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Fixed LaTeX - Ricky

Last edited by ZHero (2008-07-21 12:59:57)

The question is what do we really mean by something 1/a?  1/a is related to a by the following property:

That is, 1/a is simply some number such that when multiplied by a, it gives the value 1.  Note that all the little details about arithmetic, different properties of numbers, and their meanings (when we interrupt numbers as quantities) are not used here.

1/a is simply some number (any number) such that when multiplying it by a, we get 1.  It turns out with a few very simple algebraic properties, 1/a must be unique.  So my "any number" remark doesn't really come up in the vast majority of mathematics.  But I suppose under a really ugly and chaotic system, it could.

That being said, what is 1/0?  Well, going by the above, it is some number such that 1/0 * 0 = 1.  We now prove that this is not possible.

Let a by any number.  First, note that certainly 0 + 0 = 0.  Then multiply both sides by a.  We get that a * (0 + 0) = a*0.  Using the distributed property, a*0 + a*0 = a*0.  Subtracting a*0 from both sides, we are left with a*0 = a*0 - a*0.  However, it must be that a*0 = a*0 (multiplication is well defined).  Therefore, a*0 = a*0 - a*0 = 0.  Therefore, for any number a, we have a*0 = 0.

So there is no such number where a * 0 = 1.  Now you may say, "What about infinity"?  When we introduce infinity, so many things get destroyed that make you want to go back and say, "Ok, let's not add in infinity".  We can try to define infinity in a way that intuitively makes sense:

For any positive real number a, a * infinity = infinity, a * -infinity = -infinity, infinity * infinity = infinity, infinity * -infinity = -infinity, -infinity * -infinity = infinity.

But what about 0?  If we have 1/0 * 0 = 1 and 1/0 = infinity, then infinity * 0 = 1.  Now doesn't that just strike you as a bit odd?  0*infinity = 1, but 0*infinity = (a-a)*infinity as well.  If we were to want the distributive properly to work on infinity, then a*infinity - a*infinity = 1?  No, no, that can't be.  For the love of God!  So the distributive properly doesn't work on infinity.  Everything else is still ok...

There is a well know theorem in mathematics that if a*b = c*b, then a = c.  It should be obvious that this doesn't hold since 1*infinity = 2*infinity.  Imagine trying to solve equations that involve infinity!  Distribution doesn't work and neither do normal cancellation laws.

Things only get worse as you consider the more "pure" aspects of algebra.  And I've been trying to say away from the "obvious" problems with introducing infinity (0/0, 0^0, etc).  It's best just to leave infinity where it belongs: A thing that numbers can approach, but not be.

MathsIsFun wrote:

What a nice fun explanation!

Ricky wrote:

When we introduce infinity, so many things get destroyed that make you want to go back and say, "Ok, let's not add in infinity".

Love it

ZHero: You are right that Zero is an amazing number.

Indeed!

"A thing that numbers can approach, but not be. "
awesome

Works if you take limits:

Otherwise it's not true

Last edited by Identity (2008-07-21 21:29:47)

Jane! Not so harsh, please. You are right, though.

Identity: Close! But we need to use a one-sided limit, which has special notation:

See Limits (An Introduction): "When it is different from different sides".

hahaha, I even caught my calc teacher at school making that exact mistake, marked it wrong when I said

wrote infinity in red ink. A lot of people thanked me when I pointed out that 'does not exist' was indeed the correct answer.

The two sided rule is just a technicality thats easy to forget. Just an honest mistake, not bs. Did you get enough sleep, Jane?

Last edited by mikau (2008-07-22 03:44:17)

ZHero, don't listen to Jane.

Jane, you're starting to cross the line.  The line has been pushed back quite a lot for you because you really do post very valuable things and contribute to the forums.  You are a valued member.  But saying someone's ideas are "BS" is not acceptable here, especially when ZHero has been polite and courteous.  Not everyone has the mathematical knowledge and experience you do.

Maybe you forget why ARB was banned in the first place.  It wasn't because he was wrong, it was because he was rude and did not listen to the moderators (most of all... me...).  I'm not comparing you to ARB, I would never do such a thing, but your reply to ZHero was rude and highly uncalled for.

If I had made ZHero's post, you know what?  It would be ok if you responded like that to me.  I'm someone who should know better, and I know that you know that too.  ZHero (probably) isn't.

Please, in the future, remember where you're posting.  Put yourself in the shoes of who you're replying to.  And above all else, remember that yelling makes you look like an idiot, not like a person who knows they are right.

I can see the difference between saying 'to divide by 0 (which is NOT DEFINED)' and to just say 1/0 equals infinity!

Last edited by ZHero (2008-07-22 05:25:57)

I can see the difference between saying 'to divide by 0 (which is NOT DEFINED)' and to just say 1/0 equals infinity!

Really?  Because I can't.  Those two things mean exactly the same, yet you are saying one is not defined and one is infinity.

But in the evaluation of certain integrals, involving infinity as one of the limits, one has to make use of such things as x^-infinity=0 or likewise.
However, one may say that this again involves 'limits' which is usually omitted during writing...

They are ommitted because mathematicians, as a whole, are lazy.  It does not mean they aren't their in the heads of mathematicians who read them.

ZHero wrote:

Now i see the point of such a nasty debate!

Cool. But it should never have been "nasty".

Jane please cool down ... we really love you on this forum, but not when you get angry. It is important to be a good educator as well.


Yes, ZHero, a kind of asymptote, in this case a vertical asymptote. Have a look at Plot of 1/x


BTW I have often heard "1/0 is infinity", but when you examine it closely you find that is not correct.

1/0 is asking "How many zeros make 1?"

Let us do a mental exercise: if you stack "zeros" on top of each other, the pile does not get any higher does it? Keep stacking ... the pile stays zero. So how can we say how many zeros in 1?

But yet we know if we go from 1/0.1 to 1/0.01 we get bigger ... and 1/0.001 is bigger, and 1/0.0000001 is even larger, and so on ... so we can see that as we get towards 1/0 the answer is getting towards infinity.

The result: we can't answer 1/0, but we can see that as "x" gets smaller "1/x" heads towards infinity. So that is why we use limits.

(And then we have to be careful which way we head there, from the positive or negative side, as we would head towards +infinity or -infinity)

You are forgetting the kind of people who think they know a lot... but don't really.

Would this be an appropriate time to discuss dividing by zero in modular arithmetic? Dividing by zero definitely plays an interesting role there. I'm not saying any more until either Ricky or Mathisfun gives me the go ahead, because I'm afraid that going into detail might be confusing and hence a bad idea.

Would this be an appropriate time to discuss dividing by zero in modular arithmetic? Dividing by zero definitely plays an interesting role there. I'm not saying any more until either Ricky or Mathisfun gives me the go ahead, because I'm afraid that going into detail might be confusing and hence a bad idea.

The integers modulo n always form a ring, and in such an algebraic structure, my proof that a*0 = 0 holds.  Thus, division by zero again does not make sense.  Now you can change definitions around to make it work (for example, having 1/a no longer meaning "the multiplicative inverse of a" would do it), but that isn't exactly modular arithmetic anymore.