Why 1 / ∞ = 0 but ∞ * 0 is not equal to 1?

In summary, division by zero or by infinity is undefined because it leads to mathematical inconsistencies and confusion. While division by infinity can be thought of as tending towards zero, it is not a defined operation in mathematics and can lead to ambiguous answers. Similarly, division by zero is also undefined and can result in conflicting answers. Therefore, it is best to avoid dividing by zero or infinity in mathematical equations and calculations.
  • #36
frost_zero said:
1/∞ is not zero, it tends towards zero because the difference between numerator and denominator is huge
There's a lot of imprecise terminology there.
frost_zero said:
, it certainly is undefined in common mathematics,
That is true.
frost_zero said:
however in certain cases such as in physics it is taken as zero because the difference is huge and we are not aiming for much precision in physics
It's not a question of precision. It's a question of physicists using a shorthand ##1/\infty## instead of writing out precise statements with limits. For example, in physics "at infinity" means "far enough away that the gravitational potential differs from zero by a negligible amount".
 
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  • #37
##\frac{1}{\infty}=0##, ##\frac{2}{\infty}=0##,, ##\frac{3}{\infty}=0##,, etc. Indeterminate when multiplied back.
 
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  • #38
I believe in measure theory we accept $$\infty \cdot 0 =0; \infty \times \infty = \infty $$, in that we do accept values in $$\mathbb R \cup \{\infty \} $$.

And I guess the Riemann Sphere is the/a 1-point compactification. I believe there may be inequivalent types of 1-point compactifications.
 
  • #39
I think it's helpful to consider infinity itself to be undefined. Think of it less like a number and more like a special symbol. I say symbol and not variable because there is more than one kind of infinity, such as countable versus uncountable infinities, that the simple "1/∞ != ∞*0" formulation can't differentiate between even before you get to using limits.

It's a bit like the bar over a repeating decimal - an artifact of the limitations of our ability to express a mathematical concept without collapsing the blackboard into an infinite mass black hole from information density.
 
  • #40
InkTide said:
I think it's helpful to consider infinity itself to be undefined. Think of it less like a number and more like a special symbol. I say symbol and not variable because there is more than one kind of infinity, such as countable versus uncountable infinities, that the simple "1/∞ != ∞*0" formulation can't differentiate between even before you get to using limits.

It's a bit like the bar over a repeating decimal - an artifact of the limitations of our ability to express a mathematical concept without collapsing the blackboard into an infinite mass black hole from information density.
I'd love to have a drink at that bar ;).
 
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  • #41
PeroK said:
There's a lot of imprecise terminology there.

That is true.

It's not a question of precision. It's a question of physicists using a shorthand 1/ instead of writing out precise statements with limits. For example, in physics "at infinity" means "far enough away that the gravitational potential differs from zero by a negligible amount".
I agree with the last statement and to phrase my first statement better : as the difference in numerator and denominator increases the number becomes smaller and smaller, with 1/∞ the difference is possibly infinite which is why it is said to tend towards 0
 
  • #42
frost_zero said:
as the difference in numerator and denominator increases the number becomes smaller and smaller,
That's not true. A ratio does not directly depend on this difference. Consider ##\frac n {2n}## as ##n## increases.
frost_zero said:
with 1/∞ the difference is possibly infinite which is why it is said to tend towards 0
##\frac 1 \infty ## is simply undefined. Or, if you want to extend numbers to include ##\infty## then it may be defined to be precisely ##0##.
 
  • #43
danielassayag said:
As we know those relations are true: if a/b = c, then a = b*c and b = a/c
Following your scienario
[tex]\lim_{a\rightarrow +0} a=+0[/tex]
[tex]\lim_{a\rightarrow +0} \frac{1}{a}=+\infty [/tex]
[tex]\lim_{a\rightarrow +0} a\cdot \frac{1}{a}=1[/tex]
[tex]+0\cdot+\infty=1[/tex]
Examples of other scenarios are
[tex]\lim_{a\rightarrow +0} 2a=+0[/tex]
[tex]\lim_{a\rightarrow +0} \frac{1}{a}=+\infty [/tex]
[tex]\lim_{a\rightarrow +0} 2a\cdot \frac{1}{a}=2[/tex]
[tex]+0\cdot+\infty=2[/tex] and
[tex]\lim_{a\rightarrow +0} a^2=+0[/tex]
[tex]\lim_{a\rightarrow +0} \frac{1}{a}=+\infty [/tex]
[tex]\lim_{a\rightarrow +0} a^2\cdot \frac{1}{a}=+0[/tex]
[tex]+0\cdot+\infty=+0[/tex]
, and
[tex]\lim_{a\rightarrow +0} a=0[/tex]
[tex]\lim_{a\rightarrow +0} \frac{1}{a^2}=+\infty [/tex]
[tex]\lim_{a\rightarrow +0} a\cdot \frac{1}{a^2}=+\infty[/tex]
[tex]+0\cdot+\infty=+\infty[/tex]

I am afraid that we cannot choose unique scenario of yours.
 
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  • #44
jack action said:
Pretty simple to demonstrate and Wolfram agrees.

First, what does Wolfram says about ##0 \times \infty##: undefined

But Wolfram also says that ##\frac{1}{\infty} = 0##. It is not undefined.
There is mathematica and there is IEEE754.

IEEE754 is a standard for digital floating point arithmetic. It's purpose is to ensure that every programmed math operation has a well-defined result, even if the operation, such as division by zero, is mathematically untenable.

In IEEE754, infinity is any number that exceeds the maximum floating point value, even by a tiny fractional value, is infinity, but it also raises an overflow exception.

IEEE754 allows division by zero and other operations defined by limiting values, such as log(0), and it will default to infinity as an answer (because generall it makes the most sense), buit it will also raise an exception (it will flag the result as a possible error).

IEEE754 also has +0 and -0, i.e. signed zeros, which do not occur in ordionary arithmetic.

https://en.wikipedia.org/wiki/Signed_zero

Do not confuse mathematics with IEEE754.
 
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  • #45
diegogarcia said:
There is mathematica and there is IEEE754.

IEEE754 is a standard for digital floating point arithmetic. It's purpose is to ensure that every programmed math operation has a well-defined result, even if the operation, such as division by zero, is mathematically untenable.

In IEEE754, infinity is any number that exceeds the maximum floating point value, even by a tiny fractional value, is infinity, but it also raises an overflow exception.

IEEE754 allows division by zero and other operations defined by limiting values, such as log(0), and it will default to infinity as an answer (because generall it makes the most sense), buit it will also raise an exception (it will flag the result as a possible error).

IEEE754 also has +0 and -0, i.e. signed zeros, which do not occur in ordionary arithmetic.

https://en.wikipedia.org/wiki/Signed_zero

Do not confuse mathematics with IEEE754.
I was using the extended real number line ##\overline{\mathbb{R}}##, defined as:
$$\mathbb{R} \cup \{-\infty, +\infty\} \text{ where } -\infty < x < +\infty \text{ for all } x \in \mathbb{R}$$
 

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