaporized, or at least not dying a long, slow, lingering death of cancer, is your problem.

Solar storms affect electronics in space. If you were in space, even under ordinary circumstances, you need a fair bit of protection. The storms that damage satellites would kill human beings. The ISS requires a lot more than a sheet:

Space Radiation Analysis Group

On earth, tens of miles of atmosphere provide plenty of protection, even against solar storms. A solar storm that caused damage to devices in the atmosphere, again, would be deadly to you. A sheet on top of your house would be as effective as wrapping aluminum foil around your head.

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I think most giant dinosaurs should be extinct at that time, so my survival would be for more than 10 minutes!

So, if I can bring a few tools I'll make sure to bring a knife.

Now, I need to prove time travel occured, but not necessarily that I was there, which makes it easy.

I'll need a bottle of beer.

Yep, one green glass of Beirut Beer should last long enough before decomposing to be discovered by modern people, taking into consideration that it won't break to dust by animals or other things.

Now if I also had a really REALLY permanent marker, I could write up part of my glorious story.

Now, having enjoyed my beer and using my knife as the ultimate and most high-tech weapon around, I could live up to my 40's without getting killed.

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I wondered the same thing when I first saw that. After all, it's a number we can calculate, right? And once you calculate a number, it should be easy to check if it's an integer! So I DID THE MATH[math]^{TM}[/math]

[math]f(x)=^4 x = x^{x^{x^x}}[/math]

[math]f'(x)=^4x*^3x*(1/x+ln(x)*x^x*(1/x+ln(x)(ln(x)+1)))[/math]

That was not a fun derivative to take, but it can be done inductively.

The point is that

[math]f(pi)>10^{18}[/math]

[math]f'(pi) > 10^{10^{17.8}}[/math]

and note that [math]f''(pi) > 0[/math] (in fact, its obscenely huge, but all I need to make my point is that it's greater than 0)

Moreover, all derivatives are positive (and growing rapidly)

Whenever we use [math]pi[/math], we are actually using an approximation, [math]pi+epsilon[/math].

[math]f(pi+epsilon) = sum_{n=0}^infty f^{(n)}(pi) frac{epsilon^n}{n!}[/math]

So

[math]f(pi+epsilon) > f(pi) + f'(pi)*epsilon[/math]

The error in computing [math]f(pi)[/math] is then

[math]f(pi+epsilon) - f(pi) > f'(pi)*epsilon[/math]

Substitute our lower bound for [math]f'(pi)[/math] to get

[math]f(pi+epsilon) - f(pi) > 10^{10^{17.8}}*epsilon[/math]

[math]10^{10^{17.8}}[/math] IS HUGE! It's not a googolplex, but it's pretty massive.

To be sure [math]f(pi)[/math] is not an integer, we need the error to be less than the difference between [math]f(pi)[/math] and the next integer. Let's be absurdly generous and assume we can have an error of 1

What we derived is actually a lower bound on the error, so we need to make sure that's less than 1.

[math]10^{10^{17.8}}*epsilon<1 implies epsilon < 10^{-10^{17.8}}[/math]

That means we would need to know [math]pi[/math] to AT THE MINIMUM [math]10^{17.8}[/math] digits.

That is about a trillion times as many digits as mankind has successfully calculated. And keep in mind that it gets more and more computationally expensive to compute the next digit as you go.

Taking a higher order approximation would show that we actually need a lot more than that.

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