defending /0.

[studiot]

Division by zero????

 

Responses, they do not mean the same thing.

 

1) It can't be done.

 

2) It is forbidden.

 

3) It is undefined.

 

So what should the questioner think?

Should he stamp his foot like a petulent schoolboy?

Should he ask why can't it be done or is it not defined?

 

1) Perhaps it is like finding a helium atom with twentry protons, that can't be done because atom with twenty protons would be something else, not helium.

 

2) If it is necessary to forbid it it implies that it is possible, but some authority does not want the procedure performed.

 

3) Why is it undefined? Well perhaps because the responder does not know the answer, or perhaps because he knows a great many things it is not.

[ydoaPs]

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.

Yet infinity is not a number. This should give the OP pause. What operation can you do with two numbers to yield something which is not a number?

[John]

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?

An immediate consequence of this definition is the construction of the rational numbers breaks. This is because part of said construction is defining an equivalence relation such that given integers a, b, c and d, with b and d not equal to zero, the two ordered pairs (a,b) and (c,d) are related iff ad = bc, i.e. two rational numbers a/b and c/d are equivalent (read: equal) if and only if ad = bc. But, relaxing the restriction on b and d and using your definition here, we have

 

[math]\frac{7}{0} = 7 = \frac{7}{1}[/math]

 

which is problematic since [math]7 \times 1 \neq 7 \times 0[/math]. We also have that for two distinct integers m and n, m = n, which is a contradictory result. This is because [math]m \times 0 = n \times 0[/math], so [math]\frac{m}{0} = \frac{n}{0} \implies m = n[/math].

 

Of course, we could just find some other way to construct the rationals, but certain other problems arise.

 

For instance, given [math]\frac{7}{0} = 7[/math], we have

 

[math]\begin{array}{rcl}7 \times \frac{1}{0} & = & 7 \times \frac{1}{1} \\ \frac{1}{7} \times 7 \times \frac{1}{0} & = & \frac{1}{7} \times 7 \times \frac{1}{1} \\ \frac{1}{0} & = & \frac{1}{1},\end{array}[/math]

 

i.e. 1/0 is just another way of writing 1. This begs the question of why we need the definition in the first place, since 1/1 works perfectly well. But even ignoring that, it should be clear that if a/b = c/d, then b/a = d/c, so since we have 1/0 = 1/1, we also have 0/1 = 1/1, which is certainly not true.

 

Edit: Another problem with [math]\frac{7}{0} = 7[/math] is that we could just as easily say the following:

 

[math]\begin{array}{rcl}7 \times \frac{1}{0} & = & 7 \times \frac{1}{1} \\ 7 \times \frac{1}{0} \times \frac{1}{7} & = & 7 \times \frac{1}{1} \times \frac{1}{7} \\ 7 \times \frac{1}{0} & = & 7 \times \frac{1}{7} \\ 7 & = & 1,\end{array}[/math]

 

which is another contradiction.

Edited November 30, 2013 by John

[Didymus]

Division by zero????

 

Responses, they do not mean the same thing.

 

1) It can't be done.

 

2) It is forbidden.

 

3) It is undefined.

 

So what should the questioner think?

Should he stamp his foot like a petulent schoolboy?

Should he ask why can't it be done or is it not defined?

 

1) Perhaps it is like finding a helium atom with twentry protons, that can't be done because atom with twenty protons would be something else, not helium.

 

2) If it is necessary to forbid it it implies that it is possible, but some authority does not want the procedure performed.

 

3) Why is it undefined? Well perhaps because the responder does not know the answer, or perhaps because he knows a great many things it is not.

 

 

I'd refer you to the first page... because it's most certainly not impossible. It's, in fact, quite easy to grasp. However, some people ignore the direct example of it being done because SOME ways of checking the answer fail. Because graphs can be manipulated to the point where the question no longer makes sense, they ignore the example of it being solved.

 

Yes, your text book will say it can't be done. It's wrong.

[studiot]

 

Responses, they do not mean the same thing.

 

Please read my post again properly.

 

Then perhaps you will be in a position to comment on what I did say, rather than what I did not say.

Edited November 30, 2013 by studiot

[Dekan]

 

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.

 

Thanks imatfaal, I fully take your point about Maths being a set of rules. And in a way, that's what I'm trying to get at.

 

Maths has been invented by humans. It's a strictly human thing - animals don't use Maths. Nor do other bodies in the Universe, such as

atoms, grains of dust, moons, planets, stars and galaxies.

 

The Moon doesn't do Maths to calculate its movements, so as to stay in orbit round the Earth. The Moon isn't conscious.

 

Only conscious humans do Maths. And as you say - in Maths we set the rules! So suppose we find some particular rule causing a problem. Such as division by 0. Why can't we just change the rule?

 

I mean, surely the Universe doesn't care about human inventions like the symbol "0".

 

If we tweak our definition of 7/0, will that make the skies fall?

[studiot]

 

If we tweak our definition of 7/0, will that make the skies fall?

 

 

You are beginiing to appreciate no3 on my list (post26)

[Dekan]

(In reply to #28)

 

Thanks John, I've studied your post carefully, and it certainly shows that some seeming paradoxes can arise.

However, I wonder whether these are real paradoxes, Mightn't they result from our invented terms - "Multiplication" and "Division",

 

What does " Multiplication" mean? Isn't it just repeated addition. So "7 multiplied by 3" is the same as "7 + 7 + 7 =21"

Similarly, " Division" is just repeated subtraction. So "7 divided by 3" is the same as " 7 - 3 - 3 = 1, remainder 1"

 

So, suppose we accept that so-called "Division" is just a shorthand term for repeated subtraction.

 

What follows? Surely this:

 

7 - 0 - 0 - 0 - 0 - 0..... ( ad infinitum) = 7. Does that cause a paradox - apparently not.

 

Then why should the shorthand equivalent : 7 / 0 = 7, cause a paradox?

Edited November 30, 2013 by Dekan

[John]

If we're defining the operations in those terms, then the following is more accurate:

 

Multiplication tells us the result of summing some quantity some number of times. For example, 7*3 is the result of adding three sevens (or seven threes) together, and this equals 21.

 

Division tells us how many times we need to subtract a quantity from another quantity to reach zero. For example, 7/3 is 2 1/3 (or, if you like, 2.333...) because after subtracting two threes from seven, we must then subtract one (which is a third of three) to reach zero.

 

Now consider 7/0. This operation should tell us how many zeroes we must subtract from seven to reach zero. Now, whether we subtract three zeroes, a trillion zeroes, or a googolplex of zeroes, the result will still be seven. In fact, any real number of zeroes subtracted from seven will simply result in seven. Thus, there is no real number solution to 7/0. We might say the result must exceed any real number, i.e. 7/0 must be infinity. But as we've seen before, this leads to problems too.

Edited November 30, 2013 by John

[Endercreeper01]

x/0 is undefined because if you take the limit as n approaches 0 of x/n, then from the positive side, it approaches infinity and it approaches negative infinity from the negative side, so therefore, x/0 is undefined

[Didymus]

 

You are beginiing to appreciate no3 on my list (post26)

 

 

So if all these problems are based on the fact that it's undefined.... Define it. John's post on #12 said "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 be an infinitesimal, then and , but division by zero is still undefined."

 

... sounds like here is just closer to 0 than he thinks. The idea of it being "not 0" inherently makes it finite, therefore x/ can not be infinite because it's dividing by a finite (although undefinable) number.

Here's how I see it:

Priority 1: 0/x=0.

Priority 2: x/0=an infinite multiplication of x. Thus, x/0=[math]\infty[/math] and -x/0=[math]-\infty[/math] and 0/0 falls under priority 1 because it's an infinite multiplication of 0, which is still 0.

This also solves the 0/0=1 problem simply with a bit of priority.

The problem of it not being able to be defined because it's undefined is no longer a problem... because it's defined to simply communicate the idea that an infinite number of points that do not take up space can fit into a finite space (or infinite points on a line segment).

The idea of "video game skill does X damage divided evenly among targets in the area" hitting an area with no targets would still result in 0 damage being dealt because x/0 would pump out [math]\infty[/math]... but this would be multiplied by 0 because no targets were hit by this infinite damage. Thus, any similar examples of exceptions would be (x/0)0. Even when multiplied by an infinite number, 0 takes priority, balancing the problem and negating that exception to division by 0.

Lastly, it would be accepted that a formula can't be balanced by multiplying or dividing by infinite numbers such as 0... explaining why A times 0 and B times 0 doesn't mean A=B.

So, there ya go. All problems addressed with the sole exception of "it's not in my math book."


If anyone has any difference objections, I'd love to see them... but all objections that have been presented, so far, are debunked above.

[studiot]

 

studiot, on 30 Nov 2013 - 3:44 PM, said:

 

You are beginiing to appreciate no3 on my list (post26)

 

 

So if all these problems are based on the fact that it's undefined.... Define it.

 

Didymus, My response you quoted was addressed to Dekan, but at least you are beginning to take note of my comments.

 

I offered you three common responses to the issue of division by zero and told you that they are all used to avoid the issue but mean different things.

 

I further offered you a few short comments about these different meanings.

 

John has wasted a great deal of effort offering much longer comments about the same things.

 

Yes, the key statement is 'It is undefined' and that after three thousand years of very clever men thinking about it.

 

I also said that today's inquirers after the truth would ask 'why is it undefined, after all that clever thinking?'

 

Discussing that question could lead you to fruitful further knowledge.

 

It is up to you, Do you want to shout at me from the rooftops? In which case I am not interested.

 

Or do you want to explore the how and why we are at the current situation in mathematics and any available resolutions ?

[Endercreeper01]

So if all these problems are based on the fact that it's undefined.... Define it. John's post on #12 said "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 be an infinitesimal, then and , but division by zero is still undefined."

 

... sounds like here is just closer to 0 than he thinks. The idea of it being "not 0" inherently makes it finite, therefore x/ can not be infinite because it's dividing by a finite (although undefinable) number.

 

Here's how I see it:

 

Priority 1: 0/x=0.

Priority 2: x/0=an infinite multiplication of x. Thus, x/0=[math]\infty[/math] and -x/0=[math]-\infty[/math] and 0/0 falls under priority 1 because it's an infinite multiplication of 0, which is still 0.

 

This also solves the 0/0=1 problem simply with a bit of priority.

 

 

The problem of it not being able to be defined because it's undefined is no longer a problem... because it's defined to simply communicate the idea that an infinite number of points that do not take up space can fit into a finite space (or infinite points on a line segment).

 

The idea of "video game skill does X damage divided evenly among targets in the area" hitting an area with no targets would still result in 0 damage being dealt because x/0 would pump out [math]\infty[/math]... but this would be multiplied by 0 because no targets were hit by this infinite damage. Thus, any similar examples of exceptions would be (x/0)0. Even when multiplied by an infinite number, 0 takes priority, balancing the problem and negating that exception to division by 0.

 

Lastly, it would be accepted that a formula can't be balanced by multiplying or dividing by infinite numbers such as 0... explaining why A times 0 and B times 0 doesn't mean A=B.

 

So, there ya go. All problems addressed with the sole exception of "it's not in my math book."

 

 

If anyone has any difference objections, I'd love to see them... but all objections that have been presented, so far, are debunked above.

The difference is that even though 1/ is infinite, 1/- is negative infinity. You can have a positive and a negative . cannot be 0 because 0 is neither negative or positive.

[John]

... sounds like here is just closer to 0 than he thinks. The idea of it being "not 0" inherently makes it finite, therefore x/ can not be infinite because it's dividing by a finite (although undefinable) number.

It's not a matter of what I think. The hyperreal number system is used as part of nonstandard analysis. It's simply an example of an area where "infinitely small" makes sense, whereas it doesn't for the real numbers.

 

The problem of it not being able to be defined because it's undefined is no longer a problem... because it's defined to simply communicate the idea that an infinite number of points that do not take up space can fit into a finite space (or infinite points on a line segment).

 

[...]

 

So, there ya go. All problems addressed with the sole exception of "it's not in my math book."

 

 

If anyone has any difference objections, I'd love to see them... but all objections that have been presented, so far, are debunked above.

Division by zero is left undefined because definitions lead to logical absurdity. "Not being able to be defined because it's undefined" is nonsense, and "it's not in my math book" doesn't matter and isn't an argument anyone's made.

 

And you haven't debunked anything.

 

 

John has wasted a great deal of effort

Maybe.

Edited December 1, 2013 by John