Sooooo..... being nonprofessional, I found myself doing a double-take when
magazine articles use the word "width" when I would have expected "length",
or vice versa.

It took a couple decades, but I finally came to the conclusion that "width"
means "across the grain"...."length", along it.....irrespective of the
actual dimensions of the board (a board could be much wider than its length,
and up until my new understanding I would have reversed the words, to apply
"length" to the longest dimension).

And then p29 in the current WOODSMITH, in discussing how to make crossgrain
splines, says to "cut the splines to width (length)." aaaaack!
I do see what, in their normally helpful way, they are talking about.....but
got me to thinking.


Which led to my overthinking things and now I have a need to know.

Let's say you have a board that is 1" thick, 3" wide, 6" long.
The "ends" are where the grain shears off.
The "edges" are the thinner sides.
The "faces" are the wider sides.

But if I were to rip a thin strip off ...say 1/2" wide, so that the smaller
"board" is 1 x 1/2 x 6...have the "edges" and "faces" changed places?
Is the "edge" still the thinner (1/2") side?
If I rotate that board axially, the wider side (1") looks like a "face" to
me. Or is there some subtle grain differentiation that I don't get?

Or did I overthink myself into stupidity?

john

On 03/24/2010 11:18 AM, wrote:

Let's say you have a board that is 1" thick, 3" wide, 6" long.
The "ends" are where the grain shears off.
The "edges" are the thinner sides.
The "faces" are the wider sides.

But if I were to rip a thin strip off ...say 1/2" wide, so that the smaller
"board" is 1 x 1/2 x 6...have the "edges" and "faces" changed places?
Is the "edge" still the thinner (1/2") side?

Personally I'd say yes. The edge should be the thinnest side that shows
long grain.

Of course it gets tricky if you have pieces with a square cross-section.

Incidentally, for cabinetmaking plywood the second dimension is the
grain direction. So an 8x4 sheet has the grain going the short way.

Chris

"Chris Friesen" wrote in message
news
On 03/24/2010 11:18 AM, wrote:

Let's say you have a board that is 1" thick, 3" wide, 6" long.
The "ends" are where the grain shears off.
The "edges" are the thinner sides.
The "faces" are the wider sides.

But if I were to rip a thin strip off ...say 1/2" wide, so
that the smaller
"board" is 1 x 1/2 x 6...have the "edges" and "faces" changed
places?
Is the "edge" still the thinner (1/2") side?

Personally I'd say yes. The edge should be the thinnest side
that shows
long grain.

Of course it gets tricky if you have pieces with a square
cross-section.

Incidentally, for cabinetmaking plywood the second dimension is
the
grain direction. So an 8x4 sheet has the grain going the short
way.

Chris

Thanks, I learned something today.

--
Nonny
Suppose you were an idiot.
And suppose you were a member
of Congress.... But then I repeat myself.'

-Mark Twain
..

On 3/24/10 12:18 PM, wrote:
Sooooo..... being nonprofessional, I found myself doing a double-take when
magazine articles use the word "width" when I would have expected "length",
or vice versa.


Sorry, this is a tangent....

A pet piece of mine is when "woodworkers"-- especially ones on TV
shows-- say the word, "heighth."

They think that somehow, because length, width, and depth, have a "th"
sound at the end, then so does height.
Sometimes, I'll jokingly say, "Hey, height only has two H's."


--

-MIKE-

"Playing is not something I do at night, it's my function in life"
--Elvin Jones (1927-2004)
--
http://mikedrums.com

---remove "DOT" ^^^^ to reply

"Chris Friesen" wrote in message
news
On 03/24/2010 11:18 AM, wrote:

Let's say you have a board that is 1" thick, 3" wide, 6" long.
The "ends" are where the grain shears off.
The "edges" are the thinner sides.
The "faces" are the wider sides.

But if I were to rip a thin strip off ...say 1/2" wide, so that the
smaller
"board" is 1 x 1/2 x 6...have the "edges" and "faces" changed places?
Is the "edge" still the thinner (1/2") side?

Personally I'd say yes. The edge should be the thinnest side that shows
long grain.

Of course it gets tricky if you have pieces with a square cross-section.


Incidentally, for cabinetmaking plywood the second dimension is the
grain direction. So an 8x4 sheet has the grain going the short way.

Chris


Lowe's has some 4X4 sheets.
What is one to do?

Max

Ehh... you need to be careful for what purpose is the statement being
made.

Face and edge mean very specific things when you are considering
grain. Edge grain is very different than face grain in terms of
apperance and somewhat structurally when gluing. However, even that
has some ambiguity because you cut wood from round logs.

So if you cut a 4" square timber from the center of an 8" round log it
has face grain on 4 faces.

However, if you slice a 1" thick board from the same round log (rather
than a 4x) it has face grain on the wide "faces" and edge grain on the
thin sides. The face always shows edges of rings to varying degrees
while the edge shows to varying degrees the faces of rings. The piece
cut from the center of the log will have an edge grain with almost
completely the face of a ring. Confusing enough?

An edge to edge glue joint is super strong. A face to face glue joint
is super strong. An edge to face glue less strong. This last sentence
is conjecture on my part but it seems to bear out in my experience but
matbe because edges are usually narrow and faces usually wide so the
joint has available leverage to break it.


On Mar 24, 10:18*am, wrote:
Sooooo..... *being nonprofessional, I found myself doing a double-take when
magazine articles use the word "width" when I would have expected "length",
or vice versa.

It took a couple decades, but I finally came to the conclusion that "width"
means "across the grain"...."length", along it.....irrespective of *the
actual dimensions of the board (a board could be much wider than its length,
and up until my new understanding *I would have reversed the words, to apply
"length" to the longest dimension).

And then p29 in the current WOODSMITH, in discussing how to make crossgrain
splines, says to "cut the splines to width (length)." aaaaack!
I do see what, in their normally helpful way, they are talking about.....but
got me to thinking.

Which led to my overthinking things and now I have a need to know.

Let's say you have a board that is *1" thick, 3" wide, 6" long.
The "ends" are where the grain shears off.
The "edges" are the thinner sides.
The "faces" are the wider sides.

But if I were to rip a thin strip off ...say 1/2" *wide, so that the smaller
"board" is *1 x 1/2 *x 6...have the "edges" and "faces" changed places?
Is the "edge" still the thinner (1/2") side?
If I rotate that board axially, the wider side (1") looks like a "face" to
me. *Or is there some subtle grain differentiation that I don't get?

Or did I overthink myself into stupidity?

john

On 03/24/2010 01:18 PM, SonomaProducts.com wrote:

An edge to edge glue joint is super strong. A face to face glue joint
is super strong. An edge to face glue less strong. This last sentence
is conjecture on my part but it seems to bear out in my experience but
matbe because edges are usually narrow and faces usually wide so the
joint has available leverage to break it.

Your conjecture is false.

The fact that longer pieces confer more leverage doesn't imply anything
about the strength of the joint itself.

And besides, an edge-to-edge joint with the same glue area has at least
as much leverage potential--imagine holding onto the edges of a panel
and bending it over your knee right along the glue join.

Generally speaking a well-constructed long-grain to long-grain glue
joint is as strong as the wood itself regardless of growth ring
orientation. This is a simplification, but one that works in the vast
majority of cases.

Chris

On 3/24/2010 2:18 PM, SonomaProducts.com wrote:
Ehh... you need to be careful for what purpose is the statement being
made.

Face and edge mean very specific things when you are considering
grain. Edge grain is very different than face grain in terms of
apperance and somewhat structurally when gluing. However, even that
has some ambiguity because you cut wood from round logs.

More importantly, most "boards" are cut from the "length" of round logs.

Tree cells are generally elongated and and that elongation is aligned to
carry water and nutrients along the length of the tree trunk (log) and
limbs.

"Long grain", running along the length of elongated cells structures,
and "end grain" running across the elongated cell structure, are much
better terminology for joinery/gluing purposes, IMO.

An edge to edge glue joint is super strong. A face to face glue joint
is super strong. An edge to face glue less strong. This last sentence
is conjecture on my part but it seems to bear out in my experience but
matbe because edges are usually narrow and faces usually wide so the
joint has available leverage to break it.

Not to quibble, but I would tend to disagree somewhat with that conjecture.

Long grain to long grain, whether it be on a "face" or "edge", should
have the same relative glued strength characteristics for the most part
and, MOST IMPORTANTLY, should also the same relative dimensional
stability for joinery/gluing purposes (although the way it was cut from
the log (i.e., flat, rift, QS, etc) will cause some differences in the
latter)

Why? ... mainly because both have same elongated axis of cell structure
exposed to the chemical reaction of gluing.

CAVEAT: Not a botanist, but took two semester of botany in college,
although 40+ years ago, that's how I remember it, and I'm sticking to
it.

--
www.e-woodshop.net
Last update: 10/22/08
KarlC@ (the obvious)

Chris and Swing,

I stand corrected. I really started out just to say that width usually
means face grain and actually never said it and tangented off into
other topics. I've never done that before.

Who knows, maybe I am also wrong about the health care thing being
another government sink hole for my money with minimal return. I hope
I'm wrong.

Caveat: Not a real woodworker, I just play one on TV.

--www.e-woodshop.net
Last update: 10/22/08
KarlC@ (the obvious)

On 3/24/2010 4:28 PM, SonomaProducts.com wrote:
Chris and Swing,

I stand corrected. I really started out just to say that width usually
means face grain and actually never said it and tangented off into
other topics. I've never done that before.

LOL ... in that regard, you're more than one up on me!

I really wasn't trying NOT to be a smart ass, sorry if it came across
that way.

Who knows, maybe I am also wrong about the health care thing being
another government sink hole for my money with minimal return. I hope
I'm wrong.

Ditto ... but I like what Upscale said:

"The US people made a choice on a particular leader and that leader has
made a choice on healthcare. Cry in your soup as much as you want, but
deal with it instead of doing all this shrill whining and running around
that the sky is falling."

If you use your head for something besides a hat rack, you can't argue
with that bottom line, and the perspective.

Caveat: Not a real woodworker, I just play one on TV.

I'd also take exception to that ... your work speaks otherwise.

--
www.e-woodshop.net
Last update: 10/22/08
KarlC@ (the obvious)

You mention the different cuts, Karl. When I see them drawn in magazines, I
understand the difference among rift-, quarter-, and plain-sawn...but damned
if I ever could tell by looking at an actual piece of wood. Maybe it's
just that I've only "played at" woodworking off and off for 30 years....


Although....thinking about it, I *do* see that the oak in front of me has
those weird spots ("pickling"?) only on the "edge" and when I make banding
I prefer to put the "face" out front of the plywood shelf.

john

For quartersawn look at the end grain. A quartersawn board will have
the rings perpendicular to the face or very close to perp. This is
where it gets its stability. Even when they just cut up a log willy
nilly, some of it will end up QS.

On Mar 24, 3:02*pm, wrote:
You mention the different cuts, Karl. *When I see them drawn in magazines, I
understand the difference among rift-, quarter-, and plain-sawn...but damned
if I ever could tell by looking at an actual piece of wood. * Maybe it's
just that I've only "played at" woodworking *off and off *for 30 years....

Although....thinking about it, I *do* see that the oak in front of me has
those weird spots ("pickling"?) only on the "edge" *and when I make banding
I prefer to put the "face" out front of the plywood shelf.

john

On 03/24/2010 04:02 PM, wrote:
You mention the different cuts, Karl. When I see them drawn in magazines, I
understand the difference among rift-, quarter-, and plain-sawn...but damned
if I ever could tell by looking at an actual piece of wood.

Look at the end grain and you should be able to clearly see the growth
rings. The angle of the growth rings relative to the surface determines
whether its quarter or plainsawn. If the ring is less than 45 degrees
relative to the face it's plainsawn, more than 45 is quartersawn.

Rift-sawn is sometimes also called *******-sawn, and means that the
rings are 30-60 degrees relative to the face. This can be useful for
legs as it means you get similar figure on all four sides.

Chris

On 3/24/2010 5:02 PM, wrote:
You mention the different cuts, Karl. When I see them drawn in magazines, I
understand the difference among rift-, quarter-, and plain-sawn...but damned
if I ever could tell by looking at an actual piece of wood. Maybe it's
just that I've only "played at" woodworking off and off for 30 years....

http://community.klipsch.com/forums/.../sawn_wood.jpg
http://i29.tinypic.com/wla3m.jpg

Although....thinking about it, I *do* see that the oak in front of me has
those weird spots ("pickling"?) only on the "edge" and when I make banding
I prefer to put the "face" out front of the plywood shelf.

Those weird spots are medullary rays and characteristic of quarter and
rift sawn, particularly noticeable in red and white oak.

You will notice that often plain or flat sawn woods do have a very
apparent quarter sawn edge. AAMOF, I often seek these pieces out when
I want to show the edge in a drawer/dust divider where the rest of the
wood is quarter sawn, as you can see he

http://www.e-woodshop.net/images/MSB5.JPG

If the wood is thick enough, say 3 or 4", you can actually make a wider
quarter sawn board out of it by ripping it to thickness.

If you really have an eye for picking out and buying rough lumber, you
can save money by buying rough stock and seeking out a real thick plain
or flat sawn board (which are generally cheaper) that exhibits the
typical quarter sawn medullary rays on the edge, and then rip it to the
appropriate thickness in the shop for some nice quarter sawn boards.

Lots of graphic info here on all the above:

http://tinyurl.com/ykqu527

--
www.e-woodshop.net
Last update: 10/22/08
KarlC@ (the obvious)

In article ,
Max wrote:
"Chris Friesen" wrote in message
news
On 03/24/2010 11:18 AM, wrote:

Let's say you have a board that is 1" thick, 3" wide, 6" long.
The "ends" are where the grain shears off.
The "edges" are the thinner sides.
The "faces" are the wider sides.

But if I were to rip a thin strip off ...say 1/2" wide, so that the
smaller
"board" is 1 x 1/2 x 6...have the "edges" and "faces" changed places?
Is the "edge" still the thinner (1/2") side?

Personally I'd say yes. The edge should be the thinnest side that shows
long grain.

Of course it gets tricky if you have pieces with a square cross-section.


Incidentally, for cabinetmaking plywood the second dimension is the
grain direction. So an 8x4 sheet has the grain going the short way.

Chris


Lowe's has some 4X4 sheets.
What is one to do?


Do what a purported disconnected number recording at MIT once advised:

"The number you have called is imaginary,
Please rotate your phone 90 degrees, and try again."

"Robert Bonomi" wrote in message
...
In article ,
Max wrote:
"Chris Friesen" wrote in message
news
On 03/24/2010 11:18 AM, wrote:

Let's say you have a board that is 1" thick, 3" wide, 6" long.
The "ends" are where the grain shears off.
The "edges" are the thinner sides.
The "faces" are the wider sides.

But if I were to rip a thin strip off ...say 1/2" wide, so that the
smaller
"board" is 1 x 1/2 x 6...have the "edges" and "faces" changed places?
Is the "edge" still the thinner (1/2") side?

Personally I'd say yes. The edge should be the thinnest side that shows
long grain.

Of course it gets tricky if you have pieces with a square cross-section.


Incidentally, for cabinetmaking plywood the second dimension is the
grain direction. So an 8x4 sheet has the grain going the short way.

Chris


Lowe's has some 4X4 sheets.
What is one to do?


Do what a purported disconnected number recording at MIT once advised:

"The number you have called is imaginary,
Please rotate your phone 90 degrees, and try again."

But of course. Why didn't I think of that.

Max (hitting himself in the forehead)

"Robert Bonomi" wrote in message
...
In article ,
Max wrote:
"Chris Friesen" wrote in message
news
On 03/24/2010 11:18 AM, wrote:

Let's say you have a board that is 1" thick, 3" wide, 6" long.
The "ends" are where the grain shears off.
The "edges" are the thinner sides.
The "faces" are the wider sides.

But if I were to rip a thin strip off ...say 1/2" wide, so that the
smaller
"board" is 1 x 1/2 x 6...have the "edges" and "faces" changed places?
Is the "edge" still the thinner (1/2") side?

Personally I'd say yes. The edge should be the thinnest side that shows
long grain.

Of course it gets tricky if you have pieces with a square cross-section.


Incidentally, for cabinetmaking plywood the second dimension is the
grain direction. So an 8x4 sheet has the grain going the short way.

Chris


Lowe's has some 4X4 sheets.
What is one to do?


Do what a purported disconnected number recording at MIT once advised:

"The number you have called is imaginary,
Please rotate your phone 90 degrees, and try again."

Cute. You mean multiply it by exp^(i \PI /2). I've never seen the word
imaginary and degrees in the same sentence--no, never before. Obviously
undergraduate MIT students... I guess you could use the cord for the
initial side of the angle of rotation---but, what if you only have a cell
phone?

Bill

On 3/24/2010 5:23 PM, Chris Friesen wrote:
Rift-sawn is sometimes also called *******-sawn

As in "That ******* sawn my wood rong!"
?
:-)

--
See Nad. See Nad go. Go Nad!
To reply, eat the taco.
http://www.flickr.com/photos/bbqboyee/

In article ,
Bill wrote:

"Robert Bonomi" wrote in message
...
In article ,
Max wrote:
"Chris Friesen" wrote in message
news On 03/24/2010 11:18 AM, wrote:

Let's say you have a board that is 1" thick, 3" wide, 6" long.
The "ends" are where the grain shears off.
The "edges" are the thinner sides.
The "faces" are the wider sides.

But if I were to rip a thin strip off ...say 1/2" wide, so that the
smaller
"board" is 1 x 1/2 x 6...have the "edges" and "faces" changed places?
Is the "edge" still the thinner (1/2") side?

Personally I'd say yes. The edge should be the thinnest side that shows
long grain.

Of course it gets tricky if you have pieces with a square cross-section.


Incidentally, for cabinetmaking plywood the second dimension is the
grain direction. So an 8x4 sheet has the grain going the short way.

Chris


Lowe's has some 4X4 sheets.
What is one to do?


Do what a purported disconnected number recording at MIT once advised:

"The number you have called is imaginary,
Please rotate your phone 90 degrees, and try again."

Cute. You mean multiply it by exp^(i \PI /2).

No, I don't 'mean' that.

If I'd said that, it would *not* have been acurate reportage of the story
_as_I_heard_it_.

I've never seen the word
imaginary and degrees in the same sentence--no, never before.

Admittedly, it is a complex subject.

But, even with 'pure' imaginaries, it's still a matter of degree.

what's sqrt(-i), for example? "imaginary in the 2nd degree"? *GRIN*

Robert Bonomi wrote:
In article ,
Bill wrote:
"Robert Bonomi" wrote in message
...
In article ,
Max wrote:
"Chris Friesen" wrote in message
news On 03/24/2010 11:18 AM, wrote:

Let's say you have a board that is 1" thick, 3" wide, 6" long.
The "ends" are where the grain shears off.
The "edges" are the thinner sides.
The "faces" are the wider sides.

But if I were to rip a thin strip off ...say 1/2" wide, so that the
smaller
"board" is 1 x 1/2 x 6...have the "edges" and "faces" changed places?
Is the "edge" still the thinner (1/2") side?
Personally I'd say yes. The edge should be the thinnest side that shows
long grain.

Of course it gets tricky if you have pieces with a square cross-section.


Incidentally, for cabinetmaking plywood the second dimension is the
grain direction. So an 8x4 sheet has the grain going the short way.

Chris

Lowe's has some 4X4 sheets.
What is one to do?

Do what a purported disconnected number recording at MIT once advised:

"The number you have called is imaginary,
Please rotate your phone 90 degrees, and try again."
Cute. You mean multiply it by exp^(i \PI /2).

No, I don't 'mean' that.

If I'd said that, it would *not* have been acurate reportage of the story
_as_I_heard_it_.

I've never seen the word
imaginary and degrees in the same sentence--no, never before.

Admittedly, it is a complex subject.

But, even with 'pure' imaginaries, it's still a matter of degree.

what's sqrt(-i), for example? "imaginary in the 2nd degree"? *GRIN*


sqrt(-i) = exp(-PI/4) or exp(3PI/4). Not "purely imaginary" at all--the
real part is +/- sqrt(2)/2. It's as plain as a point on the unit circle.
You could double-check that if a = sqrt(2)/2 - sqrt(2)/2*i, or b=-a,
then a^2=b^2=-i. Simple trig and De Moivre's Theorem...

Bill

On Wed, 24 Mar 2010 21:04:19 -0500, the infamous
(Robert Bonomi) scrawled the following:

In article ,
Max wrote:
"Chris Friesen" wrote in message
news
On 03/24/2010 11:18 AM, wrote:

Let's say you have a board that is 1" thick, 3" wide, 6" long.
The "ends" are where the grain shears off.
The "edges" are the thinner sides.
The "faces" are the wider sides.

But if I were to rip a thin strip off ...say 1/2" wide, so that the
smaller
"board" is 1 x 1/2 x 6...have the "edges" and "faces" changed places?
Is the "edge" still the thinner (1/2") side?

Personally I'd say yes. The edge should be the thinnest side that shows
long grain.

Of course it gets tricky if you have pieces with a square cross-section.


Incidentally, for cabinetmaking plywood the second dimension is the
grain direction. So an 8x4 sheet has the grain going the short way.

Chris


Lowe's has some 4X4 sheets.
What is one to do?

Lay her down, roll her over, and do it again?


Do what a purported disconnected number recording at MIT once advised:

"The number you have called is imaginary,
Please rotate your phone 90 degrees, and try again."

I like it! I wonder how many people actually tried that. g

--
If we attend continually and promptly to the little that we can do, we
shall ere long be surprised to find how little remains that we cannot do.
-- Samuel Butler

In article ,
Bill wrote:
Robert Bonomi wrote:
In article ,
Bill wrote:
"Robert Bonomi" wrote in message
...
In article ,
Max wrote:
"Chris Friesen" wrote in message
news On 03/24/2010 11:18 AM, wrote:

Let's say you have a board that is 1" thick, 3" wide, 6" long.
The "ends" are where the grain shears off.
The "edges" are the thinner sides.
The "faces" are the wider sides.

But if I were to rip a thin strip off ...say 1/2" wide, so that the
smaller
"board" is 1 x 1/2 x 6...have the "edges" and "faces" changed places?
Is the "edge" still the thinner (1/2") side?
Personally I'd say yes. The edge should be the thinnest side that shows
long grain.

Of course it gets tricky if you have pieces with a square cross-section.


Incidentally, for cabinetmaking plywood the second dimension is the
grain direction. So an 8x4 sheet has the grain going the short way.

Chris

Lowe's has some 4X4 sheets.
What is one to do?

Do what a purported disconnected number recording at MIT once advised:

"The number you have called is imaginary,
Please rotate your phone 90 degrees, and try again."
Cute. You mean multiply it by exp^(i \PI /2).

No, I don't 'mean' that.

If I'd said that, it would *not* have been acurate reportage of the story
_as_I_heard_it_.

I've never seen the word
imaginary and degrees in the same sentence--no, never before.

Admittedly, it is a complex subject.

But, even with 'pure' imaginaries, it's still a matter of degree.

what's sqrt(-i), for example? "imaginary in the 2nd degree"? *GRIN*


sqrt(-i) = exp(-PI/4) or exp(3PI/4). Not "purely imaginary" at all--the
real part is +/- sqrt(2)/2. It's as plain as a point on the unit circle.
You could double-check that if a = sqrt(2)/2 - sqrt(2)/2*i, or b=-a,
then a^2=b^2=-i. Simple trig and De Moivre's Theorem...

I must be doing something wrong:
( sqrt(2)/2 - sqrt(2)/2*i ) **2

=(sqrt(2)/2)**2 -*((sqrt(2)/2*sqrt(2)/2*i) - (sqrt(2)/2*i)**2
= 2/4 -2*(2/4*i) - 2/4*(i*i)
= 2/4 - 4/4*i + 2/4
= 1-i

I'm tempted to make a facitous remark about an 'off by one' error.


Admittedly, I haven't played with this stuff for 30+ years, but I've got
a vague recollection of 'j' (the 'hyper-imaginary'??) as sqrt(-i).



Of course, on my first exposure to the imaginary exponential, I immediately
and rather vehemently questioned "exp(2*pi*i) = 1". As follows:

exp(0) = 1 by definition
exp(2*pi*i) = exp(0) 2 thins equal same thing, equal to each other
(2*pi*i) = 0 if bases (non-zero, and non-multiplicative-
identity) equal, exponents equal.

This rather upset the 8th grade math teacher. He *knew* my reasoning
was incorrect, but pointing out _where_ the flaw was was not obvious.

Simple reasoning on complex issues can lead one to into trouble. grin

Once it was clarified that '2*pi' was angular measure, things clarified
It is true that 2*pi == 0 (plus 1 revolution), even though neither of
the multiplicands is zero.

sqrt(-i) = exp(-PI/4) or exp(3PI/4). Not "purely imaginary" at all--the
real part is +/- sqrt(2)/2. It's as plain as a point on the unit circle.
You could double-check that if a = sqrt(2)/2 - sqrt(2)/2*i, or b=-a,
then a^2=b^2=-i. Simple trig and De Moivre's Theorem...

I must be doing something wrong:
( sqrt(2)/2 - sqrt(2)/2*i ) **2

=(sqrt(2)/2)**2 -*((sqrt(2)/2*sqrt(2)/2*i) - (sqrt(2)/2*i)**2
= 2/4 -2*(2/4*i) - 2/4*(i*i)



Robert, Nice Try! The line above should be 2/4 -2*(2/4*i) +2/4*(i*i)

= 2/4 -4/4i -2/4 = -i, as advertised. You had me worried for a moment.



= 2/4 - 4/4*i + 2/4
= 1-i

I'm tempted to make a facitous remark about an 'off by one' error.


Admittedly, I haven't played with this stuff for 30+ years, but I've got
a vague recollection of 'j' (the 'hyper-imaginary'??) as sqrt(-i).



Of course, on my first exposure to the imaginary exponential, I
immediately
and rather vehemently questioned "exp(2*pi*i) = 1". As follows:

exp(0) = 1 by definition
exp(2*pi*i) = exp(0) 2 thins equal same thing, equal to each other
(2*pi*i) = 0 if bases (non-zero, and non-multiplicative-
identity) equal, exponents equal.

Yes, you assumed that the function e^x is 1-to-1. And it is definitely not.
It wraps the imaginary axis around the unit circle infinitely often.
Indeed,
it maps every horizontal strip of height 2PI in the complex plane onto the
whole complex plane minus the point 0. I use this property of e^x as a
building block to help construct other mappings which are infinity-to-one.
Of course, there is nothing special abought e^x in this context, any
exponential
function a^x (a0, a!=1) may be written in the form e^(kx) for some real
number
k so similar properties hold for a^x. What is special about e^x is that
d/dx (e^x) = e^x, and only constant multiples of it have this property (of
being equal to their derivative on a suitable domain).



This rather upset the 8th grade math teacher. He *knew* my reasoning
was incorrect, but pointing out _where_ the flaw was was not obvious.




...Not obvious to an 8th grade math teacher. They have their hands full
trying to convince students that (a+b)^2 is not a^2 + b^2!!! Tough job!

Actually, my 8th grade math teacher was one of my favorite teachers, K-12.




Simple reasoning on complex issues can lead one to into trouble. grin

Once it was clarified that '2*pi' was angular measure, things clarified
It is true that 2*pi == 0 (plus 1 revolution), even though neither of
the multiplicands is zero.



It is of course Not true that 2*pi =0. What is true is that e^(2pi*i) =
e^0=1.

What is also true is that, for real values x, e^(ix) = cos(x) + i sin (x),
which I see is what you were referring to by "plus 1 revolution" above.


Best,
Bill

In article ,
Bill wrote:


sqrt(-i) = exp(-PI/4) or exp(3PI/4). Not "purely imaginary" at all--the
real part is +/- sqrt(2)/2. It's as plain as a point on the unit circle.
You could double-check that if a = sqrt(2)/2 - sqrt(2)/2*i, or b=-a,
then a^2=b^2=-i. Simple trig and De Moivre's Theorem...

I must be doing something wrong:
( sqrt(2)/2 - sqrt(2)/2*i ) **2

=(sqrt(2)/2)**2 -*((sqrt(2)/2*sqrt(2)/2*i) - (sqrt(2)/2*i)**2
= 2/4 -2*(2/4*i) - 2/4*(i*i)

Robert, Nice Try! The line above should be 2/4 -2*(2/4*i) +2/4*(i*i)

ARGH!!! like I said, "I must be doing something wrong' wry grin

I inverted the sign on the last term _and_ left the i^2 in. one or the
other, but not both. Takes more than 2 i's to find it.

= 2/4 -4/4i -2/4 = -i, as advertised. You had me worried for a moment.

Of course, on my first exposure to the imaginary exponential, I
immediately
and rather vehemently questioned "exp(2*pi*i) = 1". As follows:

exp(0) = 1 by definition
exp(2*pi*i) = exp(0) 2 thins equal same thing, equal to each other
(2*pi*i) = 0 if bases (non-zero, and non-multiplicative-
identity) equal, exponents equal.

Yes, you assumed that the function e^x is 1-to-1. And it is definitely not.

Well, It _is_, for exponents without an imaginary component. The concept
of multiplying something by itself an imaginary number of times is utterly
lacking in any intuitive foundation.

This rather upset the 8th grade math teacher. He *knew* my reasoning
was incorrect, but pointing out _where_ the flaw was was not obvious.

..Not obvious to an 8th grade math teacher.

His own fault! _He_ was the one that sprung it on me, cold, and wanted me
to agree it was true. I'd stuck my head in the classroom, after school
was out, to ask him something, and he threw that at me, from the middle of
a discussion he was having with somebody else -- "e to the two pi eye
equals one, right?" I thought for about half a second and said, firmly,
"no". He asked "why not?" And I wrote the above 'disproof' on the black-
board. Finding -where- to kick a hole in it is difficult. Every line
_does_ follow validly from the prior one. It's merely that the 'meaning'
of the numbers changed.
They have their hands full
trying to convince students that (a+b)^2 is not a^2 + b^2!!! Tough job!

HUH?????

I think my 5th or 6th grade class spent -maybe- one afternoon on that.
Re-write the '^2' as an explicit multiplication, and apply the distributive
property (a total of 3 times) to eliminate the parens, then consolidate the
like terms.

The error is also 'painfully obvious' if you simply work through a couple
of 'concrete' examples.

Admittedly, I was learning this stuff _just_before_ the "new math" teaching
started. (My younger brother, 3 years behind me, got a *very* different math
education!)

Simple reasoning on complex issues can lead one to into trouble. grin

Once it was clarified that '2*pi' was angular measure, things clarified
It is true that 2*pi == 0 (plus 1 revolution), even though neither of
the multiplicands is zero.

It is of course Not true that 2*pi =0. What is true is that e^(2pi*i) =
e^0=1.

Depends on what you mean by '='. grin type='Clintonesque'

They're both the same "direction". "equal" in that sense.

Anything derived from the point-value (ignoring the 'path' that got you
there) is the same for both.

What is also true is that, for real values x, e^(ix) = cos(x) + i sin (x),
which I see is what you were referring to by "plus 1 revolution" above.

Quoting a line from an old comic strip -- where they got it absolutely
wrong (unintentionally!) in context: "How do you tell a pilot he's 360
degrees off-course?"

"Robert Bonomi" wrote in message
communications...
Yes, you assumed that the function e^x is 1-to-1. And it is definitely
not.

Well, It _is_, for exponents without an imaginary component. The concept
of multiplying something by itself an imaginary number of times is utterly
lacking in any intuitive foundation.

Yes, think of expressions like 2^i as exp^(i *ln 2), then you can use your
intuition
a little more once you are adequately aquainted with the function e^x.



Once it was clarified that '2*pi' was angular measure, things clarified
It is true that 2*pi == 0 (plus 1 revolution), even though neither of
the multiplicands is zero.

What you REALLY mean is that: 2*pi radians ~ 0 radians!

0 and 2*Pi are complex numbers, different ones, which are over 6 units
apart in the complex plane. We save "=" for when the numbers coincide.


Quoting a line from an old comic strip -- where they got it absolutely
wrong (unintentionally!) in context: "How do you tell a pilot he's 360
degrees off-course?"

Again, The terminal side of angle of 0 degrees is the same of one of 360
degrees.
But, turning a gear or the steering wheel of your car 360 degrees or -360
degree or 0 degree
are 3 different things, right? I agree it probably doesn't matter how a
pilot turns west, if
west is the direction he is supposed to go! : )

Best,
Bill