[Tutor] Maths (slightly OT)

[Danny Yoo]

> >Hoping to hear from all the career people that can use Mathematical
> >Calculations on a regular basis.

Hi Philip,


Math is not an artificial construct, but a very human endeavor:

    http://perso.unifr.ch/rafael.nunez/

The reason people reason with exponentials is because they're part of our
common human experience.

People have mentioned compound interest to appeal to the financial side of
things.  Maybe your son isn't that much into finance.  But perhaps your
son is into fairy tales.  Show him the story about A Grain of Rice:

    http://www.lhsgems.org/MAWConx.html#grain


The story goes that there's a beggar who once does a favor to a king.
The king owes him, so he asks the beggar to choose his own reward.  The
beggar asks for a single grain of rice, and to have his amount of rice
double every day, for one hundred days.  The king, not knowing much about
exponentials, foolishly agrees.

###
>>> def count_grains_of_rice(day):
...     if day == 1: return 1
...     else: return 2 * count_grains_of_rice(day - 1)
...
>>> count_grains_of_rice(1)
1
>>> count_grains_of_rice(2)
2
>>> count_grains_of_rice(3)
4
>>> count_grains_of_rice(4)
8
###

Not too scary so far.  8 grains of rice?  Bah, that's nothing.


What do things look like on the hundredth day?

###
>>> count_grains_of_rice(100)
633825300114114700748351602688L
###

That's a honking large number.  *grin*


What does this have to do with exponentials?  It turns out that the
function we've written is equivalent to:

###
>>> def f(x):
...     return 2**(x-1)
...
>>> f(1)
1
>>> f(2)
2
>>> f(100)
633825300114114700748351602688L
###

That's exponential growth.  We use exponentials because they're one of the
standard ways we can describe such tremendous rates of change.  And these
things happen in real life, in both natural and unnatural situations.


> The obvious answer to your question is: a mathematician. Physicists also
> use a lot of mathematics, in fact a lot of mathematical research is
> motivated by problems in physics.

Computer science is one of those "unnatural" sciences, and computer
scientists use math very heavily.  The kind of mathematics that CS
students use, though, might be much different than one might expect ---
unlike Physics, CS depends less on calculus, and more on discrete math.

Programmers eventually need to understand exponentials, because programs
that take exponential time to complete are pretty useless programs.  We
need to know about exponentials in order to recognize (and avoid writing)
programs that take exponential time to solve problems.


> Having said this, let me add that mathematics is being applied to
> various subjects. People studying DNA use knot theory, biologists model
> the evolution of populations using differential equations, economists
> use game theory, sociologists make heavy use of statistics, research in
> computer science makes heavy use of category theory, and the list goes
> on.

Biologists also use discrete math quite a bit.  As a concrete example of
the kind of mathematics involved in molecular biology and computer
science, you might want to browse 'Algorithms on Strings, Trees, and
Sequences:  Computer Science and Computational Biology':

    http://wwwcsif.cs.ucdavis.edu/~gusfield/paperlist.html

The kind of mathematics that's used there is heavy on proof techniques and
reasoning about recursive data structures.  The book may not use many
exponents or integral signs, but it does have sigma summation signs
everywhere.


But computer scientists do use calculus --- in particular, it provides
very useful tools in the analysis of recurrence relations:

    http://www.math.upenn.edu/~wilf/DownldGF.html

(The program at the very top, the count_grains_of_sand() function, is an
implementation of a "recurrence relation" as code.)


Anyway, that link to the 'Literature Connections to Math Around the World'
from the very top of this post may be useful.

    http://www.lhsgems.org/MAWConx.html

Your son may still doubt you after reading some of the books listed there,
but at least he'll be mildly entertained for a few minutes.  *grin*


Talk to you later!