Add 5 to 0? No. 5x5 is a representation of 5 groups of 5.
And how many things are there in 5 groups of 5?
I think you missed his point... when you count 5 groups of 5, you start with zero, and add 5 to it 5 times. Or you do multiplication and conclude that, without counting, you know that there are 25 objects because that’s just how multiplication works.
Well, we don't. We take it to a particular digit then round off because a computer would endlessly calculate it.
No... I can use perfectly good algebra as follows, and obtain an exact (rational!) result without rounding:
Area circle 1, radius 10’ / Area circle 2, radius 20’
pi x 10^2 / pi x 20^2
10^2 / 20^2
100 / 400
1 / 4 I was multiplying by an irrational number with an infinite number of decimal digits... yet in the end we found that circle 1 is exactly 1/4th the area of circle 2 – no rounding or irrational numbers involved, because pi was simplified out of the equation.
In fact, I could generalize it and say that the formula for any two circles will be (r1 / r2)^2.
Infinite.000... still equals 0.
By the same logic infinite.999... equals 1. It isn’t an infinite sequence of digits that we need to calculate infinitely before we can know the result. It’s the concept that is expressed by them: the summation of the geometric series f(n) = 9/10^n, as n goes from 1 to infinity... calculus tells us that it converges to 1.
If only because you're real question should be "where the hell did you get infinitely many ping pong balls - let alone two lots of them and a container to hold them?"
Yes... well... that was my first question, but I was willing to let at least that much slide for the sake of the analogy.
Also, you did manage to have infinitely many times more apostrophes in that quoted sentence than you should have.:p
I’m pretty sure the answer was yes, but it was making my brain hurt and I stopped thinking about it.
If you’re into that sort of puzzle, though, perhaps you can answer one of my personal favourite brain teasers that I came up with...
How many times will the minute and hour hands on a standard 12-hour analog clock cross each other: (a) in a 12-hour period, from 6 AM to 6 PM; (b) in a 24-hour period, from 6 AM to 6 AM.
The number '0.99999....' is only an approximation to the value "nine times one ninth", limited by decimal notation.
...what?!
The number 1 is the exact value of “nine times one ninth”.
Did you perhaps mean that the number 0.111... is an approximation of the value of one ninth limited by decimal notation? Even then I’d have to disagree on semantics: the 3 dots (or a bar printed above 1 in the repeating decimal 0.1) are an extension to the decimal notation that allow it to describe the exact value of the fraction. It makes it act in dis-intuitive ways, sometimes, though, particularly if people don’t understand what’s going on. Namely, when you multiply it by 9 and get another repeating decimal that’s exactly equal to 1...
Hey, moron: what is the set of all positive integers? Is it an infinite set? It starts at 1. What is that, half-infinite? Are you really that retarded?
So you get to break the rules of decimal notation (pretending that algorithms can be written as numeric literals) but I don't?
Long division and long multiplication are algorithms which can be written as numeric literals, and if you’ve even taken high-school mathematics you surely know that they can and quite often do produce repeating decimals: decimals which repeat infinitely.
Putting an infinite number of 9’s, “with a 1 at the end”, is nonsense. If the 9’s end anywhere, they weren’t infinite – so where does your 1 come from? You’ve created a contradiction in terms: infinity which isn’t infinite.
It’s the same as an infinitesimal. By any standard of measurement, it’s zero – in fact, adding any finite number of them would still be zero – but mathematically it can’t (quite) be zero because adding an infinite number of them isn’t zero.
So it’s zero, except that... it isn’t quite zero. It’s infinitesimally small, and it should be zero as far as we can tell, but there’s still an infinitesimally small chance it can occur.
But, since the area of the diagonal of the square is zero, the probability that the dart lands exactly on the diagonal is zero. So, the dart will almost surely not land on the diagonal.
“Almost surely not” is just another way of saying the probability is vanishingly small. Vanishingly small just means that if you take it to infinity, the probability goes to zero, and yet, just like the dart must land on the square somewhere, and it can land on the diagonal just as easily as it can land anywhere else, the probability of it landing on the point on which it does land cannot be exactly zero or that event would have been impossible. It is vanishingly small: Mathematically it must be zero, but empirical evidence proves that it is not nonexistent.
Another example of something which is “vanishingly small” is a differential, or in this case it’s called an “infinitesimal” (but the concept is the same):
In common speech, an infinitesimal object is an object which is smaller than any feasible measurement, hence not zero size, but so small that it cannot be distinguished from zero by any available means.
A differential is, for any practical purposes, zero. However, an infinite number of zeros added together still makes zero; while the whole theory of integration is adding an infinite number of infinitesimals and coming up with something that is both finite and nonzero.
No, it is assuming something else: That multiplying a number with an infinite number of decimal digits by 10 results in another number that has an infinite number of decimal digits, and that the “infinite” number of digits in the first number is fundamentally no different from (i.e. not “1 more” than) the “infinite” number of digits after you’ve multiplied it by 10.
For correct understandings of the concept of infinity, this assumption is correct.
The identical assumption is made in the following:
It works in any base number system. It is the concept of an infinite series which converges to a finite value, and that is not a flaw in the math system. Rather... it’s a feature.
Because once you get down smaller than the smallest subatomic particle in the universe, more fractions lose meaning and you might as well round up or down.
Never!
That’s the whole point. In real life you have to. In mathematics, you never have to round. It’s 9s all the way down. It’s a concept more than anything else, and it represents the same idea that 1 does.
No; it is true, and in fact the distance traveled by the ball will also be finite and calculable, because each bounce is some constant percentage smaller than the previous.
Suppose the ball bounces to 1/2 its original height with each bounce, and you toss it upward from the ground with just enough velocity initially that it reaches a height of 0.25 meters. It travels 0.5 meters in total (0.25 meters up and 0.25 meters back down) before its first bounce, travels 0.25 meters on that bounce, 0.125 meters on the next, etc. How far does it travel in total before it stops bouncing?
The answer is 1 meter.
This image visualizes exactly that... and since this is Slashdot I will also point out that the distance it travels is the repeating decimal 0.11111... in binary.
I’m too confused by the impossibilities in the analogy to even begin to see what you’re trying to say about infinity.
E.g. the two containers would have to be infinitely large, there is no possible way that you could sort them in a minute – hell, it’d take infinitely long just to find the ball numbered “1” in the one that you’re attacking in an orderly fashion – and going infinitely fast at the end, as you suggested, is impossible; furthermore, if your containers are infinite in size, how is “full” a useful adjective when describing them?
Infinity is not a number. What’s more, even if infinity was a number, the function could not be simultaneously +infinity and -infinity at the same point. That is why the function is undefined at 0.
The probability, upon your initial choice, of picking the right door is 1/100. Now, eliminate 98 wrong doors. The probability that either of those two doors is correct is 1/2.
No... the probability of your door being right is still 1/100 because you picked it before any of the wrong doors were identified, and because the person who opened the 98 wrong doors intentionally wouldn’t open yours.
Look at it this way:
There is a 1% chance that you picked the correct door; but there is a 99% chance that you forced Monty to open everything except the right door, and the one you picked.
More interestingly, if Monty opens 999,998 doors at random and the 999,998th door happens to be the one with the car behind it... are you allowed to pick that door, or have you lost the game?
In fact, the parent doesn't understand infinity, because with an infinite number of doors,, the probability that you picked the car is exactly zero, not "vanishingly small", and the odds are not "very, very high" but exactly 1 that you picked a goat to start with.
The odds that you pick the door with the car are just as good as the odds that you pick any other specific door. Yet, you must pick one of the doors...
No matter which door you pick, the probability of you picking that door were “vanishingly small”. That is just how we describe what happened when mathematically speaking the odds should be “exactly zero”, yet the event occurs...
The people who wrote the software in the first place. They want to produce software that isn’t buggy and exploitable, and the only way to find exploitable bugs is to be actively looking for them and to be good at exploiting them.
They need good software crackers (in both senses of the word: skilled and working for them) working on betas to find vulnerabilities in the software so that the vulnerabilities can be fixed before the alpha of the software is released.
Note that it specifically says that they won’t be dealing with 0-day exploits (critical exploits in existing, already-released software products). They want to find these before they release, and to do that, they have to hire crackers.
Slashdot Mirror
Add 5 to 0? No. 5x5 is a representation of 5 groups of 5.
And how many things are there in 5 groups of 5?
I think you missed his point... when you count 5 groups of 5, you start with zero, and add 5 to it 5 times. Or you do multiplication and conclude that, without counting, you know that there are 25 objects because that’s just how multiplication works.
Well, we don't. We take it to a particular digit then round off because a computer would endlessly calculate it.
No... I can use perfectly good algebra as follows, and obtain an exact (rational!) result without rounding:
Area circle 1, radius 10’ / Area circle 2, radius 20’
pi x 10^2 / pi x 20^2
10^2 / 20^2
100 / 400
1 / 4
I was multiplying by an irrational number with an infinite number of decimal digits... yet in the end we found that circle 1 is exactly 1/4th the area of circle 2 – no rounding or irrational numbers involved, because pi was simplified out of the equation.
In fact, I could generalize it and say that the formula for any two circles will be (r1 / r2)^2.
Infinite .000... still equals 0.
By the same logic infinite .999... equals 1. It isn’t an infinite sequence of digits that we need to calculate infinitely before we can know the result. It’s the concept that is expressed by them: the summation of the geometric series f(n) = 9/10^n, as n goes from 1 to infinity... calculus tells us that it converges to 1.
If only because you're real question should be "where the hell did you get infinitely many ping pong balls - let alone two lots of them and a container to hold them?"
Yes... well... that was my first question, but I was willing to let at least that much slide for the sake of the analogy.
Also, you did manage to have infinitely many times more apostrophes in that quoted sentence than you should have. :p
I’m pretty sure the answer was yes, but it was making my brain hurt and I stopped thinking about it.
If you’re into that sort of puzzle, though, perhaps you can answer one of my personal favourite brain teasers that I came up with...
How many times will the minute and hour hands on a standard 12-hour analog clock cross each other:
(a) in a 12-hour period, from 6 AM to 6 PM;
(b) in a 24-hour period, from 6 AM to 6 AM.
The number '0.99999....' is only an approximation to the value "nine times one ninth", limited by decimal notation.
...what?!
The number 1 is the exact value of “nine times one ninth”.
Did you perhaps mean that the number 0.111... is an approximation of the value of one ninth limited by decimal notation? Even then I’d have to disagree on semantics: the 3 dots (or a bar printed above 1 in the repeating decimal 0.1) are an extension to the decimal notation that allow it to describe the exact value of the fraction. It makes it act in dis-intuitive ways, sometimes, though, particularly if people don’t understand what’s going on. Namely, when you multiply it by 9 and get another repeating decimal that’s exactly equal to 1...
Zero is not a positive integer.
You’re either trolling or too retarded to waste any more time arguing with.
Hey, moron: what is the set of all positive integers? Is it an infinite set? It starts at 1. What is that, half-infinite? Are you really that retarded?
And where did your 9's come from?
Long division followed by long multiplication?
So you get to break the rules of decimal notation (pretending that algorithms can be written as numeric literals) but I don't?
Long division and long multiplication are algorithms which can be written as numeric literals, and if you’ve even taken high-school mathematics you surely know that they can and quite often do produce repeating decimals: decimals which repeat infinitely.
Putting an infinite number of 9’s, “with a 1 at the end”, is nonsense. If the 9’s end anywhere, they weren’t infinite – so where does your 1 come from? You’ve created a contradiction in terms: infinity which isn’t infinite.
It’s the same as an infinitesimal. By any standard of measurement, it’s zero – in fact, adding any finite number of them would still be zero – but mathematically it can’t (quite) be zero because adding an infinite number of them isn’t zero.
So it’s zero, except that ... it isn’t quite zero. It’s infinitesimally small, and it should be zero as far as we can tell, but there’s still an infinitesimally small chance it can occur.
Well... that depends on what the guy who found the exploitable bug is planning on doing with it if you don’t buy it...
(and if he’s threatening to sell it to highest bidder if you won’t buy it, that is blackmail/extortion, and quite illegal)
http://en.wikipedia.org/wiki/Almost_surely#Throwing_a_dart, in particular:
But, since the area of the diagonal of the square is zero, the probability that the dart lands exactly on the diagonal is zero. So, the dart will almost surely not land on the diagonal.
“Almost surely not” is just another way of saying the probability is vanishingly small. Vanishingly small just means that if you take it to infinity, the probability goes to zero, and yet, just like the dart must land on the square somewhere, and it can land on the diagonal just as easily as it can land anywhere else, the probability of it landing on the point on which it does land cannot be exactly zero or that event would have been impossible. It is vanishingly small: Mathematically it must be zero, but empirical evidence proves that it is not nonexistent.
Another example of something which is “vanishingly small” is a differential, or in this case it’s called an “infinitesimal” (but the concept is the same):
In common speech, an infinitesimal object is an object which is smaller than any feasible measurement, hence not zero size, but so small that it cannot be distinguished from zero by any available means.
A differential is, for any practical purposes, zero. However, an infinite number of zeros added together still makes zero; while the whole theory of integration is adding an infinite number of infinitesimals and coming up with something that is both finite and nonzero.
No, it is assuming something else: That multiplying a number with an infinite number of decimal digits by 10 results in another number that has an infinite number of decimal digits, and that the “infinite” number of digits in the first number is fundamentally no different from (i.e. not “1 more” than) the “infinite” number of digits after you’ve multiplied it by 10.
For correct understandings of the concept of infinity, this assumption is correct.
The identical assumption is made in the following:
1/3 = 0.333...
(1/3) * 10 = 3.333...
(1/3) * 10 - 1/3 = 3.333... - 0.333...
(1/3) * 10 - 1/3 = 3
It works in any base number system. It is the concept of an infinite series which converges to a finite value, and that is not a flaw in the math system. Rather... it’s a feature.
a = 9/10^1 + 9/10^2 + 9/10^3 + 9/10^4 + ...
What does the series converge to?
It’s no different than watching a clock counting off seconds and believing there have also been an infinite number of seconds in the universe’s past.
Because once you get down smaller than the smallest subatomic particle in the universe, more fractions lose meaning and you might as well round up or down.
Never!
That’s the whole point. In real life you have to. In mathematics, you never have to round. It’s 9s all the way down. It’s a concept more than anything else, and it represents the same idea that 1 does.
No; it is true, and in fact the distance traveled by the ball will also be finite and calculable, because each bounce is some constant percentage smaller than the previous.
Suppose the ball bounces to 1/2 its original height with each bounce, and you toss it upward from the ground with just enough velocity initially that it reaches a height of 0.25 meters. It travels 0.5 meters in total (0.25 meters up and 0.25 meters back down) before its first bounce, travels 0.25 meters on that bounce, 0.125 meters on the next, etc. How far does it travel in total before it stops bouncing?
The answer is 1 meter.
This image visualizes exactly that... and since this is Slashdot I will also point out that the distance it travels is the repeating decimal 0.11111... in binary.
I’m too confused by the impossibilities in the analogy to even begin to see what you’re trying to say about infinity.
E.g. the two containers would have to be infinitely large, there is no possible way that you could sort them in a minute – hell, it’d take infinitely long just to find the ball numbered “1” in the one that you’re attacking in an orderly fashion – and going infinitely fast at the end, as you suggested, is impossible; furthermore, if your containers are infinite in size, how is “full” a useful adjective when describing them?
Infinity is not a number. What’s more, even if infinity was a number, the function could not be simultaneously +infinity and -infinity at the same point. That is why the function is undefined at 0.
The probability, upon your initial choice, of picking the right door is 1/100. Now, eliminate 98 wrong doors. The probability that either of those two doors is correct is 1/2.
No... the probability of your door being right is still 1/100 because you picked it before any of the wrong doors were identified, and because the person who opened the 98 wrong doors intentionally wouldn’t open yours.
Look at it this way:
There is a 1% chance that you picked the correct door; but there is a 99% chance that you forced Monty to open everything except the right door, and the one you picked.
Consider this scenario instead: There are two contestants.
Is it in Contestant A’s best interest to stick with their original choice, or to “steal” Contestant B’s door?
More interestingly, if Monty opens 999,998 doors at random and the 999,998th door happens to be the one with the car behind it... are you allowed to pick that door, or have you lost the game?
In fact, the parent doesn't understand infinity, because with an infinite number of doors,, the probability that you picked the car is exactly zero, not "vanishingly small", and the odds are not "very, very high" but exactly 1 that you picked a goat to start with.
The odds that you pick the door with the car are just as good as the odds that you pick any other specific door. Yet, you must pick one of the doors...
No matter which door you pick, the probability of you picking that door were “vanishingly small”. That is just how we describe what happened when mathematically speaking the odds should be “exactly zero”, yet the event occurs...
Erm, yeah, I meant the release version.
The people who wrote the software in the first place. They want to produce software that isn’t buggy and exploitable, and the only way to find exploitable bugs is to be actively looking for them and to be good at exploiting them.
They need good software crackers (in both senses of the word: skilled and working for them) working on betas to find vulnerabilities in the software so that the vulnerabilities can be fixed before the alpha of the software is released.
Note that it specifically says that they won’t be dealing with 0-day exploits (critical exploits in existing, already-released software products). They want to find these before they release, and to do that, they have to hire crackers.
Pff... you thought that was bad?
0.111... = 1