Why can't you divide by zero?

[Angst]

The answer I've gotten has usually been "you just can't" or "you can" (depending on whether people know their math.)

I have this subjective nonmathematician's gut feeling (I'm a freaking musician) that 1/0 equals a number with an infinite value ("Infinity" isn't a number is it? I think I'm told it's just a property) since the closer the demoninator gets to 0, the larger the number becomes; why is it not so that the quotient becomes immeasurely large?

Has there been any attempts to divide by zero? When was it established that it was impossible?

 

[Gatsby]

Mathematics is a a human-created modelling tool for describing the world. Like all human attempts to conceptually model reality it is not perfect, and dividing 1 by 0 causes the whole logical structure of mathematics to collapse (e.g. dividing 1 by 0 can lead to the conclusion that 1=2=3=4 etc). Your gut feeling might be true on some fundamental metaphysical level - I for one suspect that it is - but for practical and technical purposes it's more trouble to divide by zero than it's worth.

 

[Gatsby]

Double post

 

[Angst]

That's a pretty throughout and sensible answer. Thanks man!

 

[rugbyLEAGUEfan]

(e.g. dividing 1 by 0 can lead to the conclusion that 1=2=3=4 etc).

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.

 

[Leoreth]

The best way to understand this is to remember that

a) every basic mathematical operation (addition, subtraction, multiplication, division) has one and only one unambiguous result
b) division is the inversion of multiplication

So if you write 6/3 you basically write "with what number do I have to multiply 3 to get 6?" and the answer is of course 2. 6/3 = 2 and 3*2 = 6.

Now if you write 6/0 the question you ask is "with what number do I have to multiply 0 to get 6?". And there is no answer to that. There isn't any number x so that 0*x = 6. Therefore 6/0 is undefined (which is a better way to say it than to say it's "forbidden" or something like that).

 

[rugbyLEAGUEfan]

That's crystal. Nice one

 

[ArneHD]

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.

 

[classical_hero]

 

[MagisterCultuum]

It sounds like that reasoning would also mean that you cannot take the roots of any even power though.

 

[Flying Pig]

That's not a 'basic mathematical operation', though, because it's not a straight division, addition, multiplication or subtraction.

 

[LegionSteve]

I like Leoreth's explanation

As a software developer though, all I need to know is that dividing by zero results in an infinte amount of Derp.

 

[Josu]

The best way to understand this is to remember that

a) every basic mathematical operation (addition, subtraction, multiplication, division) has one and only one unambiguous result
b) division is the inversion of multiplication

So if you write 6/3 you basically write "with what number do I have to multiply 3 to get 6?" and the answer is of course 2. 6/3 = 2 and 3*2 = 6.

Now if you write 6/0 the question you ask is "with what number do I have to multiply 0 to get 6?". And there is no answer to that. There isn't any number x so that 0*x = 6. Therefore 6/0 is undefined (which is a better way to say it than to say it's "forbidden" or something like that).

 

[Leoreth]

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.

I hope your math professor said that a specific context, because otherwise it doesn't make sense.

What you might be referring to are limits. For example, if x -> 0 then 1/x -> infty (to avoid the problems with writing math expressions: this means if x approaches 0, 1/x approaches infinity (i.e. grows without any bounds). "Approaches" is important because x never becomes 0. So limits are not to be confused with actual division.

Depending on from which "direction" (positive or negative) x approaches zero you would end up with positive and negative infinity.

It sounds like that reasoning would also mean that you cannot take the roots of any even power though.

Took me a while to get what you mean, but no.

The mathematical operation "square root" is defined as yielding only positive results: sqrt(4) = 2. But sqrt(4) = -2 is false, even though (-2)² = 4. Therefore, taking the square root is not the inverse of squaring.

What you're referring to is solving a quadratic equation. But x² = 4 isn't solved by taking sqrt(4). x² = 4 has two solutions, sqrt(4) and -sqrt(4).

 

[Verarde]

Leoreth's explanation is excellent.

Personally, I always thought of it as "infinity" because 0 would go into X an infinite amount of times.

 

[luiz]

I have this subjective nonmathematician's gut feeling (I'm a freaking musician) that 1/0 equals a number with an infinite value ("Infinity" isn't a number is it? I think I'm told it's just a property) since the closer the demoninator gets to 0, the larger the number becomes; why is it not so that the quotient becomes immeasurely large?

You'll never get to zero, but your gut feeling is correct in the sense that the limit of 1/x as x approaches 0+ is indeed infinity.

 

[Lord Gay]

Why can't you divide by zero?

I got nothing.

 

[GoodGame]

I see it like this: arithmetic is discrete operations (including division) on discrete number sets.
Dividing by zero leads to a violation of the definition of arithmetic.
Dividing by zero turns a discrete arithmetic operation on a discrete number set into an infinite set (see quote), violating the definition.

Since it's not calculus, there's no asymptote, so there's no "approaching" infinity. It's just illogical. As in an illogical error that calculators and computers don't like.

How do we define division? The ratio r of two numbers a and b:
r=a/b

is that number r that satisfies
a=r*b.

Well, if b=0, i.e., we are trying to divide by zero, we have to find a number r such that
r*0=a. (1)
But
r*0=0
for all numbers r, and so unless a=0 there is no solution of equation (1).

Now you could say that r=infinity satisfies (1). That's a common way of putting things, but what's infinity? It is not a number! Why not? Because if we treated it like a number we'd run into contradictions. Ask for example what we obtain when adding a number to infinity. The common perception is that infinity plus any number is still infinity. If that's so, then
infinity = infinity+1 = infinity + 2

which would imply that 1 equals 2 if infinity was a number. That in turn would imply that all integers are equal, for example, and our whole number system would collapse.

http://www.math.utah.edu/~pa/math/0by0.html

 

[BasketCase]

Division is the answer to "how many of X will fit into one Y". When you divide fifteen by three, the answer (five) says that it takes five threes to make fifteen.

Why can't you divide fifteen by zero? Because there is no way to make a fifteen out of zeros. It's impossible. No matter how many zeros you pile together, you can't ever get them to add up to fifteen.

 

[Murky]

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.