why was I not told 1/0 = infinity

[robertliguori]

Question that’s not worth its own thread: is 0*infinity also undefined? Because it seems that all integration is taking the sum of an infinite number of areas which individually equal zero, yet different integrals give different answers.

[Achernar]

@robertliguori

Question that’s not worth its own thread: is 0*infinity also undefined? Because it seems that all integration is taking the sum of an infinite number of areas which individually equal zero, yet different integrals give different answers.

What you’re saying is technically not true. It’s okay to think of an integral as the sum of an infinite number of things when you want to be informal, but in reality that’s not what it is. It’s the same deal with a series - you may think of it as the sum of an infinite number of terms, but in reality, it’s the limit of partial sums, and a partial sum is the sum of a finite number of terms.

[Jabba]

@robertliguori

Question that’s not worth its own thread: is 0*infinity also undefined? Because it seems that all integration is taking the sum of an infinite number of areas which individually equal zero, yet different integrals give different answers.

Yes, as are infinity/infinity and infinity - infinity.

[Ilsa_Lund]

(and saying ‘the limit is infinity’ is another shorthand, since it treats infinity like a number).

Err, isn’t limf(x), x->c=inf not true, as the function becomes unbounded?

[raygirvan]

@Achernar

You mean 0^0, right, not 0/0?

Aargh! Never trust memory. [crawls away under stone]

[Ilsa_Lund]

@raygirvan

Aargh! Never trust memory. [crawls away under stone]

0/0 is indeterminate.

[ultrafilter]

@Maximum C

This sort of quasi-math is a result of deep philosophical problems with Calculus. Well, not problems as far as getting a consistent system is concerned- problems as far as the underlying math making any sense.

A few posts above people were asking about why x/x = 1. They had the impression this was true for all x, when it’s only strictly true for x=!0. The problem is probably rooted in the Calc concept of infintesimals. That is to say, Calc deals with problems of continuity by assuming that there exist numbers (which the call differentials) that “approach zero” or “get smaller without bound.” You know these boys as “dx” and “dy”.

The whole key to Calculus (other than recognizing that differentiation and integration are the same thing in reverse) is treating differentials as > 0 for purposes of division, but =0 for purposes of addition.

The confusion about x/x=1 comes directly out of this. If you’ve ever differentiated by hand, then you know that you make liberal use of dx/dx=1 (or maybe it was called “h” nowadays… I forget) when you’re reducing the equation. When you simplify the equation, say to “x + dx” maybe, the teacher will tell you just to drop the dx because it’s zero. Strictly speaking, it ISN’T zero, it’s just an infintesmial. You can see how this could be confusing.

There’s a sort of muddy argument Lebniez and Newton used to explain this. Newton went on about “Mouns” and other philosophical nonsense, too, but that’s neither here nor there. The notion is that these infintestimals represent a change in the value of x, and they can be as small as you want - but they will never be zero. So you are free to divide by them. But how can you then just drop them when you’re adding? How come dx acts like zero in addition?

Because there’s always two ways to represent a number in a decimal system. Every real number can be written two ways. Some examples: 1 = 0.999… 1.33 = 1.329999… or 2345 = 2344.99999… See the pattern? I won’t prove this is true here, someone else can do that (the proof is simple). The point is that, conceptually, there is a “value” in between nothing and something in math. What I’m saying, VERY ROUGHLY, is that 1 - 0.999… = dx.

So the confusion about x/x=1 ends up cutting to the very root of what I think is the most facinating and deep subject in mathematics…

Any comments from people who know more about this?

-C

This post would’ve been accurate about 150 years ago, before the logical problems with infinitesimals got them tossed out of the foundations. Now it’s all done with limits. The notation dy/dx is left over from the early days, but it doesn’t have the intuitive meaning of the ratio of two infinitesimals. It’s just the value of a limit.

Now, in the 1960s, somebody did come up with a logically sound version of the infinitesimals, and you can base calculus on this. It comes out being fairly simple compared to the limit version, but it can be confusing, because it requires you to understand the difference between standard and non-standard reals. That’s a little bit more than the average college freshman is ready for.

And even if you do have infinitesimals, 1 - .9… = 0.

[kniz]

@kunilou

In other words, divide that cake into 0 pieces. It’s a meaningless statement.

To me it means:
[indent]:mad: [sup]“Keep your friggin hands off the cake!”[/sup][/indent]

[ultrafilter]

@Ilsa_Lund

Err, isn’t limf(x), x->c=inf not true, as the function becomes unbounded?

This expression is shorthand for “as x approaches c, f(x) increases without bound”.