Finally, started blogging again, this time with purpose. I and a few of my friends will be looking into how mathematicians of the past could have found what they did. The basic structure of these will be:
1) Hey, I wonder… How many …? What is the size of … for …? How do …. and …. relate? or are they unrelated?
2) Using mathematics we have learnt so far, completely rigorously, trying to show that a function will behave in a certain way at certain points
3) Numerical analysis, once all our deductions have been used it might be useful to analyse what we do.
4) Graphing and changing all the variables. ‘Maths by Computer’ saving the day, also with the above.
5) Discussing results, talking about how to make algorithms faster, using what we learn, spotting patterns and making conjectures and then going back to step 2, until we have a formula and have rigorously proved why it is the case. - Or realized that this is, unfortunately, beyond us, and look it up, have a laugh and good think, and move on.
Before you get all impressed, or thinking we’re over-achievers, these results should be fairly trivial, and within our grasp, we could look them up at any point, but we are trying to do the research ourselves. One day, we may be at a point, where we cannot just look it up, it is just practice for then :)
I have done some, just not written up yet, will type those up in LaTeX and then move on to doing the others.