Tuesday, March 22, 2011

The (Crappy ?) Elementary Algebra Operations With Zero and Infinity

Engineers know "Infinity" doesn't exist, but Mathematicians haven't gotten the memo. That's fine, there is a lot they can do with "Infinity."

Engineers DO however know very much that "Zero" exists. We run into it all the time, and here's how we usually run into it:

"It's not in the budget."

Elementary Algebra is the algebra that most people know/love/loathe. There are other and more interesting algebras, but we won't get into them today.

Today, we go "elementary", and these are some of the rules that make people's heads swim:

a x 0 = 0

0 / a = 0

a / \infty = 0

a ^ (-\infty) = if a^2 > 1

0 ^ a = 0

a ^ \infty = 0 if a^2 < 1

a x \infty = \infty                            <=== Hey, look, it's "Multiplication by Infinity"

a  / 0  = \infty                             <=== Dave Richeson, take note (See the Replies section)

\infty / a = \infty

a  ^  \infty  =   \infty   if  a^2  >  1

a  ^ - \infty  =   \infty   if  a^2  <  1

\infty  -  a  =   \infty

0  x  \infty  =  indeterminate

0  x  0  =  indeterminate

\infty  /  \infty  =  indeterminate

0  ^  0  =  indeterminate

\infty  ^  0  =  indeterminate

\infty  -  \infty  = indeterminate

???

Musical interlude:



Keep on loving what is true and the world will come to you, you can find it in yourself.
at

10 comments:

  1. Phil WarnellMarch 23, 2011 at 2:59 AM

    Hi Steven,

    a/0 is undefined

    a/infinity is undefined

    It’s not necessary to deal with the rest as infinity is a concept, not a number, making it a measure of quality and therefore not quantitatively definable except in comparison with itself; one of which Cantor refereed to being its cardinalies. That is to say something divided by infinity equals zero is to mistake a limit as being a truth, rather than a convenience of practice; which defines what being the difference between an earthly engineer and one not so limited:-)

    Best,

    Phil

    ReplyDelete
  2. Steven ColyerMarch 23, 2011 at 5:35 AM

    I think you're right Phil, and the book I took those formulas from, which is Calculus Refresher (and which I will add to the post) says otherwise. Authors are allowed to make mistakes, aren't they?

    ReplyDelete
  3. Phil WarnellMarch 23, 2011 at 6:15 AM

    Hi Steven,

    Everyone is allowed to make mistakes and do, yet only those like yourself is wisdom gained in the process by having learned from them:-) I may not be familiar with all the notation and intricacy of mathematics, yet I have strived over the years to examine its foundations as best as I can.

    Best,

    Phil

    ReplyDelete
  4. Steven ColyerMarch 23, 2011 at 6:25 AM

    You're up early, Phil. Insomnia? Going to the salt mines today to slay the dragons or some such? I have terrible insomnia. I think I suffer from thinkitosis ... thinking too much. :-)

    Yup, foundations are way important to me too. Sure enough. Or, we could stop trying to solve the mysteries of the Universe and be happy being couch potatoes, I suppose. Most people in my world are. At least they're off the streets not making trouble. That's ONE good thing about TV and computers. And the other good thing is .... is .... well, when I think of one I'll get back to you. :-)

    ReplyDelete
  5. Phil WarnellMarch 23, 2011 at 6:52 AM

    Hi Steven,

    “Going to the salt mines today to slay the dragons or some such? “

    So now I not only find you wise yet also intuitive, no wonder you are no longer a practicing engineer :-) Unfortunately that is my primary role at work, which is to attempt to have the illogical find that understanding doesn’t conflict with compassion, yet the two are actually inseparable. In my later years I have realized for the most part it is a thankless task and often fraught with difficulty and sometimes danger. Never the less, like the leopard I’ve long ago accepted I’m unable to chance my stripes and if I could I’d be all the more miserable. So I live for quality moments and try not to have them quantified.

    Best,

    Phil

    ReplyDelete
  6. Steven ColyerMarch 23, 2011 at 7:35 AM

    Phil wrote ....
    In my later years I have realized ...

    Well now you see Phil for the first time I have caught you speaking an untruth. We are in our middle years, sir.

    Granted, there's always the possibility (that you are correct and) we can go to sleep tonight and wake up dead in the morning ... many around our age have and frankly that's the way my Dad and I would prefer to go .... but there's nothing we can do about that so I don't fret about it and anyway what the heck, I've had a good run.

    I'd like a few more laps around the track but knowing me that will never change regardless of age. Anyway, the world ain't done with us yet and medicine gets better and better at an astounding rate.

    On the other hand, you DO live in a country with socialized medicine, so, I'll remember you well my friend.

    ReplyDelete
  7. Phil WarnellMarch 24, 2011 at 5:28 AM

    Hi Steven,

    “We are in our middle years, sir.”

    That could be true if one attempts to quantify things rather than qualify them and yet even statistically we are entering our last third of the journey. Anyway I have no great expectation either way and try to remember that no matter how long I live it will always be in the now. So I’m more concerned with my nows and those of the ones I hold dear that will experience the ones after me.

    Best,

    Phil

    ReplyDelete
  8. Steven ColyerMarch 24, 2011 at 7:21 AM

    Good words, Phil. Yes, that's how I live my life, in the "now." The past is set in stone and we humans have the hubris to think the future isn't either. Oh, I guess "It's all about ME!!!" Please.

    So we thinkinoids split time into 3 parts, Past, Present and Future, when in truth I get the strong feeling it is all of ONE thing.

    I couldn't prove it, I believed in Free Will all my life, until one day I deduced on two simple postulates (I forget the other one) that Karma, not Free Will, must be true for our Universe to have any meaning.

    And boy oh boy did that tick me me off when I deduced that. I HATE being a cog in a gear, part of some large clockwork.

    On the other hand (there's always another hand), I take great joy in knowing that however this Universe is constructed, at least we have be given the illusion of Free Will.

    And that ain't bad!
    :-)
    Right, Douglas Adams?

    Time is an illusion, lunchtime doubly so.
    ... Douglas Adams

    ReplyDelete
  9. Phil WarnellMarch 24, 2011 at 7:26 AM

    Hi Steven,

    I’m glad we agree that it is the now which stands as being the most relevant. However I’m not convinced that we are merely puppets on a stage as I’ve discovered as to be convinced there is something that is positively what David Z. Albert referred to as a “private will”. The question this raises is this private will what acts as the element that maintains our illusion or rather what has conscious being able to actually have affects of choice mould the future. I’m not however suggesting this being in a profound way universally, yet in a diminishing way relevant to one’s own circumstance and influence. Anyway, not to beat you over the head, yet if you ever make the time to read Albert’s book “Quantum Mechanics and Experience” perhaps you may after feel there may be sound reason to believe you are more than some self delusional puppet unwittingly playing his role..

    Best,

    Phil

    ReplyDelete
  10. Steven ColyerMarch 24, 2011 at 8:58 AM

    Thanks, Phil, I'll definitely put that book on my bucket list. To which I would return the favor and ask if you've ever read Julian Barbour's "The End of Time." I haven't but it's already on my list, and rapidly approaching the top.

    But first ... Fractals! :-)

    Meaning, it's time I stopped talking about them, and started digging into their actual Maths.

    Diophantus would not be amused, but he did the best he could in the times in which he lived, and we're all better off for his parents doing the one thing that humans seem to do so well and giving him to first themselves and then, to the ages.

    ReplyDelete