With that being said, I think the math is coming! Prof Heap briefly mentioned how the speed of algorithms are measured with graphs and how they look like the graphs of polynomials. One thing that surprised me is how the speed of algorithms are measured by the worst speed. It doesn't feel like the sturdy math and it reminds me of psychology class where we learned the brain uses heuristics to be faster and sacrifices accuracy.
Sunday, October 26, 2014
Week 7: Logic and Math!
Even though it's called mathematical logic according to Wikipedia, it sure doesn't feel like math I'm used to. Set theory with all of it's implications and De Morgan's rules, it feels much more like learning a new language than learning math.
Friday, October 17, 2014
(I donno) How to Write Proofs
I have a lot of difficulty understanding how these proofs were figured out. I know that most of the proofs in the world just explain how they get from point A to Z but don't explain how they thought of going through all the other 24 points.
In class today, we learned that \for all natural numbers, n² + n is even.
It was done using proof by cases where n² + n had to be even regardless of if n was odd of if n was even.
However, one person pointed out what many were probably thinking in class:
if n² + n is factorized into n(n+1), this intuitively explains that it must be even because either n or n+1 must be even and any natural number multiplied by an even number is always even.
The problem is that I don't know how to prove this without using the proof by cases and substituting n with 2k+1 or 2k.
In class today, we learned that \for all natural numbers, n² + n is even.
It was done using proof by cases where n² + n had to be even regardless of if n was odd of if n was even.
However, one person pointed out what many were probably thinking in class:
if n² + n is factorized into n(n+1), this intuitively explains that it must be even because either n or n+1 must be even and any natural number multiplied by an even number is always even.
The problem is that I don't know how to prove this without using the proof by cases and substituting n with 2k+1 or 2k.
Saturday, October 11, 2014
End of Week 5: What's the History of Teaching Proofs
I don't think I'm supposed to write down the complete solutions to a problem which leaves me with the question about what to write about. Nothing meaningful usually comes out of my head especially concerning math related topics so I usually look for inspiration while I procrastinate while watching educational videos on Youtube.
Cool.
Professor Heap has mentioned on numerous occasions on how Euclid established the tradition of proofs.After watching this video, I wonder that if the foundations of proofs that we have started learning in class had been established over 2300 years ago, then are we learning the exact way that every single mathematician has learned proofs? Are we learning the same stuff that Newton did in the 17th century or a royal mathematician lost to history in the Mali empire did?
Cool.
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