[paul cotter]

Not so, Alan. you state that xⁿ>x if n>0.  X to power of 1/2 <x. It should say xⁿ>x if n>1. And x raised to the power of zero=1

[alancalverd]

Apologies. What a twit I am. No excuse.

[paul cotter]

Thank you, Alan, for your generous acceptance of my correction. This is the way it should be and I wish it would also apply to some of our prolific nonsense posters.

[paul cotter]

Just noticed i'm in error too. If x and n are both <1, xⁿ> x. Eg if x=0.5 and n=0.5 then xⁿ>x. Apologies for a rash reply.

[hamdani yusuf]

I'm afraid the answer is yes.

If you rotate a hyperbola x² - y² = 1 around the y axis, you form a body with finite volume but infinite surface area, so you can paint an infinite surface by pouring in a finite can of paint, but not by spraying or brushing it.

Economists make their living by talking about the can of paint and ignoring the brush.

Can the paint have zero thickness?

[alancalverd]

No, by definition - that's why you can't cover it by spraying. But obviously if you pour it in from a can, it will have finite thickness. Ask any government minister (except the Treasury).

[hamdani yusuf]

No, by definition - that's why you can't cover it by spraying. But obviously if you pour it in from a can, it will have finite thickness. Ask any government minister (except the Treasury).

What's the minimum thickness?

[alancalverd]

The question is meaningless since you can't cover the surface with paint.

[hamdani yusuf]

The question is meaningless since you can't cover the surface with paint.

This statement would also be meaningless.

I'm afraid the answer is yes.

If you rotate a hyperbola x² - y² = 1 around the y axis, you form a body with finite volume but infinite surface area, so you can paint an infinite surface by pouring in a finite can of paint, but not by spraying or brushing it.

Economists make their living by talking about the can of paint and ignoring the brush.

[alancalverd]

Exactly the point I was making. You can fill a finite volume but you can't cover an infinite surface. The hyperboloid has both.

But see reply #5 for a correction to the maths - I was recalling a lesson from about 60 years ago!

[varsigma]

Ok. Whew.

A whole thread about infinity and no mention of the Banach-Tarski paradox.

Oops.

[paul cotter]

Hmm, set theory. Not for me, although I like to expand my mathematical knowledge I find set theory boring and lacking practical application, certainly in engineering. As regards infinity I think it is pointless discussing it as it has no tangible existence. What concerns me is the limit of some function as a variable approaches infinity(basic maths but extremely useful). 

[varsigma]

Hmm, set theory. Not for me, although I like to expand my mathematical knowledge I find set theory boring and lacking practical application, certainly in engineering.

I can think of at least one practical application in logic where sets and set theory are useful. We called it programming logic in the course.

[paul cotter]

I don't doubt what you say, at all. I am a nuts and bolts, transients and slew rate, resonance and skin effect type of engineer(retired). I don't rate software writing as real engineering, though many will disagree.