Your life story in six words.

Started by Caity, September 07, 2008, 08:22:08 PM

Previous topic - Next topic

0 Members and 1 Guest are viewing this topic.

Izzy1337

Izzy was recently accused of vampirism...
Fire is my Element, and I love to play with it; Water is my Passion, and I live to play with it.




Mordred

Today, braindead took on new meaning.
*See Pillory*

Zaer Darkwail

Too much school paperwork single week

Qt


Izzy1337

Sometimes, murder should be more legal...
Fire is my Element, and I love to play with it; Water is my Passion, and I live to play with it.




Twilightfiredragon

Ex-racer missing his favourite fast car.

Lockdown Lucy

Thwarted supervillain returns to strike again.

InariShiftskin

A mindless slog through empty hours.

Mordred

Scrambling to return to level zero.
*See Pillory*

Twilightfiredragon


Mordred

Contemplating leap from -1 to +1.

*as the wind causes the rickety stairway to sway with creaks and groans.. hehehe.. and that gorgeously wicked shadow below licks it's insatiable chops as it waits for this mad vampire to fall.. and the mad vampire wonders where his lovely pool of blue jell-o is now, eh? :P*
*See Pillory*

Mordred

Feelings shadow's full power once again. :D
*See Pillory*

WhiteTigerForever

Quote from: Mordred on October 04, 2011, 07:15:31 PM
Contemplating leap from -1 to +1.

*as the wind causes the rickety stairway to sway with creaks and groans.. hehehe.. and that gorgeously wicked shadow below licks it's insatiable chops as it waits for this mad vampire to fall.. and the mad vampire wonders where his lovely pool of blue jell-o is now, eh? :P*

oh noooooz you didn't...take this!!! XD


1 is not actually defined to be greater than zero, it can be proven. The real numbers are an ordered field (technically, a complete ordered field, but completeness is not needed in this proof). If you are unfamiliar with the field axioms and the definition of a total ordering, you should look those up first. In fact, since there are different ways of defining a total ordering, I'll just give you the definition I'll be using:

If S is a set, define a subset P (called the positive set) by the following three conditions:

1.) Elements of P are closed under addition
2.) Elements of P are closed under multiplication
3.) If x is an element of S, then one and only one of the following are true: a is in P, -a is in P, a = 0

The third condition is called the law of trichotomy. -a is defined to be the additive inverse of a and 0 is defined to be the additive identity. We define the order relation > by

a > b if and only if a - b is an element of P.

Now, before proving this, I have to start with a lemma. I need to prove that if a is a non-zero real number, then a^2 > 0 (meaning it's in the positive set).

Suppose a is a non-zero real number. By the law of trichotomy, either a > 0 or a < 0. If a > 0, then by closure of multiplication in the positive set, a^2 > 0. If a < 0, then -a > 0. So,

a^2 = aa = (-a)(-a) = (-a)^2, which is in the positive set, again by closure. Thus, a^2 > 0 in all cases where a is not equal to zero.

Note that I didn't justify the step aa = (-a)(-a). This can be proven as well, but this is taking long enough as it is, so I'll let you figure it out (it's not that hard if you use the distributivity axiom). Now, onto the proof.

We first want to show that 1 is not equal to zero (bear in mind that 1 is defined to be the multiplicative identity). We'll do this by assuming the opposite and arriving at a contradiction. Suppose that 1 = 0. If a is a non-zero real number, we have a = a*1 = a*0 = 0, which is a contradiction (because we assumed a was non-zero). Note that the step a*0 = 0 was also unjustified. Again, this isn't difficult to prove using the field axioms, but I'd rather not get any more off track.

So, since 1 is not equal to zero, then the law of trichotomy says that either 1 > 0 or 1 < 0. Now, by the definition of multiplicative identity, 1^2 = 1*1 = 1. However, our lemma says that given any non-zero real number a, a^2 > 0. Thus, 1 > 0.    hee hee hee ;D
Not accepting new roles but Photoshop riches await you instead.

Izzy1337

Can you translate that from Math?
Fire is my Element, and I love to play with it; Water is my Passion, and I live to play with it.




WhiteTigerForever

*giggles* You made me smile big! :D
Not accepting new roles but Photoshop riches await you instead.

Izzy1337

Thanks, glad I can help someone.
Fire is my Element, and I love to play with it; Water is my Passion, and I live to play with it.




Mordred

*Simply plans on spreading unlimited chaos* :P
*See Pillory*

WhiteTigerForever

ooooohhhhh dearest Mordred... we could have great fun discussiog "chaos"  XD  ;D or 0Kelvin..XD
Not accepting new roles but Photoshop riches await you instead.

Ephemera

“I bleed myself to be your drink:  Is not the blood of poets—ink?”   ―William Soutar
My Ink Blood Spilled | Who I Am | Where I Am (A/A) | Intro | Avi Source
My Poly/Kink/Random Blogs | My Drawer | My Concupiscence (O/O)
I, with a deeper instinct, choose a man who compels my strength, who makes enormous demands on me, who does not doubt my courage or my toughness, who does not believe me naive or innocent, who has the courage to treat me like a woman.” ~Nin  Working on: Ink Blood Spilled

WhiteTigerForever

#2469
Wrote things in the wrong place.

Not accepting new roles but Photoshop riches await you instead.

Ephemera

“I bleed myself to be your drink:  Is not the blood of poets—ink?”   ―William Soutar
My Ink Blood Spilled | Who I Am | Where I Am (A/A) | Intro | Avi Source
My Poly/Kink/Random Blogs | My Drawer | My Concupiscence (O/O)
I, with a deeper instinct, choose a man who compels my strength, who makes enormous demands on me, who does not doubt my courage or my toughness, who does not believe me naive or innocent, who has the courage to treat me like a woman.” ~Nin  Working on: Ink Blood Spilled

Izzy1337

...Anything that's of use to anyone.
Fire is my Element, and I love to play with it; Water is my Passion, and I live to play with it.




Mordred

Desperately trying to clear brain fog. :(
*See Pillory*

Izzy1337

Fire is my Element, and I love to play with it; Water is my Passion, and I live to play with it.




Mordred

Time to re-embrace the elusive shadow.
*See Pillory*