Killjoy

Here's another trick that can work. 1) Take your original number (the one you want to find the square root of) - lets call it x1 2) Produce a new number thus: x2 = x1-(x1^2-x1)/(2*x1) 3) Produce a third number thus: x3 = x2-(x2^2-x1)/(2*x2) [iMPORTANT: Notice the second term in (x2^2-x1) (and only this term) is x1 - this is not a typo - it has to be the original number all the way through] 4) Repeat. The values of 'x' get closer to the square root of the original number. You can repeat as often as you want. The more you repeat the more accurate the answer, but you'll find that 4 or 5 repeats will be accurate enough for most purposes

If the average man is 175cm and the average woman is 168cm, the probability that the average man is taller than the average woman is 1. I think the problem is poorly phrased. Here are some possible alternatives:- What is the probability that a randomly selected woman is taller than the average man? What is the probability that a randomly selected woman is taller than a randomly selected man? Of course it could be a trick question.

I think you would need a metric to define ds in your first definition, which would then become circular.

WRT point 1: Thank you WRT points 2&3: Sorry - my bad WRT point 4: The name "Stokes' Theorem" sometimes refers to a specific case of a more general theorem which, rather confusingly, is also called "Stokes' Theorem". This more general theorem relates what happens in an N-dimensional hypervolume to what happens on the orientable (N-1)-dimensional hypersurface of that hypervolume, all embedded in an M dimensional manifold. Depending on the values of M and N, and what kinds of functions are being considered you get the various different theorems mentioned, and many others. If N=1 you recover the fundamental theorem of calculus. I'll try to find a better reference and post it.

This one gets my vote by and far. Not a whole lot gets done without this being true. I guess that's why it's been named "fundamental" eh? I agree that the fundamental theorem of calculus should be included. There is also a generalisation called Stoke's Theorem which, while perhaps a little advanced to be called "fundamental", actually incorporates several concepts in physics and mathematics - including the fundamental theorem (and Green's theorem, and the divergence theorem), is used to derive equations such as the wave equation and the diffusion equation, and is of central importance in differential geometry, general relativity, and many other fields.

What are people's opinions on which are the most important theorems in mathematics? By important I mean those with the most significant consequences both in pure and applied mathematics (not just obscure results that happen to be interesting to a few individuals). I would suggest the the Pythagoras Theorem should be up there. What else?

Is this the same as (2)?

In the absence of a diagram my question may be a bit hard to follow, but I'll do my best... Imagine I have a barbell floating in empty space - ie two equal weights connected by a rigid bar (a single rigid pole would do as well, but the barbell might be clearer). I start it spinning on an axis that passes through the center of mass of the system (ie, middle of the bar) but is neither parallel nor perpendicular to to bar. My intuitions as to what will happen lead in contradictory directions:- 1) Centrifugal force will cause the spinning barbell to straighten out so that the axis of spin becomes perpendicular to the bar; 2) Conservation of kinetic energy and angular momentum will mean that the barbell will continue to spin on the same axis as it started. Which is right and why is the other one wrong?