Nope. Any number with a recurring pattern is automatically rational, and pi is not.
This isn't too hard to see. For example, if pi repeated after 1.5 trillion digits, we could write its value (where [1.5tril] represents all those repeating digits:
pi = 3.14159[1.5tril]..14159[1.5tril]..14159[..]
Then multiply this number by 10^(1.5 trillion).
10^(1.5trillion) * pi = 314159[1.5tril].14159[..]
such that the repeating part starting with.14159 still follows the decimal point.
Then subtract the top equation from the bottom one, so the repeating part gets subtracted away.
(10^(1.5 trillion) - 1) * pi = 314159[1.5tril]
Then just divide both sides by (10^(1.5 trillion) - 1) and you've written pi as a ratio of two integers.
A stable equilibrium is one that's robust to little nudges - it tends to go back towards the equilibrium point all by itself.
An unstable equilibrium is more like a ball on top of a hill. It's in equilibrium because it isn't going anywhere; but the equilibrium is unstable because a small push will destroy this particular equilibrium permanently.
That's the concept underlying the big spiral graph, by the way.
Let's see. ..why CAN'T they fly Coach: the rest of us do.
Because this would show that we, as a nation, treat our wealthy businesspeople better than we treat our elected representatives.
Then again, if **I** was running things, I'd have Congresscritters and Senators living in general-issue family housing on any of the local military bases in the DC area.
Then you'd also have only people running for office who don't care what kind of housing the government gives them and their families. That would be the ones who are independently wealthy and can afford to turn it down. Oops.
Some dork making a hundred grand a year is too out of touch with reality to represent the average American who is making 15 thousand a year. It should work like the Peace Corps...while you're in office, all you get is what the average American makes per year.
This is so not a good idea. Problems:
Do you really want a semi-impoverished congressman who has to constantly worry about feeding his family? I want my representative to be able to devote his time to figuring out his constituents' interests and keeping himself informed.
Making the position a low-salaried job will obviously deter qualified people from running. In fact, ironically, paying Congressmen little or nothing is probably the best way to fill the government with wealthy businessmen who don't have to worry about how much money they make.
For people as important as representatives, dignity matters. Paying the people who make our laws as though they were construction workers or secretaries is undignified and unfair to them.
Senators and reps would be even more subject to influence by gifts and perks from lobbyists.
Were a law to be made 'that no man should hold an office who had not a private income sufficient for the subsistence and prospects of himself and family...all offices would be monopolized by the rich; the poor and the middling ranks would be excluded and an aristocratic despotism would immediately follow.'
-John Adams (quoted by David McCullough)
When paperbacks started costing > 9 dollars, I stopped buying them. It hurt to decrease my favorite entertainment, but with my scifi/fantasy appetite of 2-4 paperbacks a weekend, I just couldn't afford it.
Dude... haven't you got a library? The original source of free books...
Re:Freecell Solitaire...
on
Awari Solved
·
· Score: 1
For a less contrived unsolvable example, pick game 11982.
Re:Arbiters... Something doesn't make sense.
on
Clockless Computing
·
· Score: 1
I don't think I agree with the first poster re: Asparfame's question. An arbiter could just test its two input channels at some two arbitrary moments in time. There's nothing inherently wrong with this collision-detection strategy - it gives imperfect results, but that's pretty much unavoidable.
The problem is not with collision detection, but collision resolution. I'm pretty sure you're correct: a dumb strategy, like "alternate left and right" or just "always pick left" would never lock up. So somebody did oversimplify. But the dumb strategy also wouldn't work very well. Maybe the left instruction is useful right now, while the process that generated the right instruction is stalled for some other reason anyhow - we'd like to detect which instruction is better to run first. I imagine an intelligent strategy actually has to look at the instructions and weight them somehow. And any such intelligent and fair detection strategy risks valuing the two instructions equally.
If numbers are too tough, then you shouldn't be playing with that much money anyway.
It's not that anybody can't tell the difference between the bills, it's that making it easier to tell the difference is a good thing. Color makes bills more distinguishable at a quick glance, from a distance, in low light, whatever.
It's weird that so many engineer-types here don't seem to understand this. You want color and numbers for the same reason that you want for loops and while loops. You can get by with just one of them, but it makes things needlessly difficult for the reader.
Slashdot Mirror
Nope. Any number with a recurring pattern is automatically rational, and pi is not.
This isn't too hard to see. For example, if pi repeated after 1.5 trillion digits, we could write its value (where [1.5tril] represents all those repeating digits:
Then multiply this number by 10^(1.5 trillion).
such that the repeating part starting with .14159 still follows the decimal point.
Then subtract the top equation from the bottom one, so the repeating part gets subtracted away.
Then just divide both sides by (10^(1.5 trillion) - 1) and you've written pi as a ratio of two integers.
A stable equilibrium is one that's robust to little nudges - it tends to go back towards the equilibrium point all by itself.
An unstable equilibrium is more like a ball on top of a hill. It's in equilibrium because it isn't going anywhere; but the equilibrium is unstable because a small push will destroy this particular equilibrium permanently.
That's the concept underlying the big spiral graph, by the way.
Because this would show that we, as a nation, treat our wealthy businesspeople better than we treat our elected representatives.
Then again, if **I** was running things, I'd have Congresscritters and Senators living in general-issue family housing on any of the local military bases in the DC area.
Then you'd also have only people running for office who don't care what kind of housing the government gives them and their families. That would be the ones who are independently wealthy and can afford to turn it down. Oops.
This is so not a good idea. Problems:
Do you really want a semi-impoverished congressman who has to constantly worry about feeding his family? I want my representative to be able to devote his time to figuring out his constituents' interests and keeping himself informed.
Making the position a low-salaried job will obviously deter qualified people from running. In fact, ironically, paying Congressmen little or nothing is probably the best way to fill the government with wealthy businessmen who don't have to worry about how much money they make.
For people as important as representatives, dignity matters. Paying the people who make our laws as though they were construction workers or secretaries is undignified and unfair to them.
Senators and reps would be even more subject to influence by gifts and perks from lobbyists.
Were a law to be made 'that no man should hold an office who had not a private income sufficient for the subsistence and prospects of himself and family...all offices would be monopolized by the rich; the poor and the middling ranks would be excluded and an aristocratic despotism would immediately follow.' -John Adams (quoted by David McCullough)
When paperbacks started costing > 9 dollars, I stopped buying them. It hurt to decrease my favorite entertainment, but with my scifi/fantasy appetite of 2-4 paperbacks a weekend, I just couldn't afford it.
Dude... haven't you got a library? The original source of free books...
For a less contrived unsolvable example, pick game 11982.
I don't think I agree with the first poster re: Asparfame's question. An arbiter could just test its two input channels at some two arbitrary moments in time. There's nothing inherently wrong with this collision-detection strategy - it gives imperfect results, but that's pretty much unavoidable.
The problem is not with collision detection, but collision resolution. I'm pretty sure you're correct: a dumb strategy, like "alternate left and right" or just "always pick left" would never lock up. So somebody did oversimplify. But the dumb strategy also wouldn't work very well. Maybe the left instruction is useful right now, while the process that generated the right instruction is stalled for some other reason anyhow - we'd like to detect which instruction is better to run first. I imagine an intelligent strategy actually has to look at the instructions and weight them somehow. And any such intelligent and fair detection strategy risks valuing the two instructions equally.
If numbers are too tough, then you shouldn't be playing with that much money anyway.
It's not that anybody can't tell the difference between the bills, it's that making it easier to tell the difference is a good thing. Color makes bills more distinguishable at a quick glance, from a distance, in low light, whatever.
It's weird that so many engineer-types here don't seem to understand this. You want color and numbers for the same reason that you want for loops and while loops. You can get by with just one of them, but it makes things needlessly difficult for the reader.