Abstract Confusions

Complexity is not a cause of confusion. It is a result of it.

Street Fighting Mathematics – Preview

I read this news from MIT, I haven’t read this book yet, and thus this is a preview.

An exciting book written on educated guessing, problem solving  and rough calculations by MIT Prof. Sanjoy Mahajan. Prof. Sanjoy Mahajan wrote Street Fighting Mathematics – The art of educated guessing and opportunistic problem solving book out of years of experience he got teaching Street Fighting Mathematics for math undergraduates. You can read the Open Course Ware material for street fighting mathematics here. You can download and read an early draft version of Street Fighting Mathematics – in PDF from MIT (the book is considerably improved and has more materials 150+ pages).

Antidote to Rigor Mortis

This book is all about applying the mathematics principles in situations dominated by fear and pressure,like  a street fighting perhaps. You don’t have to follow rules in street fighting, so why not do so in fast problem solving too?

Traditional mathematics teaching is largely about solving exactly stated problems exactly, yet life often hands us partly defined problems needing only moderately accurate solutions. This engaging book is an antidote to the rigor mortis brought on by too much mathematical rigor, teaching us how to guess answers without needing a proof or an exact calculation.

Just perfect. I have earlier noted, how mathematics, with few abstract tools, becomes difficult for engineers and physicist in “stray dogs, rigor mortis and mathematics”. Prof. Sanjoy Mahajan’s approach is the perfect antidote to rigor mortis. As mentioned in the forward –

Most of us took mathematics courses from mathematicians – Bad Idea!

So, Prof. Sanjoy Mahajan’s gets into more practical role and attempts to shed light on problem solving. By doing so, he shows insights into gleaning gems.

Tools for Opportunistic Problem Solving

Prof. Sanjoy Mahajan’s build, sharpen and demonstrate six different tools to make calculated guessing and fast, dirty problem solving. These six tools are

  1. dimensional analysis
  2. easy cases
  3. lumping
  4. picture proofs
  5. successive approximation and
  6. reasoning by analogy

These tools can be used to solve problems varying from every day problem solving to top notch engineering. There are six different chapters explaining all these concepts.

Prof. Mahajan starts every chapter with setting up the launch pad,  explaining important concepts and finally sheds loads of light on complex mathematical problems. One such example, from his book and present in the courseware, he explains what is picture proofs and why you need them to understand.

Have you ever worked through a proof, understood and confirmed each step, yet still not believed the  the theorem? You realize that the theorem is true, but not why it is true.

Sum of odd numbers up to a given number ‘n’ is given by a formula n^2 . Now, how do you proof this? Most of the time, we use induction hypothesis, starting from n=1, 2 and assume for n to proof for n + 1 . Let us imagine these as the following pieces.

S_1image ,

S_2 = image + image = image

And hence, S_3 = image = image

So, if you can count the unit squares, you can now, intuitively understand that S_3 = 3^2 = 9 .

There are many such wonderful insights. After reading this, you will never see math as you used to.

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