what is the cube root of -1?
michel paul
"What I cannot create, I do not understand."
"Computer science is the new mathematics."
===================================
mok...@earthtreasury.org
> In Algebra 2 the typical text exercises will ask the kids to specify
> domains for expressions such as x^(1/3). I always tell the kids to feel
> free to use things like Wolfram Alpha as resources. One of my students
> pointed out that Alpha states the the domain of
> the non-negative reals but that our text specifies the domain as all
> reals. I found that really interesting, and we started exploring.
Wolfram Alpha is correct as far as it goes, but has left out the most
interesting part, presumably because most people think that complex
numbers are too hard. Your text is almost correct, but not quite.
If you use the non-negative reals as the domain for principal values of
fractional powers, you get a consistent, simple system where the answers
are consistent and many of the functions are continuous and infinitely
differentiable. However, you get only one nth root for each argument.
If you use all of the reals, you get inconsistent results from wildly
discontinuous functions. In particular, negative numbers have odd roots,
but no even roots. The number of roots of a number is variable among 0 1
2.
In the complex domain, everything is consistent and holomorphic
(infinitely differentiable in a much stronger sense than in the real
domain) almost everywhere. Every number other than 0 has n distinct nth
roots. However,
z0^z1 is in general infinitely-valued. For example, with i=.0j1, as in the
J language, i^i can be evaluated as ^i*^.i (exponential of i times ln i),
and by de Moivre's theorem, i has infinitely many logarithms, all
differing by multiples of 2*i*o.1 (conventionally, two pi i).
z0^z1 has an essential singularity at 0 0, where every finite complex
value occurs in every neighborhood of 0 0.
> If you google
> (-1)^(1/3)<https://www.google.com/webhp?sourceid=chrome-instant&ix=sea&ie=UTF-8&ion=1#hl=en&safe=off&output=search&sclient=psy-ab&q=(-1)%5E(1%2F3)&pbx=1&oq=&aq=&aqi=&aql=&gs_sm=&gs_upl=&fp=7f381450be81b469&ix=sea&ion=1&ix=sea&ion=1&bav=on.2,or.r_gc.r_pw.r_cp.r_qf.,cf.osb&fp=7f381450be81b469&biw=1134&bih=802&ix=sea&ion=1>,
> you get -1.
>
> Lots of calculators in class these days will give you -1, but I've seen
> many over the years where raising a negative base to a fractional power
> raises a domain error.
>
> Alpha says
> (-1)^(1/3<http://www.wolframalpha.com/input/?i=%28-1%29%5E%281%2F3%29>)
> is 0.5 +0.86602540378443864676372317075293618347140262690519 i.
>
> Python says
>>>> (-1)**(1/3)
> (0.5000000000000001+0.8660254037844386j)
J, using default print precision:
_1^%3
0.5j0.866025
> Both Mathematica and Sage say that (-1)^(1/3) is simply that ...
> (-1)^(1/3),
> but if you ask them for a decimal approximation, both will give the
> complex approximation.
>
> Alpha also gives you a graph showing the complex location.
>
> I found all this delightful. We usually don't expect Alg 2 to need to get
> into DeMoivre's Theorem or Euler notation, but when you have resources
> like
> this permeating the environment that we math teachers should be
> encouraging
> kids to use, they're going to be running into some real math occasionally.
Riemann surfaces for multiple-valued complex functions, too.
"I am never forget the day I am given my first original paper to write. It
is on Analytic and Algebraic Topology of Local Euclidean Metrization of
Infinite Differentiable Riemann Manifolds. Bozhe moi! This, I know from
nothing!"--Tom Lehrer, Lobachevsky
Nowadays, this is perfectly ordinary second-year grad school stuff. (The
proof that every holomorphic function has a Riemann surface was in my
first-year class on Complex Variables. I also took General Topology and
Algebraic Topology. But I didn't get the topology of Riemann surfaces
because I quit math to go into the Peace Corps.) The basic concepts of
Riemann surfaces can be made clear at a high-school level. For example,
square root is a two-valued function. Starting wherever you like, pick one
root, and go in a circle around the origin, maintaining continuity. (For
example, we can start at 1 with the root 1, or at _1 with the root 0j1.)
When you get back to your starting point (1 or _1), you have the other
root (_1 or 0j_1). Go around again, and you get back to the root you first
thought of. Do this for every complex value, and stitch the curves
together into surfaces, preserving continuity. You get two copies of the
complex plane less the origin. In polar coordinates, the square root has
half the angle of the argument in either direction. Similarly, you get
three sheets for cube roots. In both cases, 0 is an exceptional point.
The principal value of each of square root, cube root, etc. is a
discontinuous function created by choosing a cut line. Each sheet of the
Riemann surface for the function is cut along that line, and one of the
cut sheets is chosen. It has a discontinuity where its edges are stitched
together. A number of programming languages have chosen to use the same
cut lines and principal values for complex functions. They include Common
LISP, Ada, and the more recent versions of FORTRAN, all using the
mathematically sound choices in the ANSI/ISO APL standard (also used in
J). For example, the cut line for square root is along the negative real
axis, so that values arbitrarily close to _1 have principal square roots
that can differ by about 0j2.
%:_1+0 0j_0.0000001
0j1 5e_8j_1
> - Michel
>
> --
> ===================================
> "What I cannot create, I do not understand."
>
> - Richard Feynman
> ===================================
> "Computer science is the new mathematics."
>
> - Dr. Christos Papadimitriou
> ===================================
--
Edward Mokurai
(默雷/धर्ममेघशब्दगर्ज/دھرممیگھشبدگر
ج) Cherlin
Silent Thunder is my name, and Children are my nation.
The Cosmos is my dwelling place, the Truth my destination.
http://wiki.sugarlabs.org/go/Replacing_Textbooks
kirby urner
Not much questioning whether "cube" is the only/best shape where 3rd powering is in play. A blind spot I'd say.
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Maria Droujkova
Michael Paul Goldenberg
got a few typos in the translation, but nothing major. Thanks for
posting this stuff.
Quoting Maria Droujkova <drou...@gmail.com>:
> Michel, this is one of the best illustrations of the joys of computer math
> I've seen lately.
>
> This can be visualized by pretty young students, with tools:
> http://naturalmath.wikispaces.com/Imaginary+numbers+for+young+kids
> And especially the first set of images at
> http://naturalmath.wikispaces.com/Multiply+signed+numbers
>
>
> Cheers,
> Maria Droujkova
> 919-388-1721
>
> Make math your own, to make your own math
>
>
>
>
> On Fri, Feb 17, 2012 at 12:31 AM, michel paul <python...@gmail.com>wrote:
>
>> In Algebra 2 the typical text exercises will ask the kids to specify
>> domains for expressions such as x^(1/3). I always tell the kids to feel
>> free to use things like Wolfram Alpha as resources. One of my students
>> pointed out that Alpha states the the domain of
>> x^(1/3)<http://www.wolframalpha.com/input/?i=domain+of+x%5E%281%2F3%29>as
>> the non-negative reals but that our text specifies the domain as all
>> reals. I found that really interesting, and we started exploring.
>>
>>
>>
>
> --
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**************************
Michael Paul Goldenberg
6655 Jackson Rd Lot #136
Ann Arbor, MI 48103
734 644-0975 (c)
734 786-8425 (h)
mike...@umich.edu
rationalmathed.blogspot.com
"A public education system is based on the principle that you care
whether the kid down the street gets an education." Noam Chomsky
**************************
Maria Droujkova
Bradford Hansen-Smith
| Maria, attached is a folding circle approach that might be useful with young children in understanding imaginary numbers. Brad --- On Fri, 2/17/12, Maria Droujkova <drou...@gmail.com> wrote: |
|
|
|
|
Maria Droujkova
Bradford Hansen-Smith
| Of course Maria, I am interested to know how your mathematical mind understands this activity. Brad |
--- On Sun, 2/19/12, Maria Droujkova <drou...@gmail.com> wrote: |
|
|
kirby urner
mok...@earthtreasury.org
McIntyre was a professor of geology at Pomona College teaching how to do
this sort of thing in APL for many years. We can also do it in an
assortment of other geometries, for shapes that do not tile the space, and
in other dimensions, including fractal dimensions. Let me know if you want
to write those lessons for the Sugar Labs program on Replacing Textbooks.
On Sun, February 19, 2012 6:25 pm, kirby urner wrote:
> Apropos:
>
> https://groups.google.com/group/r-buckminster-fuller-synergetic-geometry/browse_thread/thread/7806d78601beca6b
>
>
> X-link / X-Ref intra Google Group
>
> *gdc*
>
> On Fri, Feb 17, 2012 at 11:17 AM, kirby urner <kirby...@gmail.com>
> wrote:
>
>> Not much questioning whether "cube" is the only/best shape where 3rd
>> powering is in play. A blind spot I'd say.
>
Troy A. Peterson
mok...@earthtreasury.org
> Okay, that blows my mind, I just had to say it.
Fantasy writer Terry Pratchett once complained (jokingly) about an online
conversation that I was involved in in the newsgroup alt.fan.pratchett,
that no matter how many drugs he might take he could not come up with
anything that was as wild and crazy as genuine math and physics. In that
vein, he opened a book with the line
In the beginning there was nothing, which exploded.
kirby urner
> We can do it in all 230 3-D crystallographic groups, if you like. Donald
> McIntyre was a professor of geology at Pomona College teaching how to do
> this sort of thing in APL for many years. We can also do it in an
> assortment of other geometries, for shapes that do not tile the space, and
> in other dimensions, including fractal dimensions. Let me know if you want
> to write those lessons for the Sugar Labs program on Replacing Textbooks.
>
I have an interesting relationship with Sugar.
I do voluntary promotion of brands, including of those with
significant market presence already, so I sometimes look like a shill
for Red Bull, Costco, or Unilever (to name some names in my deck).
With OLPC, I've lent the benefit of my lens, a collection on Flickr
numbering around 12K, thekirbster, and more (a small subset featuring
the XO etc):
http://www.flickr.com/photos/17157315@N00/4222225222/ (co-marketing
with Flextegrity, another brand I've given a good spin).
Most famous are probably these:
http://worldgame.blogspot.com/2009/01/saving-children.html
Negroponte is a subject of a GOSCON slide, if you want to search,
probably best a needle in some haystack for now.
Kirby
Maria Droujkova
On Tue, February 21, 2012 7:28 pm, Troy A. Peterson wrote:Fantasy writer Terry Pratchett once complained (jokingly) about an online
> Okay, that blows my mind, I just had to say it.
conversation that I was involved in in the newsgroup alt.fan.pratchett,
that no matter how many drugs he might take he could not come up with
anything that was as wild and crazy as genuine math and physics. In that
vein, he opened a book with the line
In the beginning there was nothing, which exploded.
kirby urner
> In Algebra 2 the typical text exercises will ask the kids to specify domains
> for expressions such as x^(1/3). I always tell the kids to feel free to use
> things like Wolfram Alpha as resources. One of my students pointed out that
> Alpha states the the domain of x^(1/3) as the non-negative reals but that
> our text specifies the domain as all reals. I found that really interesting,
> and we started exploring.
>
> If you google (-1)^(1/3), you get -1.
>
> Lots of calculators in class these days will give you -1, but I've seen many
> over the years where raising a negative base to a fractional power raises a
> domain error.
>
Just jumping in again to share about Pythonware, your resource for sharing
a Python console on-line (or just use one solo).
Cutting and pasting a recently shared session (PSF chairman on the
other end):
In [6]: pow(-1,1/3)
Out[6]: (0.5000000000000001+0.8660254037844386j)
In [7]: import cmath
In [8]: r = pow(-1,1/3)
In [9]: s = r.conjugate()
In [10]: s
Out[10]: (0.5000000000000001-0.8660254037844386j)
In [11]: s**3
Out[11]: (-1-3.885780586188048e-16j)
Was double checking the (correct) theory that -1 has these three 3rd
roots, two complex conjugates and -1.
I know that's revealed elsewhere here as well.
http://mail.python.org/pipermail/edu-sig/2012-February/010561.html
Kirby
kirby urner
> Just jumping in again to share about Pythonware, your resource for sharing
> a Python console on-line (or just use one solo).
>
Sorry, that's PythonAnywhere. I also have Pythonware on my mind because
of this order of clothing with the Python logo stitched on. We had a vendor
for that stuff but a recent sweep of the web suggested I'd need to "scratch my
own itch" or "eat my own dog food" as colorful geek language has it.
<aside>
"Geek" and "gypsy" and sometimes confused with good reason. Both
associate with carnivals, an atmosphere of fun for some, but also danger
and desperation simultaneously, the geek being desperately down on
his luck, so much so that he bites the head of a chicken in the side
shows. Traveling carnivals or circuses are typically associated with
gypsy folk. So it's not for nothing that they're entangled.
</aside>
It's not that I couldn't find some articles with the logo on it, just not
the items I wanted. The two fleeces I have, for example, were
not on tap.
Another logo-related project: kids at art colony type encampments,
such as we have at OPDX, draw airplane tail fin logos for some
of the NGOs they're learning about. More about that on the
Bucky Fuller list (Koski's) if interested:
http://groups.google.com/group/r-buckminster-fuller-synergetic-geometry/msg/21dccbb7719a99bb
Kirby
>
> http://mail.python.org/pipermail/edu-sig/2012-February/010561.html
>
> Kirby
Donald Cohen
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809 Stratford Dr.
Champaign, IL 61821-4140
Tel. 217-356-4555
Fax: 1 217 356 4593
Email: doncohe...@gmail.com
Don's Mathman website URL: http://www.mathman.biz
Our Shaklee website:
http://marilynanddoncohen.myshaklee.com
mok...@earthtreasury.org
> See mathematica version of the cube root of -1 and its graph at
J version:
]w=._1^%3 NB. Cube root of negative one ( ] displays result)
0.5j0.866025
]r=.w,(+w),_1 NB. Three cube roots
0.5j0.866025 0.5j_0.866025 _1
r^3
_1j3.88578e_16 _1j_3.88578e_16 _1 NB. Imaginary parts are near 0.
_1=r^3 NB. Are cubes of these values within fuzz of 1? (1=true)
1 1 1
load 'plot' NB. Library
NB. Now split roots into real and imaginary parts; transpose;
NB. box; make dot plot
'marker'plot ;/ |: +.r
(plot.png image attached)
michel paul
Yes, solving for cube root of -1 in Mathematica does yield the 3 roots, one of them being real,
but simply evaluating the cube root of -1 defaults to the first complex root as primary:
If the primary root was -1, ComplexExpand would yield -1. And it makes a lot of sense that this would be the primary cube root, as (-1)^(1/3) = (e^(i pi))^(1/3) = e^(i pi/3).
I love that. Euler notation rocks.
I've been thinking that there's no reason to withhold this from students until Calculus. They could use this much earlier.
For example, this year in studying the unit circle and the special angles, I had the students do it all in Euler notation.
You can derive the cos and sin of a sum very easily, and in the process they're getting complex numbers as well.
What strikes me is that we teach the kids the Fundamental Theorem of Algebra in Alg 2, then we hide it from them.
The texts force them to express everything in terms of the reals. With the proper computational tools in hand, there's no need to do that.
- Michel
Christian Baune
In response to last :
Most student and people box themselves and theirs minds into what they believe is necessary to solve.
Doing so, they do not only hide "complexity" or "noise" but completely avoid what could give them everything on a plate.
Maths are made easier but in a funy way it make accomplishment far harder since it requires you to process more instead of understanding a few!
This is a choice that had been made and it's hard to start over.
mok...@earthtreasury.org
> In response to last :
> Most student and people box themselves and theirs minds into what they
> believe is necessary to solve.
> Doing so, they do not only hide "complexity" or "noise" but completely
> avoid what could give them everything on a plate.
>
> since it requires you to process more instead of understanding a few!
There is an excellent little book with the title Mathematics Made
Difficult. Its thesis is that learning just a bit of harder math makes
wide swaths much easier to state, to prove, and to understand.
> This is a choice that had been made and it's hard to start over.
> Le 23 mars 2012 13:56, "michel paul" <python...@gmail.com> a �crit :
>
>> Yes, solving for cube root of -1 in Mathematica does yield the 3 roots,
>> one of them being real,
Have you seen the J solution that I posted?
>> but simply evaluating the cube root of -1 defaults to the first complex
>> root as primary:
>>
>> If the primary root was -1, ComplexExpand would yield -1. And it makes a
>> lot of sense that this would be the primary cube root, as (-1)^(1/3) =
>> (e^(i pi))^(1/3) = e^(i pi/3).
>>
>> I love that. Euler notation rocks.
0=1+^0j1*1p1
>> I've been thinking that there's no reason to withhold this from students
>> until Calculus. They could use this much earlier.
Yes, we teach math topics in a sequence derived from the history of their
discovery, not in any logical order of ideas.
>> For example, this year in studying the unit circle and the special
>> angles, I had the students do it all in Euler notation.
>>
>> You can derive the cos and sin of a sum very easily, and in the process
>> they're getting complex numbers as well.
Saunders Mac Lane pointed out in Mathematics Form and Function that it is
trivial to demonstrate the sum and difference formulas if you have the
basic idea that a rotation is a linear function, and that a quarter circle
rotation carries x to y, and y to -x. Analyze a rotation into components,
and apply two rotations.
This is equivalent to multiplying by two complex numbers of magnitude 1.
The sum and difference trig formulas are just components of the complex
multiplication formula.
>> What strikes me is that we teach the kids the Fundamental Theorem of
>> Algebra in Alg 2,
With a great deal of handwaving rather than proof. It is easy to
demonstrate that a polynomial of odd degree with real coefficients has at
least one real root, but factoring a polynomial of even degree into
monomials for real roots together with quadratics for complex conjugate
roots is much harder, and proving that you can always do it is harder
still.
>> then we hide it from them.
In the complex domain it is straightforward to prove that any polynomial
bounded away from 0 is a constant, so every polynomial of degree 1 or
higher has a root.
>> The texts force them to express everything in terms of the reals. With
>> the
>> proper computational tools in hand, there's no need to do that.
Have you seen my treatment of complex numbers in Algebra: An Algorithmic
Treatment? The original APL edition omitted them, because early APLs did
not have a complex type. J has excellent facilities for handling complex
numbers. (The j notation, for example 0j1, is derived from electrical
engineering usage, because i had long been used to mean current.)
NB. The J comment symbol is NB. (nota bene).
_1^0.5 NB. _ is the negative sign, distinct from -
0j1
+.0j1 NB. Real and imaginary parts of a complex number
0 1
0 j. 1 NB. Complex number formed from real and imaginary parts
0j1
+0j1 NB. Complex conjugate
0j_1
*:0j1,+0j1 NB. Square the square roots of _1
_1 _1
|0j1,+0j1 NB. Complex magnitude r
1 1
1p1 NB. pi
3.14159
1p2 NB. pi squared
9.8696
1p2%6 NB. Famous value
1.64493
+/%*:>:i.2000 NB. Sum of reciprocals of squares
1.64443
+/%*:>:i.2000000
1.64493
r. 0.5p1 NB. Euler/r theta notation; Complex number of magnitude 1,
angle pi/2
6.12323e_17j1
0j1=r. 0.5p1 NB. Equality with tolerance; 1 means true
1
2 r. 0.3333333333p1
1j1.73205
(2 r. 0.3333333333p1)^3 NB. Approximate cube root of _8
_8j2.51328e_9
*. 3j4 NB. Yields r and theta for complex number
5 0.927295
I derive executable definitions for complex arithmetic. (In J, + - * %)
The original text included programs for Newton-Raphson iterative equation
solvers, which I translated to J.
>> - Michel
>> On Wed, Mar 21, 2012 at 11:53 AM, Donald Cohen
>> <doncohe...@gmail.com>wrote:
>>
>>> See mathematica version of the cube root of -1 and its graph at
>>> cuberootof-1.jpg[image: Inline image 1]
--
Donald Cohen
Don
On Fri, March 23, 2012 11:06 am, Christian Baune wrote:
> In response to last :
> Most student and people box themselves and theirs minds into what they
> believe is necessary to solve.
> Doing so, they do not only hide "complexity" or "noise" but completely
> avoid what could give them everything on a plate.
>
> Maths are made easier but in a funny way it make accomplishment far harder
> since it requires you to process more instead of understanding a few!
There is an excellent little book with the title Mathematics Made
Difficult. Its thesis is that learning just a bit of harder math makes
wide swaths much easier to state, to prove, and to understand.
> This is a choice that had been made and it's hard to start over.
--
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michel paul
On Mar 23, 2012 8:06 AM, "Christian Baune" <progr...@gmail.com> wrote:
>
> Most student and people box themselves and theirs minds into what they believe is necessary to solve.
Right. It leads to over-memorization and all kinds of superficiality.
> Doing so, they do not only hide "complexity" or "noise" but completely avoid what could give them everything on a plate.
> Maths are made easier but in a funy way it make accomplishment far harder since it requires you to process more instead of understanding a few!
>
> This is a choice that had been made and it's hard to start over.
Yes, I've had some interesting experiences in pointing out areas where we really could start over. I've noticed some differences between common computational math usages vs. our standard secondary math curriculum. The most recent observation was this (-1)^(1/3) issue.
Another observation I had awhile ago is that the standard curriculum emphasizes a distinction between 'log', referring to common base 10 logs, and 'ln', referring to natural base e logs. However, what I've noticed as a standard in various computational languages is that 'log' is by default natural. : ) And I've noticed that it is common to designate a base 10 log as 'log10'. It also seems to be a standard for log to be available as a log(n,base).
When I brought up this observation at a math dept meeting, that the default for 'log' in many commonly available computational contexts is not base 10 but e, I was met with disbelief. "Since when?" "Who thinks they have the right to just change things?"
All I could say was, well, I'm sorry, but that's just the way things seem to be in the world outside of high school math. And it makes a lot of sense.
Why do we insist on making students first learn that 'log' means base 10, forcing them to then relearn later that it is actually base e? Why not first let them explore the concept of logs to any base using a log(n,b) function? Don't restrict them to base 10 and base e - let them explore getting results in terms of any base. THEN, after exploring, let them know that log(n) without a specified base is by default natural. Then, if we can then help them develop the understanding that log(n,b) = log(n)/log(b), it seems like that much would provide a better framework than what we typically do.
-- Michel
michel paul
On Mar 23, 2012 10:44 AM, <mok...@earthtreasury.org> wrote:
>
> There is an excellent little book with the title Mathematics Made
> Difficult. Its thesis is that learning just a bit of harder math makes
> wide swaths much easier to state, to prove, and to understand.
This is hilarious! And insightful. So far I've only seen the wikipedia article, but the example they give makes you both laugh and go 'hmmm..' . Very cool.
> > Le 23 mars 2012 13:56, "michel paul" <python...@gmail.com> a écrit :
> >
> >> Yes, solving for cube root of -1 in Mathematica does yield the 3 roots,
> >> one of them being real,
> Have you seen the J solution that I posted?
Yes. I find J intriguing. I haven't used it yet, but I'm interested in what you and Kirby have had to say about it.
>...
> Have you seen my treatment of complex numbers in Algebra: An Algorithmic
> Treatment?
Yes, this is great. I really like what you've created with Iverson's book as a whole. It's useful to look through even if one is not necessarily programming with J. Good for mathematical/computational thinking per se.
michel paul
mok...@earthtreasury.org
> This is great!
>
> Wow, you know what? Years ago, must have been back when I first started
> conference. I've always liked that book, and still keep it here on my desk
> to refer to. And so ... turns out that's your book! Right? Amazing how
> small the world is. : )
That's a large part of my job in the Replacing Textbooks program at Sugar
Labs: making the world smaller by seeking out the right people and
bringing them together. The same with many others in many other such
programs.
> - Michel
>
>
> On Fri, Mar 23, 2012 at 6:42 PM, Donald Cohen
> <doncohe...@gmail.com>wrote:
>
>> Michael, see my A Map To Calculus
>> <http://www.mathman.biz/html/map.html>at the bottom click on (1+i)^n
>> (spiral), then go to "points on the spiral
>> (1+i)^n written different ways"
>>
>> Don
>>
>>
>> On Fri, Mar 23, 2012 at 12:43 PM, <mok...@earthtreasury.org> wrote:
>>
>>> On Fri, March 23, 2012 11:06 am, Christian Baune wrote:
>>> > In response to last :
>>> > Most student and people box themselves and theirs minds into what
>>> they
>>> > believe is necessary to solve.
>>> > Doing so, they do not only hide "complexity" or "noise" but
>>> completely
>>> > avoid what could give them everything on a plate.
>>> >
>>> > Maths are made easier but in a funny way it make accomplishment far
>>> harder
>>> > since it requires you to process more instead of understanding a few!
>>>
>>> There is an excellent little book with the title Mathematics Made
>>> Difficult. Its thesis is that learning just a bit of harder math makes
>>> wide swaths much easier to state, to prove, and to understand.
>>>
>>> > This is a choice that had been made and it's hard to start over.
>>> > Le 23 mars 2012 13:56, "michel paul" <python...@gmail.com> a
>>> �crit
>> *"Learning, Living and Loving mathematics.."- the core of Don's teaching
>> and books, **observed by Seth Nielson.*
>> *The Math Program*
>> *Don Cohen -The Mathman*
>> 809 Stratford Dr.
>> Champaign, IL 61821-4140
>> Tel. 217-356-4555
>> Fax: 1 217 356 4593
>> Email: doncohe...@gmail.com
>> Don's Mathman website URL: http://www.mathman.biz
>> See Don's *new* *clickable* A Map to
>> Calculus<http://www.mathman.biz/html/map.html>with student works and
>> sample problems from Don's books at every node
>> Our Shaklee website:
>> http://marilynanddoncohen.myshaklee.com
>>
>>
>>
>> --
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>>
>
>
>
> --
> ===================================
> "What I cannot create, I do not understand."
>
> - Richard Feynman
> ===================================
> "Computer science is the new mathematics."
>
> - Dr. Christos Papadimitriou
> ===================================
>
Donald Cohen
Troy A. Peterson
Linda Fahlberg-Stojanovska
Probably not the place for this question, but I doubt the general interest :) and thought that perhaps the people who contributed here might be.
When you solve a homogeneous linear differential equation you use the rules:
Now I understand the proofs of these – but to most students, it seems to be a lot of manipulations of manipulations of fake notation (e.g. e^(a+bi) is what??)
Question: Is there a relatively easy reason (easy to present/remember) that real roots (and real parts of roots) give you e^(ax) and complex roots give you trig functions.
Perhaps it is related to the way the trig functions change signs when you differentiate?
Thanks for any insight into this … Linda
kirby urner
>>> For example, this year in studying the unit circle and the special
>>> angles, I had the students do it all in Euler notation.
>>>
>>> You can derive the cos and sin of a sum very easily, and in the process
>>> they're getting complex numbers as well.
>
This rings a bell with me. I wanted my teenagers in Martian Math
to have orbiting planets to play with, maybe change color, size,
rate of revolution. How to implement in Visual Python?
[ The explosion of interest in Fractals once Mandelbrot and the
IBM team had them figured out, stemmed from their easy
comprehensibility once complex number multiplication is
modeled as rotation. "Add the angles, multiply the lengths".
z = z*z + c. It's a lot like needlepoint and you're trying to
figure out the color for each dot in the matrix, based on how
fast it spirals out, or if it converges ]
Vectors ala Gibbs / Heaviside is what we spend time on,
with their addition, subtraction, and scalar multiplication.
But vectors, unlike quaternions, don't multiply. Complex
numbers, on the other hand, do, and yet are vector like.
So if we want a clock hand to just go round and round, we
can accomplish that by repeatedly multiplying the
"clock hand" by a fixed complex number, a delta. Have
the magnitude be 1. Theta changes, radius does not.
Voila, a planet in orbit:
http://4dsolutions.net/ocn/python/orbits.py
Kirby
David Chandler
sum formulas for sines and cosines. I naively launched down that road
with a student last week and hit a logical roadblock. My starting
point was i^2=-1, and the assumption that complex numbers are a field,
so you have the distributive law. From there you can do complex
multiplication in rectangular form. Next I translated two complex
numbers of the form a+ib into polar notation, then multiplied. This
leads directly to the cosines and sines of sums formula, except there
is no identification of the expressions with the sum of angles yet.
If you knew that these expressions represented the sum formulas it
would be easy to justify Euler notation, because it is easy to argue
that the "cis" function behaves like an exponential function in
following the laws of exponents, but you can't get this without
already recognising that cosAcosB-sinAsinB is an expansion of
cos(A+B) and sinAcosB+cosAsinB is an expansion of sin(A+B).
I looked up "angle addition formulas complex numbers" and it seems
that to derive the angle addition formulas from euler's notation you
have to postulate euler's notation, which breaks the tie-in with the
original field axioms.
Bottom line: I can go from properties of i to polar notation (cis) to
euler notation, but I need the angle additiona properties to make the
connection. Alternatively I can postulate Euler's notation, get the
angle addition formulas as an easy corollary, then try to figure out
how this is consistent with the original rectangular notation. As a
consequence, I have typically used the Euler notation "derivation"
only as a "rederivation", sort of like a mnemonic for quickly coming
up with the angle sum formulas, but I haven't integrated it into my
main presentation of these topics.
Any suggestions?
--David Chandler
Oleg Gleizer
If f_1 and f_2 are solutions of a homogeneous ODE , then so are c_1 f_1
+ c_2 f_2 for all complex constants c_1 and c_2. Zero function is a
solution of a homogeneous ODE, and so on. You can easily check that the
set of all the solutions of a homogeneous ODE forms a linear space. The
dimension of the space equals to the order of the equation. You can see
it by rewriting the equation as a system of first order homogeneous ODEs
and exponentiating the matrix. In order to do that, you first bring the
matrix of the system to the Jordan normal form, then exponentiate, then
bring it back to the original basis. If all the eigenvalues of the
matrix, which happen to be the roots of the characteristic equation of
the original ODE, are different,
then your system splits into a direct sum of first order systems of the
form dx_i/dt = r_i t, hence you get a bunch of exponents e^(r_i t). Then
you mix them up coming back to the original basis. If the Jordan normal
form is not diagonal, then you get a bunch of t^j e^(r_i t), where the
max power of j equals to the size of the corresponding Jordan block. All
this machinery works naturally over complex numbers. If the original
coefficients were real, it's a matter of good taste to bring the
solutions back to the real form, at which stage trigs pop up.
Please let me know if this was sufficient. I can elaborate if needed.
Very Truly Yours,
Oleg Gleizer.
On 4/21/12 12:32 AM, Linda Fahlberg-Stojanovska wrote:
> Probably not the place for this question, but I doubt the general interest
> :) and thought that perhaps the people who contributed here might be.
>
> When you solve a homogeneous linear differential equation you use the rules:
>
>
>
> Now I understand the proofs of these - but to most students, it seems to be
> a lot of manipulations of manipulations of fake notation (e.g. e^(a+bi) is
> what??)
>
>
>
> Question: Is there a relatively easy reason (easy to present/remember) that
> real roots (and real parts of roots) give you e^(ax) and complex roots give
> you trig functions.
>
> Perhaps it is related to the way the trig functions change signs when you
> differentiate?
>
>
>
> Thanks for any insight into this . Linda
>
Linda Fahlberg-Stojanovska
Thanks so much for taking the time for this explanation.
#1 It NEVER occurred to me that (of course) c_1 and c_2 could be complex
constants. (It is always amazing to me what DOESN'T occur.)
#2 Also, I very much like the relationship to eigenvalues and roots so I
will look into that. I am very slowly learning these connections and
thinking about how to "see" them for students. Perhaps with some technology.
#3 What I really like is the "good taste" comment; that is very much what I
was looking for with the question I posed. That is something reasonably easy
to say to students that help them see the "unified" solution and then how it
got to the exponential+trig expression.
Again - very much appreciated!
Linda
-----Original Message-----
From: mathf...@googlegroups.com [mailto:mathf...@googlegroups.com] On
Behalf Of Oleg Gleizer
Sent: Saturday, April 21, 2012 8:55 PM
To: mathf...@googlegroups.com
Subject: Re: Complex numbers (was Re: [Math 2.0] what is the cube root of
-1?)
Charischak Ihor
By the way Don... This is beautiful. #truly http://www.mathman.biz/html/map.html
http://www.mathman.biz/html/map.html
If I was back in middle school, this map would have motivated me to hang out at this town called Calculus.
Charischak Ihor
Pelageya Privet
She had plans for 2-3 weeks. I don't know if she is reachable, but her
absence was planned.
Marina
On Mon, Apr 23, 2012 at 8:11 AM, Charischak Ihor <ih...@clime.org> wrote:
> Hi,
> Anyone know what Maria is up to? Haven't heard back from her in a while.
> -
>
>
>
>
>
>
Sue VanHattum
Warmly,
Sue
Charischak Ihor
--
kirby urner
> Troy writes:
>
> By the way Don... This is beautiful. #truly
> http://www.mathman.biz/html/map.html
>
Hey that's a really nice math map. I posted it to our faculty lounge
on Facebook.
The Fibonacci Numbers is where we might bubble up for a visit from our
Discrete Math beat, where we're learning about generators in Python
(an executable math notation -- MN):
http://wikieducator.org/PYTHON_TUTORIALS#Generators (WikiEducator
seems to crash Shockwave in Chrome or is that just my imagination
that's crashing?)
We're deep into figurate numbers (including those expanding
omni-directionally, e.g. icosahedral numbers 1, 12, 42, 92, 162... --
great for writing small programs.
I also like Cherlin's propensity to use J, though I consider J alone,
or J and APL only, to be overly esoteric and elitist (not that I'm
against either in moderation).
Kirby
David Chandler
Any thoughts on this?
--David Chandler
michel paul
On Sat, Apr 21, 2012 at 10:42 AM, David Chandler <david...@gmail.com> wrote:
> I can go from properties of i to polar notation (cis) to
> euler notation, but I need the angle additiona properties to make the
> connection. Alternatively I can postulate Euler's notation, get the
> angle addition formulas as an easy corollary, then try to figure out
> how this is consistent with the original rectangular notation. As a
> consequence, I have typically used the Euler notation "derivation"
> only as a "rederivation", sort of like a mnemonic for quickly coming
> up with the angle sum formulas, but I haven't integrated it into my
> main presentation of these topics.
>
> Any suggestions?
Expand(1+r/n)^n using Binomial Theorem to get e^r.
What I did was first remind the students about the compound interest formula P(1+r/n)^(nt) and how lim n->oo P[(1+r/n)^n]rt turns into Pe^(rt).
As part of that discussion I had them expand (1+r/n)^n.
After simplifying and taking limits we get 1 + x + x^2/2! + x^3/3! + x^4/4! + ... , which also happens to be the Taylor series for e^x. They of course don't know that yet, but it becomes a nice preview of coming attractions.
Evaluating that for x = i theta gives us 1 + i theta - theta^2/2! - i theta^3/3! + theta^4/4! + ... , which is series(cos theta) + i series(sin theta).
In this way the only thing that has to be taken on faith is the Taylor series for sin and cos. However, it's easily illustrated using something like GeoGebra or Sage, and I just tell the students it's something they'll study in Calculus.
-- Michel
Mike Thayer
michel paul
>I wonder if this would help:
-Mike
... ... Gauss is reported to have commented that if this formula was not immediately obvious, the reader would never be a first-class mathematician (Derbyshire 2004, p. 202).
A mathematical joke asks, "How many mathematicians does it take to change a light bulb?" and answers ""
michel paul
"What I cannot create, I do not understand."
"Computer science is the new mathematics."
===================================
Linda Fahlberg-Stojanovska
Alexander Bogomolny
michel paul
michel paul
...
> http://4dsolutions.net/ocn/python/orbits.py
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