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Topic: Non-Euclidean Arithmetic
Replies: 108   Last Post: Sep 13, 2012 3:39 PM

 Messages: [ Previous | Next ]
 Clyde Greeno @ MALEI Posts: 220 Registered: 9/13/10
Re: Non-Euclidean Arithmetic
Posted: Sep 10, 2012 9:15 PM

Joe:

Glad to learn that you were amused by the interpretation that
"multiplication" of a+bi by 7i means multiplying the -b+ai "complex i(mage)
of a+bi ,by 7.

Accordingly, you might also get a chuckle out of this one. The parabolic
function -5(x-10)(x+4) has roots at 3 [+/-] 7 ... and has polynomial
formula -5x^2+30x+200 ... and parabolic formula -5(x-3)^2+155 ...
down-cupped from its vertex at (3,155).

>From the same-vertex, the up-cupped parabola, 5(x-3)^2+155 , has no roots.
So what does it mean when popular textbooks declare that 5(x-3)^2+155 has
"complex roots" 3+7i and 3-7i?

What we now call "complex numbers" were earlier called "imaginary" numbers.
For sure, it is quite complex to try to imagine any "fictitious" roots of
parabolic functions. Indeed, it is mathematically absurd ... because the
5(x-3)^2+155 parabola is not a complex function ...
and the 5(z-3)^2+155 complex function is far too complex to be a parabola.

So is there any way to imag-ine a real-istic meaning behind textbook claims
that 5x^2+30x+200 ...
... which is 5(x-3)^2+155 ... has "complex roots", 3 [+/-] 7i? Sure.
Just recognize that
5(x-3)^2+155 and -5(x-3)^2+155 are (vertical) "images" of each other. If
either has two roots, the other has none.

Thereby, to say that the roots of 5(x-3)^2+155 are 3 [+/-] 7i ... or
better, that they are (3 [+/-] 7)i ... just to say that 3[+/-] 7 are the
roots of the image of 5(x-3)^2+155. The drawing of the double-parabola
eliminates all confusion about what it means to "find imaginary roots" of
quadratics. In this context, "i" means a kind of "inversion."

Just imag-ine how that reduces the complex-ity of solving parabolic
[Yes, dozens of times.]

Cordially,

Clyde

- --------------------------------------------------
From: "Joe Niederberger" <niederberger@comcast.net>
Sent: Monday, September 10, 2012 10:36 AM
To: <math-teach@mathforum.org>
Subject: Re: Non-Euclidean Arithmetic

> Clyde says:
>>How about (3+7i) of those there a+bi complex numbers?
> If you take 3 of those there a+bi complexes, you can have the 7i(mage) of
>
> Thanks for joining in the fun! But, seriously, I get your point. Its
> interesting how that little two letter word does so much work. I'm not
> quite sure what to make *of* it though ;-)

Date Subject Author
9/1/12 Jonathan J. Crabtree
9/1/12 Paul A. Tanner III
9/2/12 kirby urner
9/3/12 Paul A. Tanner III
9/3/12 kirby urner
9/3/12 Paul A. Tanner III
9/4/12 kirby urner
9/4/12 Paul A. Tanner III
9/4/12 kirby urner
9/5/12 Paul A. Tanner III
9/5/12 Robert Hansen
9/6/12 kirby urner
9/1/12 kirby urner
9/1/12 Joe Niederberger
9/1/12 Wayne Bishop
9/1/12 Joe Niederberger
9/2/12 Robert Hansen
9/3/12 Paul A. Tanner III
9/3/12 Robert Hansen
9/5/12 Paul A. Tanner III
9/3/12 Joe Niederberger
9/3/12 Robert Hansen
9/5/12 Paul A. Tanner III
9/3/12 Joe Niederberger
9/3/12 Paul A. Tanner III
9/4/12 Joe Niederberger
9/5/12 Paul A. Tanner III
9/5/12 Joe Niederberger
9/5/12 Robert Hansen
9/5/12 Paul A. Tanner III
9/5/12 Joe Niederberger
9/5/12 kirby urner
9/5/12 Joe Niederberger
9/5/12 Robert Hansen
9/5/12 Joe Niederberger
9/6/12 Joe Niederberger
9/8/12 Robert Hansen
9/7/12 Jonathan J. Crabtree
9/8/12 kirby urner
9/8/12 Paul A. Tanner III
9/10/12 kirby urner
9/10/12 Paul A. Tanner III
9/10/12 kirby urner
9/10/12 Paul A. Tanner III
9/10/12 kirby urner
9/8/12 Robert Hansen
9/8/12 kirby urner
9/8/12 Robert Hansen
9/8/12 kirby urner
9/8/12 Joe Niederberger
9/8/12 Jonathan J. Crabtree
9/9/12 kirby urner
9/8/12 Clyde Greeno @ MALEI
9/8/12 Jonathan J. Crabtree
9/8/12 Jonathan J. Crabtree
9/8/12 Joe Niederberger
9/8/12 Joe Niederberger
9/9/12 Paul A. Tanner III
9/9/12 Robert Hansen
9/9/12 Paul A. Tanner III
9/9/12 Robert Hansen
9/9/12 Paul A. Tanner III
9/9/12 Robert Hansen
9/10/12 Paul A. Tanner III
9/10/12 Wayne Bishop
9/10/12 Paul A. Tanner III
9/9/12 Joe Niederberger
9/10/12 Clyde Greeno @ MALEI
9/9/12 Joe Niederberger
9/9/12 Paul A. Tanner III
9/9/12 Wayne Bishop
9/9/12 Paul A. Tanner III
9/10/12 Wayne Bishop
9/10/12 Paul A. Tanner III
9/9/12 Paul A. Tanner III
9/9/12 Joe Niederberger
9/10/12 Clyde Greeno @ MALEI
9/10/12 Joe Niederberger
9/10/12 Paul A. Tanner III
9/10/12 Joe Niederberger
9/10/12 Clyde Greeno @ MALEI
9/10/12 Joe Niederberger
9/11/12 Joe Niederberger
9/11/12 Paul A. Tanner III
9/11/12 kirby urner
9/11/12 Paul A. Tanner III
9/11/12 kirby urner
9/11/12 Paul A. Tanner III
9/11/12 kirby urner
9/12/12 Paul A. Tanner III
9/12/12 kirby urner
9/12/12 Paul A. Tanner III
9/12/12 kirby urner
9/13/12 Paul A. Tanner III
9/13/12 kirby urner
9/13/12 Paul A. Tanner III
9/11/12 Joe Niederberger
9/11/12 Joe Niederberger
9/11/12 kirby urner
9/11/12 Joe Niederberger
9/11/12 Joe Niederberger
9/11/12 Paul A. Tanner III
9/11/12 israeliteknight
9/11/12 Joe Niederberger
9/12/12 Paul A. Tanner III
9/12/12 kirby urner
9/12/12 Paul A. Tanner III
9/12/12 kirby urner
9/11/12 Jonathan J. Crabtree