### Friday, January 08, 2010

## JSH: Distributive property and "core error"

I hate this mathematical result that I keep pushing. And routinely I try to destroy the argument myself but always run into the distributive property. So the sadder thing about this error is that it's easy to prove. The linchpin is the distributive property:

a*(b+c) = a*b + a*c

And I've said that before many times over the years, and it has never been refuted, and that is the reason I can use the complex plane to show the error, but the hatred for this error is so great that proof has not been enough.

Easy example of the distributive property with a simple polynomial factorization:

7(x^2 + 3x + 2) = (7x + 7)(x + 2)

notice that the result holds in the field of complex numbers that 7 distributes in one way, as the distributive property holds in the field of complex numbers. Even if I use a function to try and hide things:

7(x^2 + 3x + 2) = (g(x) + 7)(x + 2)

because g(0) = 0, forced by x=0, it must be the case that the 7 distributed in one way, even on the complex plane.

It's not a factor result.

Now I've pointed this out before on math newsgroups and had people claim that it wasn't the distributive property! I'd explain how it was. They'd say it wasn't. I'd finally tire of explaining that it was, and they'd claim victory!

So here's the core error result again:

7(175x^2 - 15x + 2) = (5a_1(x) + 7)(5a_2(x) + 7)

where the a's are roots of

a^2 - (7x-1)a + (49x^2 - 14x) = 0.

The a's can be found using the quadratic formula, but are so ugly I usually don't bother to give them, but maybe it'll help for those who wonder what they look like:

a_1(x) = ((7x - 1) +/- sqrt((7x-1)^2 - 4(49x^2 - 14x)))/2

a_2(x) = ((7x - 1) -/+ sqrt((7x-1)^2 - 4(49x^2 - 14x)))/2

or vice versa.

At x=0, one of the a's is 0, while the other is -1, as a^2 + a = 0, has those solutions (you can check with the explicit equations as well if you wish). So it pays to get functions where both are 0 at x=0, as that trivially finishes the exercise.

Letting b_1(x) = a_1(x) + 1, assuming a_2(0) = 0, I have a_1(x) = b_1(x) - 1, and substituting gives:

7(175x^2 - 15x + 2) = (5b_1(x) + 2)(5a_2(x) + 7)

And with functions that equal 0 when x = 0, the distributive property FORCES a 7 through a_2(x).

So the result holds--by the distributive property.

Math people have howled at me in rage at that claim for years now, but it doesn't change it.

So on the complex plane you have this result. It holds by the distributive property.

I hate the result. Math people hate the result. People all over the world I'm sure hate the result. It's in many ways an obnoxious slap in the face of the entire human race.

It is just a mean and nasty result that I wish were not true. I hate it. You can hate it. Everybody can hate it.

But it blows up a lot of established ideas in number theory. Which is why it's so hated.

So it takes away Galois Theory and group theory. So yeah, a lot of people hate it.

A lot of brilliant people got their lives destroyed by this error, and most of them never even knew it because they died in their ignorance as it's been around for a while. It resets the history books. A lot of big math names especially are diminished to below normalcy. It screws up some physicists too. Mucks up some Nobel prize wins. Blows up a lot of established orthodoxy.

Kind of just mucks up the whole world of particle physics. Kicks physicists in the groin—but it just clobbers mathematicians.

It is a nasty mean little math result that pulls no punches. Spares few, and does not give a damn about human feelings or needs.

It steps on prestige. Squashes cherished belief and makes a lot of smart people feel really, really dumb.

Mother Nature kicked the collective ass of the human race. Reality bit. People just weren't as smart as they thought they were, and got some crucial math wrong.

Nasty.

I can understand people running away from those feelings. And seeing their colleagues running with them in their collective sprint from the truth feel a little bit better. Nothing like running with a crowd! Oh the relief!!! Make the nasty math go away. Force reality to obey our wishes.

But damn reality won't oblige. Freaking stupid reality. What does it think it is?

Stupid experiments refuse to bend to human will. So sabotage the LHC!!! That's it! Oh wait. People expect you to fix it. It's fixed now, so now what? Lie I guess. (Oh, figure out a way to break it again?)

Nasty little math result. So mean. So vicious. Nasty math. HORRIBLE mean nasty vicious mathematics.

Don't you know now you HATE MATH!

Ok, go back to figuring out how to sabotage the LHC again, and think more about how much you HATE MATH.

Damn distributive property. SCREW the distributive property, right?

Who made IT boss of the world?

a*(b+c) = a*b + a*c

And I've said that before many times over the years, and it has never been refuted, and that is the reason I can use the complex plane to show the error, but the hatred for this error is so great that proof has not been enough.

Easy example of the distributive property with a simple polynomial factorization:

7(x^2 + 3x + 2) = (7x + 7)(x + 2)

notice that the result holds in the field of complex numbers that 7 distributes in one way, as the distributive property holds in the field of complex numbers. Even if I use a function to try and hide things:

7(x^2 + 3x + 2) = (g(x) + 7)(x + 2)

because g(0) = 0, forced by x=0, it must be the case that the 7 distributed in one way, even on the complex plane.

It's not a factor result.

Now I've pointed this out before on math newsgroups and had people claim that it wasn't the distributive property! I'd explain how it was. They'd say it wasn't. I'd finally tire of explaining that it was, and they'd claim victory!

So here's the core error result again:

7(175x^2 - 15x + 2) = (5a_1(x) + 7)(5a_2(x) + 7)

where the a's are roots of

a^2 - (7x-1)a + (49x^2 - 14x) = 0.

The a's can be found using the quadratic formula, but are so ugly I usually don't bother to give them, but maybe it'll help for those who wonder what they look like:

a_1(x) = ((7x - 1) +/- sqrt((7x-1)^2 - 4(49x^2 - 14x)))/2

a_2(x) = ((7x - 1) -/+ sqrt((7x-1)^2 - 4(49x^2 - 14x)))/2

or vice versa.

At x=0, one of the a's is 0, while the other is -1, as a^2 + a = 0, has those solutions (you can check with the explicit equations as well if you wish). So it pays to get functions where both are 0 at x=0, as that trivially finishes the exercise.

Letting b_1(x) = a_1(x) + 1, assuming a_2(0) = 0, I have a_1(x) = b_1(x) - 1, and substituting gives:

7(175x^2 - 15x + 2) = (5b_1(x) + 2)(5a_2(x) + 7)

And with functions that equal 0 when x = 0, the distributive property FORCES a 7 through a_2(x).

So the result holds--by the distributive property.

Math people have howled at me in rage at that claim for years now, but it doesn't change it.

So on the complex plane you have this result. It holds by the distributive property.

I hate the result. Math people hate the result. People all over the world I'm sure hate the result. It's in many ways an obnoxious slap in the face of the entire human race.

It is just a mean and nasty result that I wish were not true. I hate it. You can hate it. Everybody can hate it.

But it blows up a lot of established ideas in number theory. Which is why it's so hated.

So it takes away Galois Theory and group theory. So yeah, a lot of people hate it.

A lot of brilliant people got their lives destroyed by this error, and most of them never even knew it because they died in their ignorance as it's been around for a while. It resets the history books. A lot of big math names especially are diminished to below normalcy. It screws up some physicists too. Mucks up some Nobel prize wins. Blows up a lot of established orthodoxy.

Kind of just mucks up the whole world of particle physics. Kicks physicists in the groin—but it just clobbers mathematicians.

It is a nasty mean little math result that pulls no punches. Spares few, and does not give a damn about human feelings or needs.

It steps on prestige. Squashes cherished belief and makes a lot of smart people feel really, really dumb.

Mother Nature kicked the collective ass of the human race. Reality bit. People just weren't as smart as they thought they were, and got some crucial math wrong.

Nasty.

I can understand people running away from those feelings. And seeing their colleagues running with them in their collective sprint from the truth feel a little bit better. Nothing like running with a crowd! Oh the relief!!! Make the nasty math go away. Force reality to obey our wishes.

But damn reality won't oblige. Freaking stupid reality. What does it think it is?

Stupid experiments refuse to bend to human will. So sabotage the LHC!!! That's it! Oh wait. People expect you to fix it. It's fixed now, so now what? Lie I guess. (Oh, figure out a way to break it again?)

Nasty little math result. So mean. So vicious. Nasty math. HORRIBLE mean nasty vicious mathematics.

Don't you know now you HATE MATH!

Ok, go back to figuring out how to sabotage the LHC again, and think more about how much you HATE MATH.

Damn distributive property. SCREW the distributive property, right?

Who made IT boss of the world?