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Einfach
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Einfach
1,448 posts
Nomad

http://i1124.photobucket.com/albums/l577/gemcrafteinfach/0largestnumber.png

This graph, shown above, is the graph:
y=-1/x
This graph covers the values of all rational numbers (except for 0).

Its derivative (the slope of the tangent line at that particular point) is y=1/x^2, a graph which, for all rational numbers, has a positive y-value, meaning the graph is constantly increasing in value.

As x increases, y constantly increases in value, and y covers all rational numbers. As x approaches infinity, y gets closer and closer to 0. One may logically conclude that 0, therefore, is the largest real number.

One may argue that you are excluding the value x=0, because that cannot be said to have a definitive y-value value at that x-value.. However, look at the derivative, which is 1/x^2. As x approaches 0 in the derivative, y approaches infinity, regardless if you take a positive or negative 0. Therefore, at x=0, the value must be increasing, whatever the corresponding value of y is.

Discuss...

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driejen
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driejen
486 posts
Nomad

But we're using the limit - not actually dividing by 0.

Limits

You cannot circumnavigate dividing by zero by using the limits, as it still gives you a value of undefined.

We get around this by using the limit of the derivative, which is infinity whether you take the positive or negative side of 0.

No its negative infinity from the negative side.

What is (0.000000000000...1)^2? A positive number.
What is (-0.00000000000...1)^2? A positive number.

You weren't squaring the whole value, only the denominator.

So a positive divided by a positive = a positive, right, so either way, it's a positive infinity.
Zero isn't a positive value, squaring it does not make it a positive value. The idea that x^2 always results in a positive value does not account for zero.
Einfach
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Einfach
1,448 posts
Nomad

You cannot circumnavigate dividing by zero by using the limits, as it still gives you a value of undefined.


As x approaches 0, it becomes closer and closer to infinity.

No its negative infinity from the negative side.


Let me go and graph 1/x^2 so you can see that it is ONLY positive.

You weren't squaring the whole value, only the denominator.


You only square the denominator. It's 1/(x^2) - the derivative. Order of operations is implied.

Zero isn't a positive value, squaring it does not make it a positive value. The idea that x^2 always results in a positive value does not account for zero.


But the limit...taking a value arbitrarily close to 0 on either side results in a positive value.

*going out and getting that graph...
Joe96
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Joe96
2,226 posts
Peasant

If infinity is the opposite of 0, then the lines should curve away. However, your equation is negative and that is why it curves towards 0. -1/x

driejen
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driejen
486 posts
Nomad

We get around this by using the limit of the derivative, which is infinity whether you take the positive or negative side of 0.
Ah sorry, I did not look at the graph of the derivative properly, ye it tends to infinity at both sides. There must be some problem with how the limits are used, because it obviously leads to a seemingly fallacious conclusion that 0 is the greatest number and that somehow infinity=-infinity for the original graph but I will have to look more into how limits work before I discuss this further.
Einfach
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Einfach
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Nomad

[IMG]http://i1124.photobucket.com/albums/l577/gemcrafteinfach/1overxsquared.png[/IMG]

To settle the dispute, this is the derivative, which is 1/x^2. As you can see, it clearly does not cross the x-axis, especially at 0, and it approaches positive infinity as x approaches 0 - not negative infinity.

Einfach
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Einfach
1,448 posts
Nomad

http://i1124.photobucket.com/albums/l577/gemcrafteinfach/1overxsquared.png

Moe
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Moe
1,714 posts
Blacksmith

The number 0 is the smallest non-negative integer.


From Wikipedia.

Of course if can prove something like 1<0, then... well we have a problem.
Einfach
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Einfach
1,448 posts
Nomad

The number 0 is the smallest non-negative integer.

From Wikipedia.

Of course if can prove something like 1<0, then... well we have a problem.


And thus the paradox results... Your challenge is pointing out the problem in the premises / thought process of my argument. Obviously, no one would truly believe that 0 is the largest number, but you must disprove my thought process.
Einfach
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Einfach
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Nomad

You are confusing derivatives aren't you? As X approaches Y (usually zero), there is an infinite amount of elements to choose.


No - not an infinite number of elements. Only 1 - positive infinity.
driejen
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driejen
486 posts
Nomad

I don't know if perhaps this is an oversimplification of limits but reading from this; http://www.mathsisfun.com/calculus/limits-infinity.html, it tells me that limits are used for approaching a value but not getting an actual value. That as x approaches 0, y approaches infinity, but we cannot say for sure that y=infinity when x=0.

I haven't found a more descriptive explanation to why the limits might suggest a wrong answer or could be exploited in the way that you have so I'll continue looking around for a little while longer.

My maths teacher used to love dwelling on mathematical fallacies and it was kind of fun to think about, but I haven't come across exploting limits like this until now. It seemed like just another one of those simple divide by zero errors at first. Obviously the problem lies somewhere along the lines of, infinity apparently 'increasing' to negative infinity. I wouldn't conclude that 0 is the highest number though, because y=0 is just an asymptote, however you could argue that since the numbers seem to go full circle from the upper limits to the lower limits, all numbers are the highest number or that there is no highest number for any two numbers can be shown to be greater than each other.

Einfach
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Einfach
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Nomad

Infinity is a very special idea. We know we can't reach it, but we can still try to work out the value of functions that have infinity in them.


I knew at this point that this website was not intended for anyone past 4th grade to read.
Einfach
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Einfach
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Nomad

The website also said that 1/infinity was "We don't know" and saying that you cannot literally divide 1 by infinity and then jumping to the conclusion "We don't know" is much more fallacious than this argument (although I think that saying "more fallacious" is like saying more lethal)

driejen
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driejen
486 posts
Nomad

I knew at this point that this website was not intended for anyone past 4th grade to read.

Yeah I know. Everything seems dumbed down tremendously, thats why I'm trying to find a better site but have yet to find one. I haven't touched limits in A-level maths and I am not taking a maths degree so right now I have limited knowledge on it.
Einfach
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Einfach
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Nomad

There's got to be a refutation of this, but can anyone enlighten us as to what it is?

There must be a refutation of this - either my premises or I had an error in logic or 0 really is the largest number, because otherwise, it goes against the principle of non-contradiction.

Moe
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Moe
1,714 posts
Blacksmith

And thus the paradox results... Your challenge is pointing out the problem in the premises / thought process of my argument. Obviously, no one would truly believe that 0 is the largest number, but you must disprove my thought process.


As x approaches infinity, y gets closer and closer to 0. One may logically conclude that 0, therefore, is the largest real number.


This part is where I find fault. There is no logic in your problem that could possibly lead to that conclusion. In fact when you show the equation with the above statement(as opposed to a different statement like it is) you would see that y would have no choice but to get closer to 0 as x increases, however that doesn't translate to 0 being a large number.
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