Sage Intro to Matrices

[David Chandler]

I just posted a video tutorial on YouTube for my precalculus course on using Sage for doing matrix operations.  It's at http://www.youtube.com/watch?v=GKuYyuBVXoU

--David Chandler

[Linda Fahlberg-Stojanovska]

David, I am making a wiki for sage at http://sagemath.wikispaces.com and I have a http://youtube.com/sagemath  sagemath youtube channel. Can I link or upload ? Best Linda

(won’t be this week – I am so far behind :))

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[David Chandler]

I just added it to my website.  Look at the bottom of the page at
http://mathwithoutborders.com/?page_id=11
I would also like to link to yours and any other Sage tutorials at the appropriate level for my high school students.
--David Chandler

[David Chandler]

I've also now added a link to the sws file in the YouTube description text.
--David Chandler

[David Chandler]

In anticipation of your approval I have also added a link to your YouTube "channel" to my page.  I credited you by name, but will remove your name if you prefer to remain anonymous.
--David Chandler

On Sun, Mar 24, 2013 at 10:01 AM, David Chandler <david...@gmail.com> wrote:

[kirby urner]

I enjoyed it. Applying transformation matrices to lots of vectors,
namely those pointing from the origin to a set of vertexes, a
polyhedron. Rotating polyhedrons make me happy. Colorful etc. Have
you heard of Waterman Polyhedrons? I was the guy who named them that
(kind of obvious, as a guy named Waterman sifted them out of the mix
-- they meet simple criteria: all vertexes in a CCP lattice ... ).

Anyway, hey, I wonder what people think in general of using "real
number" and "floating point" in narratives like this. You mention how
we have a choice to use integers, rational numbers, or reals. But
some student will object that reals were never on the table, floats
being an IEEE specification and not of the real number set.

Kirby

[michel paul]

On Sun, Mar 24, 2013 at 1:57 PM, kirby urner <kirby...@gmail.com> wrote:

 

I wonder what people think in general of using "real

number" and "floating point" in narratives like this.

In a computational context I like using the terms 'exact' and 'inexact' to describe types of values. I think it directly points to something important in a simple way. A CAS like Sage or Mathematica will by default try to represent values exactly. If you just want the approximation, you have to specify that.

Although I know they're not going to go away anytime soon, terms like 'real' and 'imaginary' are misleading. If the 'real' world is quantum in nature, then in a certain sense integers are the 'real' numbers, and what we call the 'real' numbers are a kind of clever idealization. Irrational 'real' numbers only exist in our physically real world as limits. We can never compute their exact values. And 'imaginary' numbers are actually quite real in representing physical rotations. So these terms originated in a context that has long since passed (except in high school). It's no wonder that students relate to complex numbers as exotic and remote entities.

In a fused mathematical/computational literacy, using the terms 'exact' and 'inexact' to describe types of representation would make a lot of sense.

--

Michel

===================================
"What I cannot create, I do not understand."

- Richard Feynman

===================================
"Computer science is the new mathematics."

- Dr. Christos Papadimitriou
===================================

[David Chandler]

The rationals are a countable subset of the reals.  Ratio representation cannot approach the reals because it is inherently tied to the rationals by definition.  Decimals can represent reals, but truncation short of infinite decimals makes them rational too.  However, decimals can conceptually "point the way to reals" in a way that ratio representation cannot.  (1.414 is rational, but 1.414..., implying that we should infinitely continue the process that gave us those digits makes this a real number.)  In that sense the decimals used in computing could be thought of as "shadows" of the real numbers.  We can set a scale of approximation and continue the process until we surpass that level of precision.  In Pascal there was a Real type and a Double type.  I think there is a comfort level in using decimal "approximations" as a way to conceptualize real numbers.  Ratios are exact but inherently limited to rational numbers.  Decimal notation is just as limited, ultimately, but they are the finger pointing at the moon.

--David Chandler

--

[John Sincak]

"God created the integers, all else is the work of man."  Leopold Kronecker

[David Chandler]

By the way even sqrt(2) (radical notation) is essentially a way to "point at" a specific (exact) real number in the sense that it requires an algorithm with an infinite number of steps to get it into a usable form.  I guess this is why the Greeks stuck with geometry.
--David Chandler

[kirby urner]

On Sun, Mar 24, 2013 at 1:57 PM, kirby urner <kirby...@gmail.com> wrote:

<< SNIP >>

> Anyway, hey, I wonder what people think in general of using "real
> number" and "floating point" in narratives like this. You mention how
> we have a choice to use integers, rational numbers, or reals. But
> some student will object that reals were never on the table, floats
> being an IEEE specification and not of the real number set.
>
> Kirby

Good replies.

I think what I'm wondering is how much time to spend if any on some of
these well known and other more exotic number types that were
developed for use with digital computers. There's no "one size fits
all" answer.

In retrospect, we could call "slide rule numbers" those that were set
with a hair line on mutually sliding sticks and made to answer
multiplication questions, or roots, by means of logs.

One had to know something about propagating error and significant
digits, which was helpful in general where measurement is involved,
delicate instruments and "noise digits", plus and minus margins of
error.

All useful concepts.

Nowadays we have arbitrary precision decimal libraries (so-called
bignums) that run different algorithms than floating point and will
indeed carry your computation out to thousands or more digits if you
require that, with integers the same way in terms of arbitrarily
extensible.

As "approximations to the reals" we may pooh pooh all such
developments as evidence of how imperfect are our mechanical devices
compared with the pure mind of the superhero mathematician.

However I'm more interested in the opposite message: that these are
among the great engineering achievements of contemporary civilization
and have absorbed insights and skills that far transcend those of any
one human.

More like this: http://youtu.be/owtK58XiPGo

These are great and powerful tools, these arbitrary precision libraries.

If we bleep over this content, we lose many opportunities to marvel
and wonder, which I think schools too often unnecessarily squander.

None of which is to say I'm opposed to talking about the "real
numbers" in connection with matrices on SAGE. That's all fine. I'm
just suggesting a branch to additional worthwhile and mathematically
informative material.

Kirby

Example interesting web page:

http://rosettacode.org/wiki/Arbitrary-precision_integers_%28included%29

Looking at bignum capabilities of various languages.