When I become King I will declare √2 to be officially 1.414 and π to be 3.14159. Until then ......
This problem is a little more apparent when it comes to painting a surface. We buy a gallon of paint so we can paint a surface of say 1000 square feet (so it says on the side of the can). But what if the surface is very rough, jagged. Then the paint doesnt seem to cover the same amount of surface area. That's because the surface goes up and down a lot. What we see when we look at the surface is a 2d 'projection' of that surface, we dont see the true 3d view where we take into account all the dips and hills. Once we start to consider those kinds of things, we find that the total surface area could easily be twice the 2d projection surface area. That would mean we would have to buy two gallons of paint instead of one. So you see one common example of this kind of problem.
So, we have a shape, Gabriel's horn, which behaves much like Pi and sqrt(s). Its one end keeps on stretching.
Um, not quite.
Its volume is finite, but its surface area is infinite.
2: I was thinking that if there is a way to make numbers such as Pi and sqrt(2) rational. In other words, is this this possible to make rational irrational? I don't think changing number system could affect the ultimate outcome. Pi would still be irrational even if we had binary system. Are such mathematical concepts independent of the systems in which they are studied? If my question is not clear, then please let me know. I would try to rephrase it. Thanks a lot.
Um, not quite.
Its volume is finite, but its surface area is infinite.
1: Is it really so?! I have been thinking about it. If it has an infinite surface area, then it must also has infinite volume. If it doesn't, then it would be a self-contraction, in my humble opinion which I know is short sighted!
2: I was thinking that if there is a way to make numbers such as Pi and sqrt(2) rational. In other words, is this this possible to make rational irrational? I don't think changing number system could affect the ultimate outcome. Pi would still be irrational even if we had binary system. Are such mathematical concepts independent of the systems in which they are studied?
1: Is it really so?! I have been thinking about it. If it has an infinite surface area, then it must also has infinite volume. If it doesn't, then it would be a self-contraction, in my humble opinion which I know is short sighted!
From physical point of view you can say when one of its side is stretched to infinity then at some stage it's bore reaches a limit when no atom etc. can pass through it and any liquid put in it won't leak beyond that point where the bore doesn't allow the tiniest atoms to pass through. But from mathematical point of view, its bore can never ever become zero which implies infinite volume.
2: I was thinking that if there is a way to make numbers such as Pi and sqrt(2) rational. In other words, is this this possible to make rational irrational? I don't think changing number system could affect the ultimate outcome. Pi would still be irrational even if we had binary system. Are such mathematical concepts independent of the systems in which they are studied? If my question is not clear, then please let me know. I would try to rephrase it. Thanks a lot.
Regards
PG
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