How To Impress With Maths
Now and then I sit together with Andreas, drinking a little glass of wine (or two) and let our thoughts fly - recalling happy and unexpected "wow"events, home runs, but also disruptions, unusual approaches, failures and what have you. Not always restricted to our work.
I am always impressed about Andreas' engagement giving applied maths lectures for kids.
Recently, I read the April-14 UK issue of Wired and a review of the book: Secrets of Mental Maths.
I have not read the book yet but in the Wired review there are a few examples that show what the book is about: get ready to amaze your friends by calculating blazingly fast in your head.
I pick two:
Square - 24^2? This is (20+4)^2, with a little algebra it turns into 20*28 + 4*4 (better 10*56 + 4*4): 576. So, instead of 24*24 I calculate (24-4)*(24+4) + 4*4.
Multiply by taking a shortcut - 53*11? 53*(10+1), again after a little manipulation its 500+(5+3)*10 + 3: 583. The shortcut: write "5(5+3)3". If 58*11 apply the overflowing 1 ("5138" goes to 638).
There are other examples that work with cutting the calculation problem into small bits, using easy numbers, complements and what have you.
Behind all these things are the polynomial functions that describe numbers:
z:=a_0+a_1*10+…. a_n*10 and the application of their ring features (expand, factor, rearrange, simplify …). You always can find forms that are of good nature for quick calculations.
If you have a symbolic computation system, like Mathematica, you can explore such schemes without cumbersome hand transformations. Kids could use that for explorative learning of some of the principles of mathematical thinking.
But such problem transformations also work on a much higher level. Take cutting in small bits.
Asymptotic Maths does exactly this: decompose the problem domain into partitions where closed form solutions are available and recombine the results in a clever way.
Adaptive Integration is such an asymptotic maths scheme in UnRisk. It is amazingly powerful,
But even more general, if we test new numerical schemes for instrument-model calculations we decompose the domain into pieces where we know that solvers are accurate and robust and recombine in a way that we have enough reasoning that the will be the cease for the whole domain.
We call this umbrella testing - if we want to test say a new QMC scheme, we test them across the frames where solutions of existing schemes (say, Adaptive Integration) are verified and validated.
And all this is made available to quants, who buy UnRisk-Q. It provides not only a vast variety of deal types, models and methods (that are bank-proof), the UnRisk Financial Language inherits all symbolic computation capabilities and more from Mathematica.
So quants can use the instant derivatives and risk universe and test and optimize their own schemes across them.
This is an underestimated benefit: a symbolic domain specific language that does quant finance and maths.
We develop an test a lot of UnRisk in UnRisk. And we become swifter and swifter at less cost.
What was great for kids is great for quants.
Picture from sehfelder