# defending /0.

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I want to know why people have such a hard time dividing by 0.

Basic concept: in A/B=C, when B gets larger, C gets smaller. When B gets smaller, C gets larger. So A/0.1=10A and A/0.0001=10000A.... Obviously when B becomes infinitely small (0), C is an infinite multiple of A.

Logical proof: if I want to save $100 and I save$0/month, how long until I reach my goal? 100/0=inf, therefore an infinite amount of time will pass and I will never reach my goal.

Objection A: But, if A is negative, you'll net "negative infinity".

Answer: yea. What's wrong with that? -A/.0001=(-10000A). Still works.

objection B: if a/0=inf and b/0=inf but a and b are different, you can't have that.

Answer: it works multiplying a and b by 0. Why not dividing?

Objection C: it doesn't work functionally/can't make it into a word problem.

Answer: how many points are there on a line segment? Infinite because points have a length of 0. Thus length A/length b (0)=inf.

How many objects can I fit in this box? The smaller the item, the more objects. If the item is infinitely small, it's just a 3d line segment problem. Volume A/0= an infinite amount of spaces that take up no space can fit in a box of any area.

So where is the problem other than 0/0? Even that isn't a problem because there are 3 conflicting rules (0/x=0, x/x=1, x/0=inf) ... And even that is solved by picking rule takes priority.

So.... What problem is there with dividing by 0?

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There isn't a problem. Some people just have a hard time understanding indeterminate forms and how to evaluate them with limits. They have a need to fix something that isn't broken simply because they don't understand advanced mathematics. The answer is that you can't divide by zero.

Edited by Daedalus
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I see it as the indeterminate denominator as what causes the problem.

0 / (X != 0) = 0

I still managed to develop a system of math with it(black hole equations) but there were unresolvable issues past the very basics. Something to do if you're bored if nothing else.

Edited by Endy0816
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So, daed... If you can't solve a problem with one method, but can solve the problem with other methods... Why blame the number just because one method can't handle the number?

0, 1 and infinity break a lot of rules. Doesn't mean they aren't useful.

The thing about indeterminate forms is that there are multiple correct answers depending on the context of the question.

The fact that multiple answers may be correct doesn't mean neither are correct.

I.e. just because sqrt 4 can either be 2 or -2 doesn't mean either one is wrong. Depends on the context. Like the fact that one word may have two meanings.

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So, daed... If you can't solve a problem with one method, but can solve the problem with other methods... Why blame the number just because one method can't handle the number?

I don't blame any number, and I don't have a problem with multiple answers being correct lol. These type of problems are evaluated using limits and calculus, and there can either be a limit or not. It's as simple as that.

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Yet "the answer is that you can't divide by 0".

... Yet there are very basic calculations that rely on dividing by 0. I have 3 examples in the original post.

Without dividing by 0, how would you express the number of points on a given line segment?

Assuming points with no length:

Length/0=infinite points.

How is that not dividing by 0?

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Infinity is not a number, and cannot be put into standard arithmetic expressions as you're trying to do here. Doing so leads to logical absurdity. For example, suppose $\frac{1}{0} = \infty$. Then we have $1 = 0 \times \infty$. We can then multiply both sides of the equation by $n \neq 1$, yielding $n \times 1 = n \times 0 \times \infty$. Simplifying, we arrive at $n = 0 \times \infty$, which by the transitivity of equality yields $n = 1$. But this is a contradiction, since we assumed $n \neq 1$. Of course, we could say that while 1/0 is defined to be infinity, we can't manipulate that equation as I've done here. But then I'd ask what the point is of making the definition in the first place.

There are some contexts in which division by zero is defined, but these don't really conform to the usual notions we have about numbers. Some examples you might enjoy are the real projective line and wheel theory. Note that these structures have some odd properties (for instance, in the article about the real projective line, the author notes that ordering doesn't really work as we'd expect, so we end up with 4 > 3 and 3 > 4 both being true).

As for the line segment example, one option is this: if we consider the length of the segment to be an interval on the real number line, then we can form a bijection from that interval to the real numbers, thus showing us that there are infinitely many points on the line segment (in fact, there are as many points on the line segment as there are real numbers--ain't infinity grand?).

Edited by John
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K, John. Let's extend that.

$A \times B = C$ therefore $\frac {C}{B}=A$

So, if we can't divide by 0 on the basis of $1 = 0 \times \infty$ being wrong... then why can we multiply by 0?

So, if $1 \times 0 = 0$ then $\frac {0}{1}=0$, which we agree is legitimate. But also $\frac {0}{0}=1$... which obviously doesn't work out. Yet $A \times B = C$ therefore $\frac {C}{B}=A$

So can we not Multiply by 0's anymore? Obviously we can, even though sometimes it doesn't check out.

Being difficult to work backwards doesn't mean it's impossible to work forwards. Specifically, how is the above example of dividing Length A by Length B to get the number of B-lengthed segments not a legitimate division? We know there are infinite points in a given line segment because each point doesn't have a length... thus Length A/0=$\infty$. Is that not dividing by 0? Or do you believe there are a finite number of lengthless points on a line segment?

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K, John. Let's extend that.

$A \times B = C$ therefore $\frac {C}{B}=A$

So, if we can't divide by 0 on the basis of $1 = 0 \times \infty$ being wrong... then why can we multiply by 0?

So, if $1 \times 0 = 0$ then $\frac {0}{1}=0$, which we agree is legitimate. But also $\frac {0}{0}=1$... which obviously doesn't work out. Yet $A \times B = C$ therefore $\frac {C}{B}=A$

This rule only holds for nonzero B. To elaborate, the full series of steps is as follows:

$\begin{array}{rcl} A \times B & = & C \\ A \times B \times \frac{1}{B} & = & C \times \frac{1}{B} \\ A \times 1 & = & C \times \frac{1}{B} \\ A & = & \frac{C}{B}\end{array}$

If B = 0, then this breaks down, since 1/B is undefined.

So can we not Multiply by 0's anymore? Obviously we can, even though sometimes it doesn't check out.

Multiplication of any real number by zero always works out.

Being difficult to work backwards doesn't mean it's impossible to work forwards. Specifically, how is the above example of dividing Length A by Length B to get the number of B-lengthed segments not a legitimate division? We know there are infinite points in a given line segment because each point doesn't have a length... thus Length A/0=$\infty$. Is that not dividing by 0? Or do you believe there are a finite number of lengthless points on a line segment?

I don't think a line is defined as "an infinite collection of points" in geometry. Rather, it's the path of a point moving in one direction, if it's even defined based on a point at all--it may be taken as a primitive object. It is true that in analytic geometry, we might define a line as the collection of points with coordinates satisfying a given linear equation, but even this definition isn't based on division by zero. Your last question was answered above, when I explained one way in which we might show that a line segment contains infinitely many points.

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This rule only holds for nonzero B.

So, perhaps, it would be appropriate that the idea of checking $\frac {N}{0}= \infty$ with $\frac {\infty}{0}=N$ simply isn't a rule that holds true any more than checking $N \times 0 = 0$ with $\frac {0}{0}=N$.

It's just Okie dokie that one simply isn't a rule that holds true... yet because the other equally doesn't hold true, we just give up on the whole thing?

If points on a line don't float your boat, go to the three dimensional version: A Box has a finite area. How many infinitely small objects can fit inside that box? Area A/Area B=number of objects that fit in the box. X/0=$\infty$

How is that not dividing by 0?

(and no, I'm not ignoring your point about number lines. Comparing infinitely small points with the infinite number of numbers between numbers is a perfectly valid illustration. IMO, a better illustration of dividing by 0 being something other than infinity actually came from a video game I was playing called Heroes of Newerth. A character's ability targets a certain area and does, say, 500 damage divided equally among the number of targets he hits. One target takes full damage. if two targets are in the area, each takes half... but if he hits zero targets, how much damage did he do? 0, obviously. In this case X/0=0. Or, less nerdy: If I make a fruit cake and divide it equally among the people who want a slice... and 0 people want any... how much cake does each person get? 1/0=0 in this situation. However, this is part of the language of math. In any language, sometimes the same thing can have different meanings with a different context. From what I see, there are two perfectly legitimate ways to illustrate X/0 and two completely legitimate answers. The fact that there are two answers depending on the context of the problem doesn't mean the problem is invalid.

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So, perhaps, it would be appropriate that the idea of checking $\frac {N}{0}= \infty$ with $\frac {\infty}{0}=N$ simply isn't a rule that holds true any more than checking $N \times 0 = 0$ with $\frac {0}{0}=N$.

I'm assuming your second equation is supposed to be $N = 0 \times \infty$, but regardless, this is true. I'm not sure what your point is with the rest. The property that $\forall n \in \mathbb{R}, n \times 0 = 0$ is a result of the definition of multiplication for real numbers. While we can use various examples to illustrate why the definition makes sense, we don't need to check it.

It's just Okie dokie that one simply isn't a rule that holds true... yet because the other equally doesn't hold true, we just give up on the whole thing?

No. I mentioned in a previous post that while dividing by zero leads to illogical conclusions when dealing with the real numbers, there are other contexts in which division by zero is defined--but in those contexts, other properties of numbers and division that we take for granted are also slightly altered.

If points on a line don't float your boat, go to the three dimensional version: A Box has a finite area. How many infinitely small objects can fit inside that box? Area A/Area B=number of objects that fit in the box. X/0=$\infty$

How is that not dividing by 0?

When dealing with the real numbers, "infinitely small" doesn't make sense. There is a number system called the "hyperreal numbers," in which infinitesimals (these are quantities smaller than any nonzero number, but not equal to zero) and infinity exist. In this system, if we let $\epsilon$ be an infinitesimal, then $\frac{1}{\epsilon} = \infty$ and $\frac{1}{\infty} = \epsilon$, but division by zero is still undefined.

(and no, I'm not ignoring your point about number lines. Comparing infinitely small points with the infinite number of numbers between numbers is a perfectly valid illustration. IMO, a better illustration of dividing by 0 being something other than infinity actually came from a video game I was playing called Heroes of Newerth. A character's ability targets a certain area and does, say, 500 damage divided equally among the number of targets he hits. One target takes full damage. if two targets are in the area, each takes half... but if he hits zero targets, how much damage did he do? 0, obviously. In this case X/0=0. Or, less nerdy: If I make a fruit cake and divide it equally among the people who want a slice... and 0 people want any... how much cake does each person get? 1/0=0 in this situation. However, this is part of the language of math. In any language, sometimes the same thing can have different meanings with a different context. From what I see, there are two perfectly legitimate ways to illustrate X/0 and two completely legitimate answers. The fact that there are two answers depending on the context of the problem doesn't mean the problem is invalid.

The examples here don't really hold water. In the video game, if there are no enemies around, then the attack clearly misses. But as it happens, we can just as easily say the full amount of damage was applied to the zero enemies, i.e. $\frac{500}{0} = 500$, which doesn't make much sense. The same applies to the cake example. You could say you gave zero cake, the entire cake, or any portion of the cake, to zero people.

The fact that division by zero can yield multiple answers is a problem, though not necessarily a fatal flaw. The bigger issue is the fact that any definition either leads to contradictions, if the usual rules of arithmetic are applied, or doesn't seem to be any more useful than just leaving it undefined in the first place, if the usual rules of arithmetic don't apply.

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No.

if$A \times B = C$, then $\frac {C}{B}=A$
With me so far?
for example: $3 \times 5 = 15$ means $\frac {15}{3}=5$ and $\frac {15}{5}=3$
if $N \times 0 = 0$, then $\frac {0}{0}=N$
If dividing by 0 is a problem because when you check the math you get wonky answers... then multiplying by 0 is just as much of a problem.
Fact is, 0 is tricky. That's why it's interesting. And that's why it's banned from so many formula.

No. I mentioned in a previous post that while dividing by zero leads to illogical conclusions when dealing with the real numbers, there are other contexts in which division by zero is defined--but in those contexts, other properties of numbers and division that we take for granted are also slightly altered.

There are certainly ways to get illogical conclusions. But there are also ways to get logical ones. Why ignore the logical conclusions we CAN obtain for fear of the illogical ones we might obtain if we don't use some common sense?

That seems to be the base of the issue. I work with a lot of engineers. And by golly, the more educated they are, the farther detached some tend to be from common sense.

... this is why I'm regularly handed blue-prints telling me to install bushings 40 feet outside of a part... because they're so tuned into hitting the bullseye perfectly that they miss the barn the target's attached to (figuratively).

Edited by Didymus
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..
if $N \times 0 = 0$, then $\frac {0}{0}=N$
If dividing by 0 is a problem because when you check the math you get wonky answers... then multiplying by 0 is just as much of a problem.
...

The problem still arises because you are dividing by zero which is undefined.

N*0 is defined. You cannot manipulate that to get to a defined division by zero because that action is specifically undefined - your "then" is estopped from happening because it is specifically forbidden. Nor can you use a division by zero to call into question the validity of the rule regarding multiplication. Multiplication and division act as inverses to each - except for zero because of the very fact that division by zero is undefined.

Maths is about as "natural" as you can get - but it is still built from a series of axiomata and logical statements that must follow from these. In the general everyday system we use division by zero is not defined.

Oh yes Zero is tricky. That's why some almost universal rules do not apply to zero

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Because 0's a boss and it don't take no (!%) from no one.

I still say "you're not allowed to because it's undefined" is a rather ridiculous reason to reject a rather clear and logical definition.

Also, I'm still feeding my crippling insomnia... so, more nitpicking:

I very well may be misunderstanding something, but:

this was posted earlier (I'm having issues getting the formula in a quote box:

----------------------------fake quote box---------------------------- . Then we have . We can then multiply both sides of the equation by , yielding . Simplifying, we arrive at , which by the transitivity of equality yields . But this is a contradiction, since we assumed ----------------------------/fake quote box----------------------------

How are you going from to ? You can drop the 1 on the left side because n x 1=n... but if , you can't just drop the n on the right side of the equation.

Edited by Didymus
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I very well may be misunderstanding something, but:

this was posted earlier (I'm having issues getting the formula in a quote box:

----------------------------fake quote box---------------------------- . Then we have . We can then multiply both sides of the equation by , yielding . Simplifying, we arrive at , which by the transitivity of equality yields . But this is a contradiction, since we assumed ----------------------------/fake quote box----------------------------

How are you going from to ? You can drop the 1 on the left side because n x 1=n... but if , you can't just drop the n on the right side of the equation.

n * 0 * infinity = (n * 0) * infinity = 0 * infinity

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Which means you're checking the formula by misusing it. Finding a place to put an n next to a 0 so you can cancel it out on one side but keep it on the other. Very dishonest, sir.

That's like saying 1+1=3

Because 0 x (1+1)=0 x 3
thus if 0=0 then 1+1=3.

Wrong. You're abusing 0. Stop it.

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The properties of the real numbers under addition and multiplication are such that it is perfectly valid to multiply both sides of an equation by a value. The positioning of n doesn't matter, as multiplication of real numbers is commutative. My line of reasoning is correct if we're treating division by zero as a valid operation subject to the usual rules of arithmetic (aside from, of course, the one that says division by zero is undefined) and infinity as a quantity that can be placed in an equation. There is nothing dishonest about it.

And that's the whole point: simply defining division by zero to be infinity leads to a logical contradiction. Defining it to be some other value will also lead to a contradiction. And thus, whatever use you might find for it, at the end of the day, such a definition would yield inconsistent mathematics, which means (if it were taken to be correct) it could be used to prove anything at all. While Gödel showed us that we cannot use mathematical reasoning to prove mathematics is consistent (unless it isn't), we do strive to avoid inconsistencies where we can.

Edited by John
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Does zero actually exist?? The concept of diving something by something that doesnt exist is nonsensicle

Well, as an abstract concept it does exist. Without it there would be difficulty describing many numbers.

Without the existence of 0 there would be many problems in mathematics.

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Let's stay on topic please - this is Didymus' rant thread against the mistreatment of zero.

Which means you're checking the formula by misusing it. Finding a place to put an n next to a 0 so you can cancel it out on one side but keep it on the other. Very dishonest, sir.

That's like saying 1+1=3

Because 0 x (1+1)=0 x 3
thus if 0=0 then 1+1=3.

Wrong. You're abusing 0. Stop it.

Not at all dishonest. Multiplication by zero is well defined - and that's what was used. What you are using immediately above is division by zero; and it results in a contradiction - which is exactly the reason why division by zero is not defined and not allowed! You have just shown why division by zero is not included in our normal mathematical system

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Mightn't all the above discussions, come from our having created the deceptive symbol "0".

This symbol is quite full-bodied. It looks similar to a real number such as "8", or "6". And this similarity seduces us into thinking that "0" is also a number. But really it's just a mark of separation - like a decimal point.

The "0" could be replaced by a decimal point, or dot. So instead of writing "2001" we could put "2..1"

Wouldn't that give the same information - without making people think that a dot " ." is a number?

Edited by Dekan
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Mightn't all the above discussions, come from our having created the deceptive symbol "0".

This symbol is quite full-bodied. It looks similar to a real number such as "8", or "6". And this similarity seduces us into thinking that "0" is also a number. But really it's just a mark of separation - like a decimal point.

The "0" could be replaced by a decimal point, or dot. So instead of writing "2001" we could put "2..1"

Wouldn't that give the same information - without making people think that a dot " ." is a number?

Initially perhaps - but 3*(spaceholder) = what? Because we must be able to say that 3*(0)=0. And with plus, and as exponent etc. and you end up with all the rules we have for zero just with a different shape.

Zero is not the same as other numbers - that is the root of Didymus' argument; he seems to say that we must treat it the same, I and other say that, within standard mathematics as we have defined it, we cannot treat it the same. If you want to treat it the same in every way, or pretend that it is still merely a placeholder for trade sums then use your own brand of mathematics; but do not insist that the amazing edifice of logic which is everyday mathematics must be changed to accommodate your predilections.

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I think the whole trouble with "0", is this:

The way it's written, makes it look similar to a number "6" or "8"

So "0" looks like a number. But it isn't a number - it's an absence of number. It means, or should mean, "nothing ",

Therefore " Divide 7 by 0" really means "Divide 7 by nothing"

And isn't that same as Don't divide 7 by anything - ie leave it unchanged.

So where's the difficulty? If you divide a number by 0, you don't divide it at all. The number stays unchanged.

Doesn't that make sense?

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I think the whole trouble with "0", is this:

The way it's written, makes it look similar to a number "6" or "8"

So "0" looks like a number. But it isn't a number - it's an absence of number. It means, or should mean, "nothing ",

Therefore " Divide 7 by 0" really means "Divide 7 by nothing"

And isn't that same as Don't divide 7 by anything - ie leave it unchanged.

So where's the difficulty? If you divide a number by 0, you don't divide it at all. The number stays unchanged.

Doesn't that make sense?

I can understand what you are saying - but it is wrong. Maths is a set of rules - play by them or don't, but don't expect other people to understand you or accept what you are saying if you play by your own set of rules.

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