Originally posted by Dave2002
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Well since I had to prove it during the course of an interview for a place at Bristol (and possibly Cambridge too) I certainly do. Rather a nice proof treating the decimal expansion as a geometric series ie 0.99999...=0.9*(1+0.1+0.01+0.001+...) -
For all epsilon > 0 there exists N such that ... etc
Anybody who doesn't accept that it is exactly equal to 1 doesn't properly understand what a limit is and therefore hasn't got the tools to deal with the reals.
What is interesting is the number of people who will accept that 0.3333333... is exactly equal to 1/3 but can't accept that 0.999999... is exactly equal to 1.
I seem to remember failing to describe what cube looks like when viewed along its diagonal on my Cambridge interview.Comment
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You can get some nice questions by considering the intersection of a couple of cubes. Imagine pushing one cube into another, and then consider the shapes you get on the boundaries.Originally posted by Vile Consort View Post
PS: 3D cubes, that is!Comment
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... cylinders too - I remember trying to work out the shape that would be left if one had an apple-corer and proceeded to core away three times at 90 degree angles, ie from north to south; from east to west; from front to back.Originally posted by Dave2002 View PostComment
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It'd be great if there was an easy to use/free/cheap 3D modelling tool which would enable this sort of thing to be tested out on screen. I believe such tools exist, but I'm not sure that any of them have the characteristics mentioned.Originally posted by vinteuil View Post
On the other hand ... I suppose your problem could be done with soft stuff, such as bread dough and indeed an apple corer - back to Play School (or whatever the latest childrens' programmes are now) type stuff. Or - even use an apple!
I don't think you get a sphere, but rather a sort of cube with rounded edges - though are there any ridges, which would lie on the planes formed by projecting the diagonals of each face of the cube across to the matching diagonal on the other side? Not sure - maybe!Comment
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... no. you don't get a sphere - you get a most interesting shape.
I was lucky, in that altho' I went to an undistinguished school in Wiltshire - and was being steered to the literature/humanities side - we had an excellent maths teacher - who was interested in this query I had which I brought to him aged 16 - and who was able to call up various sophisticated mathematical papers which had addressed this problem (and related problems - what if the cores don't start, as it were, from the sides of a cube, but from a tetrahedron - icosahedron - dodecahedron etc... ) - sadly, sadly, over the last fourty years I have lost the papers`which he provided me with...
His name was David Nelson. He was our maths teacher. Of course - we called him Ratio Nelson
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How divisive!Originally posted by vinteuil View Post... no. you don't get a sphere - you get a most interesting shape.
I was lucky, in that altho' I went to an undistinguished school in Wiltshire - and was being steered to the literature/humanities side - we had an excellent maths teacher - who was interested in this query I had which I brought to him aged 16 - and who was able to call up various sophisticated mathematical papers which had addressed this problem (and related problems - what if the cores don't start, as it were, from the sides of a cube, but from a tetrahedron - icosahedron - dodecahedron etc... ) - sadly, sadly, over the last fourty years I have lost the papers`which he provided me with...
His name was David Nelson. He was our maths teacher. Of course - we called him Ratio Nelson
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Why not just do it and see what shape emerges and work back to an explanation. Maths never was a strong point with me!Comment
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