I got to thinking the other day about how it’s stated that if we ever ecounter aliens, the easiest way for us to communicate with them would be via mathematical formulas, and it occured to me that there might be some problems with that. Obviously, their symbols for numbers and operations will be different than ours, but it also seems likely to me that there’ll be problems figuring out things if they don’t use a base ten numbering system. Yes, I realize that humans do use and think in non-base 10 systems (time, for example, is figured in non-base 10 terms), but what if because of religious/philosophical reasons they base their numerical system such that Pi (or some other fractional number), is not a fractional number in their system, but is, in fact, a “whole” number? I realize that this would skew a lot of things, so what we would consider to be 1 in our system, would be some kind of odd-ball fractional number. How easy would it be for us to wrap our minds around that? Would something like that even be possible?
Pi is a transcendental in any number base. It cannot be represented by a whole number or a rational fraction. You can have a base pi, but that’s not quite the same thing.
Ya know, Exapno, you could’ve warned us about the bad pun at the top of the page of that link.
Naaaah. I enjoyed the heck out of it!
Linky no worky for me, for some reason.
Exapno, I figured that Pi probably wouldn’t be possible to base a number system on, since it goes on forever, but since to the average layman it’s such an “odd” number, I used that, but could any fractional number with a mathematical significance be used as a “whole” base number in a numbering system? It wouldn’t necessarily have to be their version of 1, but could it fill a “slot” like 10 does for us?
In order to understand this, you have to make the distinction between a number and a representation of a number, which is something that a lot of people–even some technically literate people–have trouble with.
So for starters: [symbol]p[/symbol] is transcendental, and as such is irrational. The fact that [symbol]p[/symbol][sub]10[/sub] = 1[sub][symbol]p[/symbol][/sub] doesn’t change the fact that there are no two integers a and b such that [symbol]p[/symbol] = a/b. Irrationality and transcendentality are properties of a number, while termination is a property of a number’s representation. If you’re thinking about different number bases, you have to be able to wrap your head around that difference.
It’s worth noting that no alien race would do this. Pi is an interesting, useful number, but it won’t show up until a culture has numbers and geometry. It simply is not as useful as integers in interacting with the world around you - numbers arose from the need to count and measure and compare things. It’s not useful to be unable to describe how big your herd is, right? “We have a little less than two-thirds pi buffalo” - it’d be so useless for interacting with the world that if folks did get it into their heads to try it, they’d probably give up on numbers.
Besides, like I said, pi would never be discovered until numbers are somewhat well-understood. Especially given that primitive cultures - the kind that discover basic ideas like “numbers” - don’t, most likely, have good concepts of different bases, and the concept of a transcendental number base couldn’t come up.
I guess I don’t get this thought exercise; I’m probably missing the point badly by pointing out that it’s pretty unlikely to happen. But it’s just completely implausible to me.
Well, the Egyptians are thought to have used Pi without ever understanding the concept. Given that we’ve reordered our whole calendar based on a religious event, developed a new system of measurements because a bunch of angry folks decided to kill their king and queen, it doesn’t seem too far fetched to me that a society could do something similar with their numbering system, if a significant event/religion became associated with something like Pi, or some other mathematically unique figure. After all, a group of Greeks got pretty obsessed over certain geometrical shapes as I recall.
Well, I’d think that an alien race that has mastered interplanetary flight would have to have some sort of mathematical knowhow!!!
Yeah but the point was that they would not have a number system based on pi.
Once you’ve got 42 figgured out, what’s left to know? 
A number system based on a non-rational base would be too hard to use. The numbers you’d like to use to count things–like how many dollar bills you’ve got–are non-terminating, non-repeating decimals* in base [symbol]p[/symbol], so all of a sudden finance becomes an inexact science. That’s not good.
*Well, not really decimals, but I don’t know the generic term.
As far as I know the laws of the universe are, well, universal. 1+1=2, pi is pi, (loved the pun also) and hydrogen is what it is. If an alien race has 10, 12, or 15 digits dosen’t change this.
I don’t think the laws of mathematics change anywhere. Indeed, the interpretation of math can be altered to a certain race’s beliefs, and could affect communications between races, but fundamentally math is math. IMO
Beliefs are just that, beliefs.
(I’m sure I’m in way over my head.)
Whoa. Hold on. Who says this?
The concept about pi is that it is the ratio of the circumference of a circle to its diameter. Why would the Egyptians not understand this?
They knew this very well, and at least 4000 years ago they could calculate it by their equivalent of the value 4*(8/9)[sup]2[/sup], which is about 3.16.
You may be trying to say that they didn’t understand that it was an irrational, indeed transcendental, number. But so what? You don’t need to know that to use pi in everyday work.
The rest of your post isn’t a rational argument either, so to speak. The true history of the events you refer to is well known and always involves better, more accurate, and again, more rational, means of calculation.
It’s not quite true that our whole calendar was reordered based on a religious event. The only reordering done to the calendar was to move the Roman new year from March to January and that had little to do with religion. The need to calculate a date for the vernal equinox so that it didn’t keep moving back through the year, so as to correctly calculate Easter, was religiously inspired, but was part of a 2000 year long effort in all cultures, not just Christian, to devise a calendar of a whole number of days when the solar year is an irrational number.
Next, the French Revolution allowed for the creation of the metric system, but the need for a more easily understood measurement system was acute and well known. Jefferson proposed one himself before the Revolution. They used the Revolution as an excuse to sweep out the old nonsense and put a more rational system in place, but this is not mystic at all.
And the Greeks were far too savvy about math to base a number system on irrationals. Just the opposite. They hated the very concept.
No, changes to calendar and systems have historically been inevitably in the direction of ease, simplicity, and rationality. Not the reverse.
My understanding is that they laid out the pyramids using a drum with a stick that would poke the ground once a revolution and they did this before they figured out Pi.
You’re forgetting the adoption of the numbering of years.
I never said it was mystical in that instance, only that it was inspired by a major upheaval. And Jefferson would have had his worked out, but he was having too many financial problems to devote himself fully to the issue.
Just because the Greeks hated the concept doesn’t mean that someone else would.
It’s not so much that the year length, expressed in days, is irrational. I mean it almost certainly is — although the real length fluctuates chaotically over time anyway. If you measure the year with enough precision, you notice the annual discrepancies in the 10th (or whatever) decimal places and beyond.
The real problem occurs before that though. The ratio of the Earth’s tropical year to its day is not only not a whole number, it isn’t even a “simple” rational number with a small integer denominator — even if the actual year length were a fixed rational number having its published value. (The mean tropical year is 365.24218967 days.)
So the approximation of 365 1/4 days gets you fairly close, and is what the Julian calendar essentially assumed. Or, 365 97/400 days gets you even closer, and is the value the Gregorian calendar uses. However, a calendar that achieved a mean length of 365.24218967 days would require a pretty byzantine leap year policy. No one has cared enough about that last little bit of error to propose such a change yet.
To nitpick, π = 10[sub]π[/sub], not 1[sub]π[/sub]. The number 1[sub]b[/sub] = 1[sub]10[/sub] = 1 for any base b.
By definition, a true alien culture would think in ways utterly different than ours. So either we can say nothing about their culture or anything at all about their culture. Neither one, however, is particularly meaningful.
So if you want to argue that your aliens will want to base a numbering system on irrationals I can’t counter that with fact.
But it’s just not how human mathematicians have ever worked, in any culture in any time we know of. Perhaps you should ask a mod to move this to GD so you have more freedom to play with the notion.
BTW, Petr Beckmann, in A History of Pi, goes into Egyptian mathematics and their way of deriving pi without decimals. He says that written material on the subject probably goes back to the 2000 BCE I cited and possibly as far back as Imhotep in 3000 BCE. However, even though there have been many discoveries since the book was written in 1970, I’d need to see some hard evidence that square pyramid bases were laid out the way you describe. If you have a cite I’d be curious to see it.
I’m not trying to argue that they could/would, only trying to figure out if it would be possible for such a system to work. Obviously having Pi=1 in a mathematical system probably wouldn’t work, but soemthing similar might.
I don’t want a debate this matter, I’m wondering if it could be made to work. I don’t see how you’re reading my comments as advocating that this would be how they’d do it. Our current system of telling time is based on 5,000 year old notions, and uses a different number base than we normally do, and this is an old convention, not necessarily one based on exacting standards. It’s fairly simple, and reworking it would require lots of effort, with little gain, so no one’s bothered (and probably never will bother) to switch to a different system (excepting the French, who came up with that needlessly complex system after the Revolution that they were forced to abandon).
It was, IIRC, in an old issue of Omni magazine, but I’ve seen it several times since. Googling doesn’t bring up anything helpful. The majority of links seem to be convinced that the use of Pi by the Egyptians involves aliens or the residents of Atlantis, so I’m not even going to bother to skim those pages for anything useful.
Well, you completely missed the point of what I was saying. By the time a society has mathematics advanced enough to even consider irrational base number systems, they’ve got an advanced society and people are familiar with and used to their number system. This is something that we only really started experimenting with quite recently, as people above have shown.
So you posit an advanced culture with interstellar travel and very well-developed mathematics. But somehow they’re not invested in their number system as it is? Do you think you could get everyone in our culture to give up using a useful number system in exchange for one that isn’t useful? You can’t really do anything useful with irrational number bases. You can’t count physical objects. Tell me what would motivate some group to give up their useful number system in exchange for one that couldn’t be used for any purpose?
Do you think you could even convince humans to just start counting in hexadecimal? How would that happen? Let alone this pi-based number system, which isn’t useful for much and is pretty completely incomprehensible. What event could you imagine that would convince people to completely change the way they count things? You know how metric hasn’t ever taken off in the U.S. or displaced certain measurement systems in many other areas? This is something way more fundamental. How do you imagine one would conduct their daily lives? You couldn’t even tell the host at a restaurant how many were in your party.