Quote Originally Posted by chris-sharpe999
Gerbick did you read his bit at the bottom? 0.99... doesn't itself get closer to 1

Also, something that is infinately small IS zero. Eg take the sum of a geometric progression: Sn=a(1 - r^n)/(1-r) If you take this series to infinity you get S∞=a/(1-r) because the - r^n part becomes infinitely small, tending to 0.
Something that is infinitely small IS NOT zero. zero means a non-existence. Not an infinitely small amount.

Take a piece of bread. You break this apart into tiny little bread crumbs. You break these apart into cells and fibers. All these little cells and fibers are broken to organells and liquids. You break this down into tiny little molecules, which are so incredibly small you can't even begin to visualise what they look like. But you do accept they're there, right? Let's now break down these molecules into atoms. Again, atoms do exist right? Now we break down the atons and we have a load of protons, neutrons and electrons. Then you break those down and what you'll find is quarks, neutrino's and all that mumbo jumbo. If we go any deeper we'll find ourselves in superstring regions, but this is quite far enough.

My point is that no matter how often you cut this piece of bread, you'll still be left with a bunch of stuff, however small it may be. If you cut the bread an infinite amount of times, you'll still have a teensy weensy tiny little bit of matter, infinitely small, but it's still there. It doesn't magically disappear--which would be implied if we accept that infinitely small equals zero.

Assume you bought a loaf of sliced bread. Perhaps you have an unkempt household, perhaps you were away for a couple of days, either way, when you return home you find out that one of your slices is reduced to 0.9 of it's original size. A mouse the size of a golf ball took a chunk of your bread and converted it into little mouse poop. Suppose another mouse, ten times smaller, the size of an ordinary green pea, took a bite out of another loaf of bread. This slice is 0.99 of it's original size. Another slice of bread has been ravaged by an even smaller mouse than that, the size of a grain of sand. That slice is 0.999 it's original size. An even smaller mouse took an even smaller bite out of another slice, et cetera, et cetera.

Now imagine an infinitely small mouse, not even the size of a lone neutron, took a bite out of one of the slices of bread. Does that mean he didn't actually take a bite out of the bread? Of course he did, only the bite is so incredibly small, mathmaticians made up rules with which they can usually neglect this tiny little difference.

There are only a handful of occasions when this infinitely small number actually makes a difference, and this example is one of them. This example deals with two of the most difficult mathmatical concepts, 'infinity' and 'zero'. Both concepts cannot be defined, 'infinity' for its immense magnitude (or in this case, smallness), and 'zero' because of its nothingness. You cannot specify 'zero', because whatever way you try to define it, it will always be an abstract way of formulating 'nothing', and that purpose is defeated the moment you try to visualise or define it. 'Nothing' becomes 'something'. 'Infinity' becomes 'finity'.