Why can't you divide by zero?

[SS-18 ICBM]

It can pop up in some situations. A singularity has a volume of zero, does it not? Now say you want to calculate the energy density there.

 

[Murky]

It can pop up in some situations. A singularity has a volume of zero, does it not? Now say you want to calculate the energy density there.

No, for something to exist it must have a volume greater than zero. It could be the size of an atomic particle and would still have volume.

 

[GoodGame]

It can pop up in some situations. A singularity has a volume of zero, does it not? Now say you want to calculate the energy density there.

By theory, but in measurement?

http://suite101.com/article/calculating-density-of-black-holes-a69442

After a black hole forms, all the mass is compressed into a central point, called the singularity, which is infinitely dense. However to find the density needed to form a black hole divide the mass of the collapsing star by the volume enclosed by the event horizon.

In a black hole space and time are so distorted that ordinary rules of geometry, such as the formula for the volume of a sphere, do not technically apply. However this order of magnitude calculation will ignore that technical detail. The formula for the volume of a sphere of radius, R, is V=(4/3)piR^3. So a sphere of radius 3000 meters will have a volume of about 1e11 meters^3.

Density is the mass divided by the volume. Dividing the mass of the Sun, 2e30 kilograms, by this volume gives the density to which a solar mass star would need to be compressed to form a black hole. This gives the density of a black hole as about 2e19 kilogram/meter^3.

Using Einstein's mass energy equivalence formula, E=mc^2, the mass of 2e30 kilograms is equivalent to an energy of about 2e47 joules. Dividing this energy by the volume gives the energy density needed to form a solar mass black hole, which is 2e36 joules/meter^3.

Zeilik, M. and Gregory, S. Astronomy & Astrophysics, 4th ed. Saunders, 1998.

 

[GenMarshall]

What my math professor said is that there are two possible results for dividing by zero: negative infinity (I.E. it goes forever in the negative direction) and positive infinity (I.E. it goes forever in the positive direction) depending on how you do the calculation.

So I guess this explains why I put in 1/x in my graphing calculator:

 

[GoodGame]

That's calculus, as in a limit of a function, not arithmetic.

http://en.wikipedia.org/wiki/Limit_of_a_function

 

[dutchfire]

No, for something to exist it must have a volume greater than zero. It could be the size of an atomic particle and would still have volume.

What is the volume of an electron?

 

[uppi]

If you define infinity to be the multiplicative inverse of 0, then you can divide by zero. Or if you are a physicist and don't care about the feelings of mathematicians it is also no problem to divide by zero.

 

[madviking]

Zero, imo, is more of a concept than a number since 'infinity' is a definite concept and definitely not a number. Since zero is conceptually the inverse of 'infinity', zero inherits the conceptual nature of infinity.

It always bugs me when people say a function "approaches infinity".

NO. IT CAN'T APPROACH A CONCEPT. IT INCREASES WITHOUT BOUNDS.

But interestingly I don't think saying a function "approaches zero" is logically incorrect.

 

[dutchfire]

Since zero is conceptually the inverse of 'infinity', zero inherits the conceptual nature of infinity.

Conceptually, zero is defined as the number that you can add to a number so that the latter doesn't change. So conceptually zero is linked to addition, not to multiplication.

 

[Save_Ferris]

Zero, imo, is more of a concept than a number since 'infinity' is a definite concept and definitely not a number. Since zero is conceptually the inverse of 'infinity', zero inherits the conceptual nature of infinity.

It always bugs me when people say a function "approaches infinity".

NO. IT CAN'T APPROACH A CONCEPT. IT INCREASES WITHOUT BOUNDS.

But interestingly I don't think saying a function "approaches zero" is logically incorrect.

/thread

Abstractly, however, making something into zero parts is impossible because mass cannot be destroyed.

 

[Leoreth]

Why would you ever want/need to divide by zero? If you have 10 slices of pie but nobody wants pie then you have no need of doing any division.

That's great about the common misconception that dividing something by zero yields infinity. It means that if you give your pie to nobody, you have an infinite amount of pie!

It can pop up in some situations. A singularity has a volume of zero, does it not? Now say you want to calculate the energy density there.

A physical singularity is defined by the invalidity of classical (or even general relativist) physics though, so I don't know if the common formula for energy density even applies (I think it's generally treated as if the energy density approaches infinity - it's a limit situation again in the end).

Afaik there's even a away to "get rid" of some infinities caused by singularities in cosmological models. Don't ask me how that works though.

Zero, imo, is more of a concept than a number since 'infinity' is a definite concept and definitely not a number. Since zero is conceptually the inverse of 'infinity', zero inherits the conceptual nature of infinity.

It always bugs me when people say a function "approaches infinity".

NO. IT CAN'T APPROACH A CONCEPT. IT INCREASES WITHOUT BOUNDS.

But interestingly I don't think saying a function "approaches zero" is logically incorrect.

I think it is wrong to think of zero as the inverse of infinity. Mathematically, they aren't even inverses of each other.

And your objection is a little out of place because the expression "approaches infinity" is defined as "grows beyond all bounds". I.e. a sequence x_n is said to approach infinity if for every z there is an n so that x_n > z. So you're basically complaining about treating things as the same that are by definition the same.

Disclaimer: post made while drunk, please be generous with any errors, will revisit tomorrow.

 

[Angst]

Drunk CFCing is beast. ^^

 

[Gatsby]

I'm a little confused by this because if I select zero as the numerator I could also conclude that 1=2=3=4, yet this is allowed.

0/0 = 0 as far as I know, because dividing nothing by nothing just leaves nothing.

Same for 0/1, 0/2 and so on, dividing nothing by anything (finite?) would just give nothing because there is nothing being divided in the first place. It's when you flip those sorts of equations around that things go weird, because when that is done you can derive a mathematical 'proof' that 1=2 etc.

In situations where zero is a numerator, you can still get results consistent with everyday physical reality. But when zero is a denominator, you can get results which clash with everyday physical reality. Considering that numeracy probably originated as an adaptive response a need to keep track of everyday items, and 0 itself was largely deliberately ignored throughout much of recorded history, it is not surprising that we still have this taboo.

 

[madviking]

0/0 = 0 as far as I know, because dividing nothing by nothing just leaves nothing.

No, you have conflicting definitions in 0/0.

x/0 is always infinite (for x != 0)
0/x is always zero (for x!= 0)

This is the same reason why 0^0 is indeterminate.

 

[BasketCase]

Why would you ever want/need to divide by zero? If you have 10 slices of pie but nobody wants pie then you have no need of doing any division.

Scientific curiosity. There's a piece of every scientist that wants to see if we really CAN rip a hole in reality using the LHC......

 

[Truronian]

If you want to be able to divide by zero you need to do so in an environment set up for it, for example the extended complex plane:

http://en.wikipedia.org/wiki/Extended_complex_plane

I don't remember much about the extended complex plane other than that it allows division by zero in some cases.

 

[Leoreth]

Scientific curiosity. There's a piece of every scientist that wants to see if we really CAN rip a hole in reality using the LHC......

That assumes that mathematics is a property of the world around us we can put to the test, when instead it is a logical model derived from certain axioms (which can then be used to model the world under certain circumstances). We wanted the numbers we usually do maths with to have certain properties under certain operations, and these wouldn't hold anymore if we allowed division by zero to have a result (respectively, assigning a result to division by zero would lead to a contradiction).