You can add inifinity to the set of real numbers, and then define addition and multiplication operations on it. With this extended set you'll still be able to do calculations and come up with consistent results. Consistent means "not leading to internal contradictions". It may not be consistent with your intuitive concept of numbers, but it still works.
Note that, according to Cantor's diagonal argument, you can't even count from 0 to 1 in the real numbers.
Also note that the real numbers are not real in the sense that they represent our physical reality. Nature is essentially quantified, whereas set R has infinite detail. But you can still use R to do useful calculations. Likewise, you can calculate with imaginary numbers, and get results that match real-world phenomena.
... meaning that they are not in set R.
No, because 10^0 equals 1, not zero.
Interesting, but impossible.
Proof: Assume that numbers A and B both are zeroes, i.e.
(1) x + A = x for every x, and
(2) x + B = x for every x
Then A = A + B = B.
The first equality is by rule (2), the second equality is by rule (1).
Lerts, suppose you want to post a reply in this thread. There is some distance between your mouse pointer and the "Reply" button.
You move the mouse halfway. Distance left: 0.5 part of original distance
You move the mouse halfway again. Distance left: 0.25 part.
Again. Distance left: 0.125 part.
And so on.
Do you ever reach the "Reply" button? No.
I have now proved that you did not reply to this thread.
Both. Writing it as 2 is more convenient, but 1.9(9) is still a valid notation.
You can also write the number 16 as 17, 20, 22, 24, 31, 40, 121 or 1000 (in base 9, 8, 7, ..., 2). It's still the same number.
You move the mouse halfway. Distance left: 0.5 part of original distance
You move the mouse halfway again. Distance left: 0.25 part.
Again. Distance left: 0.125 part.
And so on.
Do you ever reach the "Reply" button? No.
I have now proved that you did not reply to this thread.
Both. Writing it as 2 is more convenient, but 1.9(9) is still a valid notation.
You can also write the number 16 as 17, 20, 22, 24, 31, 40, 121 or 1000 (in base 9, 8, 7, ..., 2). It's still the same number.
Last edited by wsinda, .
There is no last digit. If you insist on there being a last digit, then you are rounding off, thus turning it into a different number.
You seem to be uneasy with infinite decimal expansions. You are not the only one.
If the encryption hides a serious crime, the suspect will prefer to go to jail for not providing the keys. This law doesn't solve or prevent the crime.
The criminals can also use steganograhy, or any other way to hide the message in innocuous communication.
As AndroidXP remarked there is more than one way to write a certain number: 1, 2/2, exp(0), 0.999...
Even if I write 9s all my life, I could NEVER write enough decimals, so I will never write down the number that is 0.999... recurring. That's inifinity for you.
You can look at it this way: if you take the smallest positive number you can imagine, then it is still larger than the difference between 1.0 and 0.999... And if you take half of that, it's still too big. Or half of that ...
That is certain to be false alarm (unless your aim is to catch jokers). Any crook worth his salt will use encryption, or use nonspecific language, like: MEETING AGREED 15:00 SAN FRANCISCO.
They will. Terrorism is the reason to create the system. But once it works and the data is available, it will come in handy for other purposes: catching tax evaders, illegal migrants, file sharers, etc. The government will need to do that to justify the expense, cause the system won't catch any of the big criminals.
If you can't be tricked then in any case you can be tortured. You should know, you're from the country that brought us the Guildford Four.
Sure. Just calculate 1.0 - 0.99999999... Start writing down the decimals, and in a day or so you will see the answer.
Yes, each part of the fly was stationary at some instant during impact. But when its tail was stationary, its head was already going in the opposite direction. And perhaps the first few layers of atoms on the train were also stationary for a very short moment.
In all of those cases, you will be dividing by zero somewhere along the way.
Castor angle is the angle at which Beavers (Castoridae) bite chips out of the tree trunks. 
I think he meant Peugeot. They often have (had?) Pininfarina designs.
Imagine yourself sitting in the pub. A bloke then says to you: "I got a mate who has top access to that surveillance system. He's gonna plant evidence that you were collecting child porn. The cops pay you a visit, and your boss and your gf won't be happy. Now, if you just give me 5,000 quid before next Thursday..."
The useful work that they can do inside a prison is limited, and has a low economical value. If you give them decent pay, you are effectively subsidizing them with taxpayers money. And if the inmates must pay for the cost of imprisonment (food, housing, healthcare, security), they have very little left to relieve their debt.
Wrong example? You can use the $500,000 to pay back the remaining mortgage and credit card debt ($400,000 + $20,000), so you still have $80,000 left. (BTW, 500 grand is optimistic. In some areas the prices have already dropped 10-20%. And with a forced sale you will get around 90% of the actual value of the house.)
That brings me to the other part of my argument: if you don't take breaks on a long workday (or are not allowed to) then you will make more errors, which cost money and thus decrease effective productivity. I'm talking about long-term here: a short period of crunch time to make a deadline may be effective. But in the long run, excessive overtime will wear you out and harm productivity.
If you don't believe me, read Evan Robinson's article "Why Crunch Mode Doesn't Work". He backs it up with research from a century ago:
Some bosses never learn, I guess.
Either you're kidding or your brains went AWOL. The economical value of prison labor is negligible, and probably less than the cost of emprisonment. Plus, their families would have to live on welfare.
How about the Indian solution: Forcing them to sell their own kids as slaves (or, if none left, themselves).
Except that research has shown that people can only be productive for 8 hours per day. Days of 10 hours means that 2 hours are spent on coffee, chatting, private phone calls, and generally goofing off. So the boss can save cost on utilities, but he gets 20% less production for the same wages.
Michael, since you said you don't have a fancy studio, here's a suggestion: put a chair + desk somewhere out in the garden or the fields, and change the location a few times. (I've watched a lot of Monty Python sketches lately. )
It might make the background more interesting, since you don't have Jessica's "assets".
)It might make the background more interesting, since you don't have Jessica's "assets".
IIRC you can do that by changing the country here: http://www.lfs.net/?page=accountdetails
Surprisingly, these figures are not present in the Wikipedia comparison. Might be a good idea to add it.
Newsflash: If you want people to read text, you can put the letters straight on a web page. You don't have to make a movie of it and put that on YouTube.
The log of suggestions for setup improvements is here.
An earlier thread about a comment box is here.
Seems like a simple change, so it indeed is a mystery why it hasn't been done already. OTOH, there is room to create a "Setup Manager" add-on. This could also implement other features, such as renaming, filtering, searching, etc.
An earlier thread about a comment box is here.
Seems like a simple change, so it indeed is a mystery why it hasn't been done already. OTOH, there is room to create a "Setup Manager" add-on. This could also implement other features, such as renaming, filtering, searching, etc.
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