What a coincidence - I am reading Hardy's /Pure Mathematics/ and I just
finished the chapter on complex numbers, which he develops in exactly
this way. Start with a number line, and the concept of distance. Move
to points in the plane, and the concept of displacement. Then to get a
definition of 'multiplication of displacements' that satisfies
commutativity etc., you have to end up with a definition of
multiplication as add-the-angles-and-multiply-the-magnitudes. All done
without mentioning square root of anything (and actually, you can do the
multiply-the-magnitudes part by construction, using similar triangles,
so you don't need square root for that either).
Henry Rich
On 1/15/2013 5:04 PM, km wrote:
Linda,
Perhaps we should be in chat, but I do not think you need square root to
introduce complex numbers, nor do young students need to calculate with them by
hand. We have calculators now.
Students are taught early about plotting points in a coordinate plane. I would introduce
complex numbers as names for points. Point 2j8 is 2 units to the right and 8 units up from
the origin. The point halfway between two complex numbers is halfway between the origin and
their sum. Draw rays from the origin to two complex numbers. Multiplying them involves a
rotation and stretching of the second ray by amounts indicated by the first ray. "This
much" and 'this much." That is why the product of 0j1 and 0j1 is _1j0, called _1
for short. It is also why _1j0 times _1j0 is 1j0. You'll learn more about complex numbers
when you get to the sixth grade, and you can already add and multiply them on your calculator.
I concede that using 2j8 as a code for "2 and 8" may be a stretch -- but young
students are flexible!
Kip Murray
Sent from my iPad
On Jan 15, 2013, at 2:16 PM, Raul Miller <[email protected]> wrote:
Why is the j in i:2j8 more important than the i?
--
Raul
On Tue, Jan 15, 2013 at 3:00 PM, Linda Alvord <[email protected]> wrote:
I am considering the structure of mathematics education. A domain of numbers
from _2 to 2 in steps of one half could be understood and developed in
elementary school even in J. It is not really sensible until square root is
mastered. After that an imaginary numbers might show up. So %:_1 becomes
0j1 in J. It will take a while to master numbers like 2j8 and other
imaginaries. The leap to an idiom which uses 2j8 in a totally different
way is counterproductive until much later.
How can you explain the connection to imaginary numbers in this expression:
i:2j8
_2 _1.5 _1 _0.5 0 0.5 1 1.5 2
Linda
-----Original Message-----
Froch: [email protected]
[mailto:[email protected]] On Behalf Of Raul Miller
Sent: Tuesday, January 15, 2013 11:35 AM
Tr o: [email protected]
Subject: Re: [Jprogramming] atop continues to puzzle me
I am uncomfortable with this reasoning, because:
*) i:9j3 is related to complex arithmetic only by notation.
*) class becomes boring when its pace is set for someine else (when it does
not match the student's needs).
*) simplicity is good, but so are different perspectives.
Waiting to introduce imaginary numbers before introducing this notation
seems analogous to waiting to introduce polynomials before introducing
decimal numbers.
Anyways, I this notation can be optional, but if I were stuck teaching a
class using J (rather than providing guidance, pacing and structure to a
group of students who had been taught how to learn and who were being
responsible for their own education) I think I'd introduce this as an
optional notation about the same time I introduced i:
--
Raul
On Tuesday, January 15, 2013, Linda Alvord <[email protected]> wrote:
Devon, Since I think of everything going forward from kindergarten
(maybe someday we'll do more with prenatal education) I would use Y=:
_2 + 0.5 * i.9" until the middle of the second year of algebra.
I really like your solution! It would be a great next step when
teaching imaginary numbers.
Linda
-----Original Message-----
From: [email protected]
[mailto:[email protected]] On Behalf Of Devon
McCormick
Sent: Monday, January 14, 2013 2:58 PM
To: J-programming forum
Subject: Re: [Jprogramming] atop continues to puzzle me
Linda -
a nice short cut for your expression "Y=: _2 + 0.5 * i.9" is "Y=: i:2j8".
In general, providing the complex argument PjN to i: gives you N+1
points from -P to P.
Regards,
Devon
On Mon, Jan 14, 2013 at 12:24 PM, Jose Mario Quintana <
[email protected]> wrote:
To Henry:
My apologies, the worst part is that I noticed the misspelling but I
neglected to correct it; I guess watching the playoffs and writing to
the forum do not mix very well.
To Linda:
That is a nice feature; thanks for sharing it. One refreshing thing
about J is that one never seems to stop learning it. Moreover, if one
follows the forums and this one in particular one is shown (or
reminded) how capable the J system really is.
On Mon, Jan 14, 2013 at 3:10 AM, Linda Alvord
<[email protected]>
wrote:
Jose, Here's a simper version. Using the aspect ratio helps make
the derivative more obvious. I use Chrome and I don't know how
this will
look
elsewhere.
Load 'plot'
u=: -:
v=: *:
Y=:_2 + 0.5 * i.9
f=: 13 :'(] ; [:|:u@v d._2 _1 0 1 2 )y'
f
plot f Y
'aspect 1'plot f Y
Linda
-----Original Message-----
From: [email protected]
[mailto:[email protected]] On Behalf Of
Linda
Alvord
Sent: Sunday, January 13, 2013 9:21 PM
To: [email protected]
Subject: Re: [Jprogramming] atop continues to puzzle me
load'plot'
u=: -:
v=: *:
Y=: _2 + 0.01 * i.401
f=: 13 :'(] ; [:|:u@v d._2 _1 0 1 2 )y'
f
] ; [: |: u@v d._2 _1 0 1 2
plot f Y
If this is in a jijs and then run, it will shw the graph you
expect bu
t
it will provide a long J error message in a separate window. I
don't
know
how to prevent it.
Also, maybe when u@v can be replaced by ([:u v)"v that will
work
also.
Linda
-----Original Message-----
From: [email protected]
[mailto:[email protected]] On Behalf Of Jose
Mario
Quintana
Sent: Sunday, January 13, 2013 5:17 PM
To: [email protected]
Subject: Re: [Jprogramming] atop continues to puzzle me
To Raul:
I have no idea what "works reasonably well" means.
That is a very subjective statement, apparently one can make use of
(@) and (@:) within the scope of (d.) but, of course, that depends
on
one's
point of view.
But consider also:
AT=: 2 :0
u@v"v
)
+:AT*: d. 1
0 4x&p."0 0 0
+:@*: d. 1
0 4x&p.
Th> >> --- For information about J forums see
http://www.jsoftware.com/forums.htm
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--
Devon McCormick, CFA
^me^ at acm.
org is my
preferred e-mail
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