# The other day our lecturer was going through field axioms, rules of

## Main Question or Discussion Point

The other day our lecturer was going through field axioms, rules of numbers, I guess what you'd call very elementary number theory. In particular, he was explaining what you can and can't do with ∞, and he mentioned that x/∞=0.

I guess I'd just like some clarification on this. Should this not imply that 0∞=x? Why can we divide by infinity but not by zero? Is this just "how it is"?

One interpretation that makes sense to me is that infinity doesn't "go" into any number, no matter how large: is this what the expression really means?

Help appreciated :)

Have you studied limits? x/∞≠0 in actuality. It only approaches zero. Assume that infinity is 999999999999999999999 and x =1.
Notice how it's an insignificant number in almost any context. It will never strictly equal zero though.

You can say that the number asymptotically approaches zero.

Normal rules of algebra don't apply for infinity or zero.

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haruspex
Homework Helper
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Your lecturer should not have stated that x/∞ = 0. ∞ is not a number, so no arithmetic operations on it are defined. ∞ is valid as specifying a limit. So it is correct to say that the limit of x/n →0 as n →∞.

The other day our lecturer was going through field axioms, rules of numbers, I guess what you'd call very elementary number theory. In particular, he was explaining what you can and can't do with ∞, and he mentioned that x/∞=0.

Yes, bad lecturer! He should have made it very clear that +∞ is not a real number, so not all arithmetic operations and rules make sense.

However, +∞ is an extended real number. The extended real numbers are just the real numbers with +∞ and -∞. As extended real numbers, we often do define x/∞=0. This is just a definition, we could have defined it as 2 or 3 if we wanted to. The definition as 0 is chosen because it conforms to our intuition. But again: this is not division as you know it between real numbers since ∞ is not a real number.

Also, from x/∞=0 does not follows that ∞.0=x (whatever x is). Indeed, in order to establish this, we would apply the rule a/b=c implies a=bc. But this rule is only true for real numbers, and not necessarily for extended real numbers.

This is only true in the sense of limits. While it is not true to say that

$\frac{x}{\infty} = 0$,

it is correct to say:

$\lim_{n \to \infty} \frac{x}{n} = 0$.

Integral
Staff Emeritus
Gold Member

When working with the extended reals it is common to define all operations on infinity.

$$\frac x \infty = 0$$ is the common definition.

Hurkyl
Staff Emeritus
Gold Member

The lecturer may have said accurate things, and the listener may have overlooked the import of some context. :tongue:

Real analysis is better understood by using the extended real numbers. Geometry, number theory, and complex analysis benefit greatly from projective arithmetic.

I assert that it's better to (properly) learn a number system that involves infinite numbers than it is to use ad hoc means to avoid them.

In particular, he was explaining what you can and can't do with ∞, and he mentioned that x/∞=0.
Pay close attention to the "what you can and can't do" part. Among the things you can't do is convert an equation
a/b=c​
into an equation
a=bc​

You've spent a long time working exclusively in a very specific number system: you've probably gotten quite used to how, e.g., the real numbers work. In that setting, there's no harm in mentally equating the expression
a/b​
with the notion of
the unique solution to a=bx​
so long as b is known to be nonzero.

(actually, that's not true: I believe there is harm in being unable to form a concept of division in its own right)

But now you're learning about a new setting, and so you're going to have to pay careful attention to what is appropriate for the new setting. Among the things inappropriate for the new setting is replacing the notion of "division" with the notion of "solving a multiplicative equation".

You've just learned this new number system. It will take some time to build intuition about it. Pay careful attention to what you can and cannot do with these new numbers. Pay attention to (and experiment with and do exercises about) the practical applications of these numbers, and soon enough you will develop intuition about them. When you're used to the applications, the reason why the number system is the way it is will become obvious, and you will eventually wonder why you ever thought otherwise!

consider the following ->

1/1 = 1

1/10 = 0.1

1/100 = 0.01

1/1000 = 0.001

1/10000000000 = 0.0000000001

and so on...

so, if the denominator exceeds any known quantity (which we lovingly refer to as our dear Mr. ∞), we see that the value moves closer and closer to zero..... thus, the definition of limit :)

Hurkyl
Staff Emeritus
Gold Member

consider the following ->

1/1 = 1

1/10 = 0.1

1/100 = 0.01

1/1000 = 0.001

1/10000000000 = 0.0000000001

and so on...

so, if the denominator exceeds any known quantity (which we lovingly refer to as our dear Mr. ∞), we see that the value moves closer and closer to zero..... thus, the definition of limit :)
Your statement is confused -- you are mixing up sequences with limits of sequences.

The "moving value" is not a property of any fraction: a sequence of fractions could be viewed this way, though, if you think of the index as being "time".

The extended real number fraction 1/(+∞) is simply an extended real number: it does not "move". This fraction does not appear anywhere in what you have written.

Instead, the denominators are converging to +∞, and since division is continuous there, the left hand sides are converging to 1/(+∞) .