Tuesday, March 17, 2015

[Maths History] #Newton's #proof that #pi is #transcendental ?

Article (from Quora, re: Alejandro Jenkins)
How do you prove that a number is a transcendental number?

Summary
Although the concept of "transcendental number" was not defined during Newton's time, Newton's Principia contained the germ of idea that seems to lead to a relatively easy proof of the idea that p  is transcendental.  Arnol'd described how Newton showed that the trajectory of planets cannot be expressed in terms of any polynomial function of time.

[#Critique] Do pediatricians suck at #maths (#math) ?

It looks like some pediatrician is complaining that maths (that's the way we spell it in Singapore) is too hard.

Paul has two-thirds as many postcards as Sean has and the number of postcards Sean has is three-fifths the number of postcards Tim has. If the boys have a total of 280 postcards, then how many more postcards does Tim have than Paul?

He says “Obviously it’s complex, but not impossible, ...  I would use a complex algebra equation to solve it and I’d guess it’s more appropriate for an eighth and ninth grade student than one still in the fifth grade.”

Oh really?  Here's an easy solution.  No complicated ("complex" can mean a few other things) algebra, nor any algebra for that matter is needed.



To answer the question "Do pediatricians suck at maths?"  My answer is: I don’t know.


But remember: Don’t use a cannon to shoot a fly, or as the Chinese say: Why kill chicken with an ox cleaver?.  Thanks for reading this blog post.  J

Monday, March 16, 2015

[#Critique] Did #Seth #Godin know what #transcendental means in #maths?

Article
Magic and irrational (by Seth Godin)


My Comments

This is an interesting piece of inspirational waffle about p to celebrate "p day".  Hopefully this special day helps to keep this important piece of knowledge that has a long cultural history fresh in our collective human consciousness, and perhaps stem the decline in mathematical literacy of some of us.

However, his article made me wonder whether he knew what transcendental means.  It is a pretty technical concept, and even if one knew what it means, it's a challenge to put it in layman's terms and say something funny with it.  If he knew the meaning, he certainly did not show it.

But I do not blame him.  Seth Godin is known for his marketing, not for his maths.  So this is not a personal attack.  I'm just following his advice, trying to be a purple cow.  Perhaps after some "moo moo" here and "moo moo" there, and transcendental meditation, this zero of a cow could break off from its roots of limited degree, eat the pi in the sky and jump over the Mooooooooon to become a hero!

;-)

[#Critique] Is #pi #infinite?

Article / Resource
The infinite life of pi - Reynaldo Lopes



My Comments
This is a well-produced video in which the team took great care in producing an interesting narrative.  The circles were drawn imperfect and shaky, so as to catch the attention of viewers and to highlight them.  A couple of points they missed out:-

1) p being irrational, does not only have an infinite number of digits in its decimal expansion, the digits are also non-repeating.  A rational number (fraction) like 1/7 also has an infinite number of digits, but they repeat: 1/7 = 0.142857 142857 142857 ...

2) near the end of the video, the animators depicted universe < p.  I find the artistic licence disturbing.  I am sure there is some way to artistically depict the known universe having less number of atoms than number of digits of p.

To answer the question in the title of this article "Is p infinite?": No, but her number of decimal digits is.

[Technology] #Calculator that makes you #Think

Article / Resource
QAMA Calculator Examples

My Comments
This calculator aims to enforce human estimation, and cultivate number-sense.  More designers / manufacturers / inventors should do likewise by at least trying to make students think.

Monday, March 9, 2015

[IB-HL H&H_Rev6C Q11] Factors of a Complex Polynomial

Question

Introduction
     This question is taken from the Haese & Harris textbook for IB HL Mathematics and is rather challenging.  Here I present two solutions.  In the first solution, I eliminate a variable.  [ heuristics: H10. Simplify the problem, H11. Solve part of the problem]  And that allowed me to crack the rest of them problem.  In the second solution, I rephrased the problem in terms of tangents to curves. [ heuristic H09. Restate the problem in another way ] This gives another angle from which to tackle the problem.
Solution 1
We seem to get into a mess here.  How can we eliminate  k?


Solution 2



Thinking Back
     Both solutions are related in the sense that they hinge on some form of the relation maked as [*].  That led to an equation in  a.  Once a  is found,  k  can be found, and the solutions proceed similarly.  The equation [*] is a manifestation of the fact that for a repeated root, both P(x) and P’(x) share a common factor.