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The Fundamental Theorem of Algebra

 

 

 

As remarked before, in the 16th century Cardano noted that the sum of the three solutions to a cubic equation

 

 

 

x3 + bx2 + cx + d = 0

 

 

 

is ̢̢̮ââ¬Å¡Ã¬Ã¢ââ¬Ãâb, the negation of the coefficient of x2. By the 17th century the theory of equations had developed so far as to allow Girard (1595-1632) to state a principle of algebra, what we call now "the fundamental theorem of algebra". His formulation, which he didn't prove, also gives a general relation between the n solutions to an nth degree equation and its n coefficients.

 

 

 

An nth degree equation can be written in modern notation as

 

 

 

xn + a1xn̢̢̮ââ¬Å¡Ã¬Ã¢ââ¬Ãâ1 + ... + an̢̢̮ââ¬Å¡Ã¬Ã¢ââ¬Ãâ2x2 + an̢̢̮ââ¬Å¡Ã¬Ã¢ââ¬Ãâ1x + an = 0

 

 

 

where the coefficients a1, ..., an̢̢̮ââ¬Å¡Ã¬Ã¢ââ¬Ãâ2, an̢̢̮ââ¬Å¡Ã¬Ã¢ââ¬Ãâ1, and an are all constants. Girard said that an nth degree equation admits of n solutions, if you allow all roots and count roots with multiplicity. So, for example, the equation x2 + 1 = 0 has the two solutions ̢̮Ãâ¹Ã¢â¬Â Ãâ¦ÃÂ¡ÃÆÃ¢Ã¢ââ¬Å¡Ã¬Ã¢ââ¬Ãâ1 and ̢̢̮ââ¬Å¡Ã¬Ã¢ââ¬Ãâ̢̮Ãâ¹Ã¢â¬Â Ãâ¦ÃÂ¡ÃÆÃ¢Ã¢ââ¬Å¡Ã¬Ã¢ââ¬Ãâ1, and the equation x2 ̢̢̮ââ¬Å¡Ã¬Ã¢ââ¬Ãâ 2x + 1 = 0 has the two solutions 1 and 1. Girard wasn't particularly clear what form his solutions were to have, just that there be n of them: x1, x2, ..., xn̢̢̮ââ¬Å¡Ã¬Ã¢ââ¬Ãâ1, and xn.

 

 

 

Girard gave the relation between the n roots x1, x2, ..., xn, and xn and the n coefficients a1, ..., an̢̢̮ââ¬Å¡Ã¬Ã¢ââ¬Ãâ2, an̢̢̮ââ¬Å¡Ã¬Ã¢ââ¬Ãâ1, and an that extends Cardano's remark. First, the sum of the roots x1 + x2 + ..., + xn is ̢̢̮ââ¬Å¡Ã¬Ã¢ââ¬Ãâa1, the negation of the coefficient of xn̢̢̮ââ¬Å¡Ã¬Ã¢ââ¬Ãâ1 (Cardano's remark). Next, the sum of all products of pairs of solutions is a2. Next, the sum of all products of triples of solutions is ̢̢̮ââ¬Å¡Ã¬Ã¢ââ¬Ãâa3. And so on until the product of all n solutions is either an (when n is even) or ̢̢̮ââ¬Å¡Ã¬Ã¢ââ¬Ãâan (when n is odd).

 

 

 

Here's an example. The 4th degree equation

 

 

 

x4 ̢̢̮ââ¬Å¡Ã¬Ã¢ââ¬Ãâ 6x3 + 3x2 + 26x ̢̢̮ââ¬Å¡Ã¬Ã¢ââ¬Ãâ 24 = 0

 

 

 

has the four solutions ̢̢̮ââ¬Å¡Ã¬Ã¢ââ¬Ãâ2, 1, 3, and 4. The sum of the solutions equals 6, that is ̢̢̮ââ¬Å¡Ã¬Ã¢ââ¬Ãâ2 + 1 + 3 + 4 = 6. The sum of all products of pairs (six of them) is

 

 

 

(̢̢̮ââ¬Å¡Ã¬Ã¢ââ¬Ãâ2)(1) + (̢̢̮ââ¬Å¡Ã¬Ã¢ââ¬Ãâ2)(3) + (̢̢̮ââ¬Å¡Ã¬Ã¢ââ¬Ãâ2)(4) + (1)(3) + (1)(4) + (3)(4)

 

 

 

which is 3. The sum of all products of triples (four of them) is

 

 

 

(̢̢̮ââ¬Å¡Ã¬Ã¢ââ¬Ãâ2)(1)(3) + (̢̢̮ââ¬Å¡Ã¬Ã¢ââ¬Ãâ2)(1)(4) + (̢̢̮ââ¬Å¡Ã¬Ã¢ââ¬Ãâ2)(3)(4) + (1)(3)(4)

 

 

 

which is 26. And the product of all four solutions is ̢̢̮ââ¬Å¡Ã¬Ã¢ââ¬Ãâ24.

 

 

 

Descartes (1596̢̢̮ââ¬Å¡Ã¬Ã¢ââ¬Ãâ1650) also studied this relation between solutions and coefficients, and showed more explicitly why the relationship holds. Descartes called negative solutions "false" and treated other solutions (that is, complex numbers) "imaginary".

 

 

 

Over the remainder of the 17th century, negative numbers rose in status to be full̢̢̮ââ¬Å¡Ã¬Ã¢ââ¬Ãâfledged numbers. But complex numbers remained in limbo through most of the 18th century. They weren't considered to be real numbers, but they were useful in the theory of equations. It wasn't even clear what form the solutions to equations might take. Certainly complex numbers of the form a + b̢̮Ãâ¹Ã¢â¬Â Ãâ¦ÃÂ¡ÃÆÃ¢Ã¢ââ¬Å¡Ã¬Ã¢ââ¬Ãâ1 were sufficient to solve quadratic equations, but it wasn't clear they were enough to solve cubic and higher-degree equations. Also, the part of the Fundamental Theorem of Algebra which stated there actually are n solutions of an nth degree equation was yet to be proved, pending, of course, some description of the possible forms that the solutions might take.

 

 

 

Next section: The number i 8-)

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In mathematics, an imaginary number (or purely imaginary number) is a complex number whose squared value is a real number not greater than zero. The imaginary unit, denoted by i, or j, is an example of an imaginary number. If y is a real number, then i̢̢̮ââ¬Å¡Ã¬Ãâây is also an imaginary number, because:

 

 

 

e4fcad0d6bc0c9476b09cb7bd3b1f92d.png

 

 

 

Imaginary numbers were defined in 1572 by Rafael Bombelli. At the time, such numbers were thought not to exist, much as zero and the negative numbers were regarded by some as fictitious or useless. Many other mathematicians were slow to believe in imaginary numbers at first, including Descartes who wrote about them in his La G̮̩̉̉om̮̩̉̉trie, where the term was meant to be derogatory.[1]

 

 

 

Although Descartes originally used the term imaginary number to mean what is currently meant by the term complex number, the term imaginary number today usually means a complex number with a real part equal to 0, that is, a number of the form i̢̢̮ââ¬Å¡Ã¬Ãâây. Zero (0) is the only number that is both real and imaginary.

 

 

 

i is an imaginary number :lol:

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Duke Dreamer Shows

 

 

 

Duke Dreamer is a one-of-a-kind character who has delighted and astounded audiences including the President of Singapore and is featured in an upcoming Rogers Wireless TV commercial.

 

 

 

When you experience Duke's show it will include: chainsaw and egg juggling. midget in a cloud, ladder to the sky, comedy juggling (with or without fire) comedy kid inside a- lasso, comedic choreography, rope-walking, and "the human levitation".

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What is it?

 

 

 

The Thing?The Thing? rates an entry in Doug Kirby's New Roadside America (1992), and it once was featured in a Jane Pauley television special on NBC. For a one dollar fee, paid at the shop's cash register, one can enter a small outside courtyard leading to three prefab corrugated steel sheds. Inside are a variety of exhibits, including odd wood carvings of tortured souls (by "Ralph Gallagher, artist"), the "wooden fantasy" of painted driftwood (from an Alamogordo collector), framed lithographs, saddles, rifles, a covered wagon and several vintage automobiles. A sign by a 1937 Rolls-Royce makes the claim that it once belonged to Adolf Hitler. Winding corridors and exhibit halls eventually lead to The Thing?, a mummified mother-and-child tableau encased in a glass-covered coffin.

 

 

 

The concept of the exhibit references the 1950 novelty song, "The Thing", recorded by Phil Harris, with the Charles R. Grean lyrics that keep the listener from learning the true nature of the Thing in the "great big wooden box a-floatin' in the bay." The narrative song ends with these verses:

 

 

 

I wandered all around the town

 

Until I chanced to meet

 

A hobo who was lookin' for

 

A handout on the street

 

He said he'd take most any old thing

 

He was a desperate man

 

But when I showed him the...

 

He turned around and ran

 

Oh, when I showed him the...

 

He turned around and ran

 

I wandered on for many years

 

A victim of my fate

 

Until one day I came upon

 

Saint Peter at the gate

 

And when I tried to take it inside

 

He told me where to go

 

Get out of here with that...

 

And take it down below

 

Oh, get out of here with that...

 

And take it down below

 

The moral of this story is

 

If you're out on the beach

 

And you should see a great big box

 

And it's within your reach

 

Don't ever stop and open it up

 

That's my advice to you

 

'Cause you'll never get rid of the...

 

No matter what you do

 

Oh, you'll never get rid of the...

 

No matter what you do

 

Also referenced is the 1951 science fiction film, The Thing from Another World, later remade by John Carpenter as The Thing (1982). In 1992, the alternative rock band The Men scored a hit with the song "Church of Logic, Sin and Love," which tells of two lost souls who searched for a purpose, saw the billboards and attempted to find the meaning of life at The Thing?, as noted in this excerpt from the lyrics:

 

 

 

Two hours later they decided to stop at a diner,

 

'Cause they just loved the smell of eggs and coffee.

 

I just had to smoke a cigarette and wear a hat...

 

By the time that they set off again,

 

The sun was starting to set; it made the sky look red like a nuclear ray...

 

One of them said, "What do you want more than anything in this whole wide world?

 

Do you want money? Do you want sex? Or do you want all that success?"

 

I thought about that myself...

 

Then they came upon The Thing?

 

The Thing? ahead 60 miles, do not miss.

 

Not for the squeamish or depressed,

 

Not for the unbelievers truly obsessed,

 

Something you just don't wanna miss.

 

It's the kind of place where space explorers could have landed around 1963;

 

When Kennedy was in Life magazine,

 

And everything was aquamarine...

 

Aquamarine.

 

 

 

[edit] Origins

 

 

 

 

 

This is the product of me randomly mashing buttons on wikipedia...

LOTRjokesigedition-1.png

Get back here so I can rub your butt.

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flodder450

 

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PostPosted: Sun Dec 30, 2007 3:44 am Post subject: Ctrl+v game Reply with quote Report post

 

easy enough

 

you copy something an the you paste it in your post!

 

i kick off this game:

 

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PostPosted: Sun Dec 30, 2007 5:01 am Post subject: Reply with quote Report post

 

The Fundamental Theorem of Algebra

 

 

 

As remarked before, in the 16th century Cardano noted that the sum of the three solutions to a cubic equation

 

 

 

x3 + bx2 + cx + d = 0

 

 

 

is ̢̢̮ââ¬Å¡Ã¬Ã¢ââ¬Ãâb, the negation of the coefficient of x2. By the 17th century the theory of equations had developed so far as to allow Girard (1595-1632) to state a principle of algebra, what we call now "the fundamental theorem of algebra". His formulation, which he didn't prove, also gives a general relation between the n solutions to an nth degree equation and its n coefficients.

 

 

 

An nth degree equation can be written in modern notation as

 

 

 

xn + a1xn̢̢̮ââ¬Å¡Ã¬Ã¢ââ¬Ãâ1 + ... + an̢̢̮ââ¬Å¡Ã¬Ã¢ââ¬Ãâ2x2 + an̢̢̮ââ¬Å¡Ã¬Ã¢ââ¬Ãâ1x + an = 0

 

 

 

where the coefficients a1, ..., an̢̢̮ââ¬Å¡Ã¬Ã¢ââ¬Ãâ2, an̢̢̮ââ¬Å¡Ã¬Ã¢ââ¬Ãâ1, and an are all constants. Girard said that an nth degree equation admits of n solutions, if you allow all roots and count roots with multiplicity. So, for example, the equation x2 + 1 = 0 has the two solutions ̢̮Ãâ¹Ã¢â¬Â Ãâ¦ÃÂ¡ÃÆÃ¢Ã¢ââ¬Å¡Ã¬Ã¢ââ¬Ãâ1 and ̢̢̮ââ¬Å¡Ã¬Ã¢ââ¬Ãâ̢̮Ãâ¹Ã¢â¬Â Ãâ¦ÃÂ¡ÃÆÃ¢Ã¢ââ¬Å¡Ã¬Ã¢ââ¬Ãâ1, and the equation x2 ̢̢̮ââ¬Å¡Ã¬Ã¢ââ¬Ãâ 2x + 1 = 0 has the two solutions 1 and 1. Girard wasn't particularly clear what form his solutions were to have, just that there be n of them: x1, x2, ..., xn̢̢̮ââ¬Å¡Ã¬Ã¢ââ¬Ãâ1, and xn.

 

 

 

Girard gave the relation between the n roots x1, x2, ..., xn, and xn and the n coefficients a1, ..., an̢̢̮ââ¬Å¡Ã¬Ã¢ââ¬Ãâ2, an̢̢̮ââ¬Å¡Ã¬Ã¢ââ¬Ãâ1, and an that extends Cardano's remark. First, the sum of the roots x1 + x2 + ..., + xn is ̢̢̮ââ¬Å¡Ã¬Ã¢ââ¬Ãâa1, the negation of the coefficient of xn̢̢̮ââ¬Å¡Ã¬Ã¢ââ¬Ãâ1 (Cardano's remark). Next, the sum of all products of pairs of solutions is a2. Next, the sum of all products of triples of solutions is ̢̢̮ââ¬Å¡Ã¬Ã¢ââ¬Ãâa3. And so on until the product of all n solutions is either an (when n is even) or ̢̢̮ââ¬Å¡Ã¬Ã¢ââ¬Ãâan (when n is odd).

 

 

 

Here's an example. The 4th degree equation

 

 

 

x4 ̢̢̮ââ¬Å¡Ã¬Ã¢ââ¬Ãâ 6x3 + 3x2 + 26x ̢̢̮ââ¬Å¡Ã¬Ã¢ââ¬Ãâ 24 = 0

 

 

 

has the four solutions ̢̢̮ââ¬Å¡Ã¬Ã¢ââ¬Ãâ2, 1, 3, and 4. The sum of the solutions equals 6, that is ̢̢̮ââ¬Å¡Ã¬Ã¢ââ¬Ãâ2 + 1 + 3 + 4 = 6. The sum of all products of pairs (six of them) is

 

 

 

(̢̢̮ââ¬Å¡Ã¬Ã¢ââ¬Ãâ2)(1) + (̢̢̮ââ¬Å¡Ã¬Ã¢ââ¬Ãâ2)(3) + (̢̢̮ââ¬Å¡Ã¬Ã¢ââ¬Ãâ2)(4) + (1)(3) + (1)(4) + (3)(4)

 

 

 

which is 3. The sum of all products of triples (four of them) is

 

 

 

(̢̢̮ââ¬Å¡Ã¬Ã¢ââ¬Ãâ2)(1)(3) + (̢̢̮ââ¬Å¡Ã¬Ã¢ââ¬Ãâ2)(1)(4) + (̢̢̮ââ¬Å¡Ã¬Ã¢ââ¬Ãâ2)(3)(4) + (1)(3)(4)

 

 

 

which is 26. And the product of all four solutions is ̢̢̮ââ¬Å¡Ã¬Ã¢ââ¬Ãâ24.

 

 

 

Descartes (1596̢̢̮ââ¬Å¡Ã¬Ã¢ââ¬Ãâ1650) also studied this relation between solutions and coefficients, and showed more explicitly why the relationship holds. Descartes called negative solutions "false" and treated other solutions (that is, complex numbers) "imaginary".

 

 

 

Over the remainder of the 17th century, negative numbers rose in status to be full̢̢̮ââ¬Å¡Ã¬Ã¢ââ¬Ãâfledged numbers. But complex numbers remained in limbo through most of the 18th century. They weren't considered to be real numbers, but they were useful in the theory of equations. It wasn't even clear what form the solutions to equations might take. Certainly complex numbers of the form a + b̢̮Ãâ¹Ã¢â¬Â Ãâ¦ÃÂ¡ÃÆÃ¢Ã¢ââ¬Å¡Ã¬Ã¢ââ¬Ãâ1 were sufficient to solve quadratic equations, but it wasn't clear they were enough to solve cubic and higher-degree equations. Also, the part of the Fundamental Theorem of Algebra which stated there actually are n solutions of an nth degree equation was yet to be proved, pending, of course, some description of the possible forms that the solutions might take.

 

 

 

Next section: The number i Cool

 

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PostPosted: Sun Dec 30, 2007 12:50 pm Post subject: Reply with quote Edit/Delete this post Report post

 

In mathematics, an imaginary number (or purely imaginary number) is a complex number whose squared value is a real number not greater than zero. The imaginary unit, denoted by i, or j, is an example of an imaginary number. If y is a real number, then i̢̢̮ââ¬Å¡Ã¬Ãâây is also an imaginary number, because:

 

 

 

 

 

 

 

Imaginary numbers were defined in 1572 by Rafael Bombelli. At the time, such numbers were thought not to exist, much as zero and the negative numbers were regarded by some as fictitious or useless. Many other mathematicians were slow to believe in imaginary numbers at first, including Descartes who wrote about them in his La G̮̩̉̉om̮̩̉̉trie, where the term was meant to be derogatory.[1]

 

 

 

Although Descartes originally used the term imaginary number to mean what is currently meant by the term complex number, the term imaginary number today usually means a complex number with a real part equal to 0, that is, a number of the form i̢̢̮ââ¬Å¡Ã¬Ãâây. Zero (0) is the only number that is both real and imaginary.

 

 

 

i is an imaginary number Laughing

 

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PostPosted: Sun Dec 30, 2007 1:43 pm Post subject: Reply with quote Report post

 

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PostPosted: Sun Dec 30, 2007 1:45 pm Post subject: Reply with quote Report post

 

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PostPosted: Sun Dec 30, 2007 1:51 pm Post subject: Reply with quote Report post

 

Duke Dreamer Shows

 

 

 

Duke Dreamer is a one-of-a-kind character who has delighted and astounded audiences including the President of Singapore and is featured in an upcoming Rogers Wireless TV commercial.

 

 

 

When you experience Duke's show it will include: chainsaw and egg juggling. midget in a cloud, ladder to the sky, comedy juggling (with or without fire) comedy kid inside a- lasso, comedic choreography, rope-walking, and "the human levitation".

 

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PostPosted: Mon Dec 31, 2007 10:37 pm Post subject: Reply with quote Report post

 

easy enough

 

you copy something an the you paste it in your post!

 

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PostPosted: Mon Dec 31, 2007 11:12 pm Post subject: Reply with quote Report post

 

What is it?

 

 

 

The Thing?The Thing? rates an entry in Doug Kirby's New Roadside America (1992), and it once was featured in a Jane Pauley television special on NBC. For a one dollar fee, paid at the shop's cash register, one can enter a small outside courtyard leading to three prefab corrugated steel sheds. Inside are a variety of exhibits, including odd wood carvings of tortured souls (by "Ralph Gallagher, artist"), the "wooden fantasy" of painted driftwood (from an Alamogordo collector), framed lithographs, saddles, rifles, a covered wagon and several vintage automobiles. A sign by a 1937 Rolls-Royce makes the claim that it once belonged to Adolf Hitler. Winding corridors and exhibit halls eventually lead to The Thing?, a mummified mother-and-child tableau encased in a glass-covered coffin.

 

 

 

The concept of the exhibit references the 1950 novelty song, "The Thing", recorded by Phil Harris, with the Charles R. Grean lyrics that keep the listener from learning the true nature of the Thing in the "great big wooden box a-floatin' in the bay." The narrative song ends with these verses:

 

 

 

I wandered all around the town

 

Until I chanced to meet

 

A hobo who was lookin' for

 

A handout on the street

 

He said he'd take most any old thing

 

He was a desperate man

 

But when I showed him the...

 

He turned around and ran

 

Oh, when I showed him the...

 

He turned around and ran

 

I wandered on for many years

 

A victim of my fate

 

Until one day I came upon

 

Saint Peter at the gate

 

And when I tried to take it inside

 

He told me where to go

 

Get out of here with that...

 

And take it down below

 

Oh, get out of here with that...

 

And take it down below

 

The moral of this story is

 

If you're out on the beach

 

And you should see a great big box

 

And it's within your reach

 

Don't ever stop and open it up

 

That's my advice to you

 

'Cause you'll never get rid of the...

 

No matter what you do

 

Oh, you'll never get rid of the...

 

No matter what you do

 

Also referenced is the 1951 science fiction film, The Thing from Another World, later remade by John Carpenter as The Thing (1982). In 1992, the alternative rock band The Men scored a hit with the song "Church of Logic, Sin and Love," which tells of two lost souls who searched for a purpose, saw the billboards and attempted to find the meaning of life at The Thing?, as noted in this excerpt from the lyrics:

 

 

 

Two hours later they decided to stop at a diner,

 

'Cause they just loved the smell of eggs and coffee.

 

I just had to smoke a cigarette and wear a hat...

 

By the time that they set off again,

 

The sun was starting to set; it made the sky look red like a nuclear ray...

 

One of them said, "What do you want more than anything in this whole wide world?

 

Do you want money? Do you want sex? Or do you want all that success?"

 

I thought about that myself...

 

Then they came upon The Thing?

 

The Thing? ahead 60 miles, do not miss.

 

Not for the squeamish or depressed,

 

Not for the unbelievers truly obsessed,

 

Something you just don't wanna miss.

 

It's the kind of place where space explorers could have landed around 1963;

 

When Kennedy was in Life magazine,

 

And everything was aquamarine...

 

Aquamarine.

 

 

 

[edit] Origins

 

 

 

 

 

This is the product of me randomly mashing buttons on wikipedia...

 

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PostPosted: Thu Jan 03, 2008 1:48 am Post subject: Reply with quote Report post

 

*presses CTRL+V*

 

 

 

 

 

 

 

Lol, i posted that a few min ago =p

 

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