

Infinity Is Not a Number - It's a Free Man - tokenadult
http://nrich.maths.org/public/viewer.php?obj_id=2756

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jacobolus
Infinity is a number if we want it to be: we can do math in whatever kind of
topological space we want, including ones with defined ∞, such as the one
point compactifications of ℝ or ℂ, the latter of which has a nice wiki page:
<http://en.wikipedia.org/wiki/Riemann_sphere>

As wikipedia says, “arithmetic with infinity does not obey all of the usual
rules of algebra, and so the extended complex numbers do not form a field.”
But they’re quite handy for some uses.

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gizmo
In the opening paragraph she mentions that you may be convinced that 1 = 0, if
you believe infinity is number. I don't think she showed the proofs there, so
here goes:

Take equation: ∞ + 1 = x

Suppose that ∞ is a number.

Suppose ∞ ≠ x. Because ∞ + 1 = ∞, (same cardinality; you can make a
bijection), you get ∞ = x. This is a contradiction, so x and infinity must be
the same.

Suppose ∞ = x. Then the equation becomes x + 1 = x. Subtract x on both sides.
1 = 0. This too is a contradiction.

Therefore ∞ cannot be a number.

~~~
jonsen
A shorter form:

Take equation: ∞ + 1 = ∞

If ∞ is a number you can subtract it from both sides giving 1 = 0

So ∞ cannot be a number.

 _But_ then 0 too is not a number:

Take equation: 0 * 2 = 0

If 0 is a number you can divide it into both sides giving 2 = 1

So 0 cannot be a number.

0 is special. You cannot multiply and divide by it unconditionally.

∞ is special. You cannot add and subtract it unconditionally.

~~~
billswift
All you managed to do is show why division by zero is not allowed. It doesn't
show 0 is not a number.

~~~
jonsen
0 is a number all right, and division by 0 isn't proving anything. My point is
that maybe subtracting ∞ doesn't prove anything either.

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nalbyuites
Ok, I understand the math part, don't quite get the 'Free Man' part. Is it a
reference to something? Enlighten a poor soul.

~~~
tome
<http://en.wikiquote.org/wiki/The_Prisoner>

~~~
nalbyuites
Ah, that clears it up. Thank you, good Sir!

~~~
brandnewlow
Be seeing you.

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jonsen
It depends, I think. In some circumstances ∞ is just a special number, as 0 is
a special number.

Consider the operation (||) of parallel coupling of resistors

    
    
      1Ω || 1Ω = 1/2Ω   cf.   1Ω + 1Ω = 2Ω 
      1Ω || 0Ω =   0Ω   cf.   1Ω + 0Ω = 1Ω
      1Ω || ∞Ω =   1Ω   cf.   1Ω + ∞Ω = ∞Ω
    

For the operator || ∞ is the identity element as 0 is for operator +

~~~
tokenadult
Do we really consider infinite resistance a numerical value of (a measurement
of) resistance?

~~~
jonsen
No we don't consider ∞ a _numerical_ value.

The big question is: Is it a value nonetheless?

Edit: Infinite resistance is as well defined as zero resistance.

Infinite resistance equals zero conductance.

Infinite conductance equals zero resistance.

Edit2: So if you measure the conductance of an infinite resistor you get the
numerical value 0.

~~~
tokenadult
I think this reasoning ends up being something like dividing by zero. A
measurement of "infinity" can't be in any definable place on interval scale.

~~~
jonsen
Well then a measurement of zero can't be in any definable place on interval
scale either.

~~~
tokenadult
Above you wrote,

 _No we don't consider ∞ a numerical value._

So you are agreeing with the title and with the substance of the submitted
article, which I submitted for truth and for admiration of how it was written.

As for an interval scale

[http://en.wikipedia.org/wiki/Level_of_measurement#Interval_s...](http://en.wikipedia.org/wiki/Level_of_measurement#Interval_scale)

the zero point is well defined as the midpoint between -1 and 1, that is
halfway between the values defined by those numbers. (It is commonly noted
that the numerical designation of any point on such a scale is arbitrary, and
the familiar examples are the Fahrenheit and Celsius scales of temperature,
each with a different zero point, which in both cases is equidistant between
-1 and 1.) No comparable definiteness of position is enjoyed by "infinity."

Psychologist Joel Michell makes the very good point that his fellow
psychologists are probably abusing language by referring to interval scales as
"measurements,"

[http://www.questia.com/library/book/an-introduction-to-
the-l...](http://www.questia.com/library/book/an-introduction-to-the-logic-of-
psychological-measurement-by-joel-michell.jsp)

but for our purposes here it is enough to note that you, I, and the whole
world notice that zero has a place on an interval scale if such a place is
arbitrarily set, but infinity does not have such a place.

~~~
jonsen
I think infinity is well enough defined - for measurements at least - on the
reciprocal scale:

    
    
       -4   -3   -2   -1    0    1    2    3    4
      --|----|----|----|----|----|----|----|----|--
    
      -1/4 -1/3 -1/2  -1    ∞    1   1/2  1/3  1/4
      --|----|----|----|----|----|----|----|----|--
    

I certainly didn't intend to devaluate the submitted article. Just wanted to
point at some circumstances where infinity in fact is considered a distinct
and definable value; by the example of resistance being the reciprocal of
conductance.

~~~
tokenadult
That's still a division by zero error. Division by zero is no more valid in
engineering than it is in mathematics. While I was out on a walk in my
neighborhood I thought about the practical engineering problem you first
mentioned above: whatever kind of resistor you are talking about, I think a
lot less than infinite electricity could jump the gap filled by any finitely
sized resistor, even if you initially regard it as having infinite resistance
for purposes of analyzing the circuit. But I will leave further discussion of
this point to readers here who have formal training and experience with
electrical engineering, as my childhood best friend does. I'm not aware that
he considers infinity a number or that he puts it anywhere on the number line.

<http://mathforum.org/dr.math/faq/faq.divideby0.html>

<http://mathworld.wolfram.com/DivisionbyZero.html>

<http://en.wikipedia.org/wiki/Division_by_zero>

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doodyhead
It's excellently written, but I wish I could turn off the analogies. For me,
analogies only muddle the point.

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TrevorJ
Infinity is the substrate on which numbers are defined. It's the space that
numbers are carved out of.

