Are you sure about that RB? I 'm pretty sure that 0/0 is Non Defined.

 

Are you saying that a math student is taught that 0/0 is non defined, or that 0/0 IS non defined? Because there is a difference.

Is "dog/dog" non defined? I'm happy to define division using "contained within", and everything by that definition contains one of itself. One zero contains one zero.

 

It seems to boil down to semantics.

Division of x (with x non-zero) by zero is said to be undefined.

Division of zero by zero is said to be indeterminate.

An argument for the latter seems to be that if you take x/x (x non-zero) the answer is always '1'. So one reasons that likewise, 0/0 should be '1'

But 0/x (with x non-zero) is always 0. So one alternatively (in relation to the previous statement) reasons that 0/0 should therefore be zero.

The result of 0/0 is therefore indeterminate.

 

Hello

I was told that division by zero is not defined and it's an illogical operation. Therefore, in my very humble opinion, x/0 and 0/0, both are undefined as long denominator has zero there. The case of 1/sin(x) as x approaches zero has nothing to with simple 1/0 situation. That's what I was told.

Thank you all of you for helping me with this.

Regards
PG

 

Actually it isn't illogical, division of any number (other than zero) with zero is infinity. You have now entered the realm of greater and lesser infinities, not to mention negative infinities.

Humans don't handle infinities well, we're not build mentally to calculate with it, but it can be done.

What is a greater and lesser infinity? Think of ∞, then think of ∞+1. The latter is greater than the first, even though they are both ∞.

I liked Carl Sagan's concept of numerous. This is not infinity, but it is a number so large we are incapable of expressing it. It describes our universe very well.

 

I thought that \(\lim_{x \rightarrow 0}\frac{1}{x}=\infty\), not \(\frac10=\infty\), which I thought was not permitted as an operation.

Was I wrong?

 

Just to point out, both infinity and zero are conceptual numbers that can't exist in the real world.

I thought that \(\lim_{x \rightarrow 0}\frac{1}{x}=\infty\), not \(\frac10=\infty\), which I thought was not permitted as an operation.

Was I wrong?

You're not wrong, it's not permitted. However much I think it should be.

 

0/0 is undefined.

x/0, where x≠0 is undefined, not ∞

0 is real and exists very often. I have 0 Ferrari's in my driveway.

∞ is mathematically real, but is not physically realizable.

 

...
An argument for the latter seems to be that if you take x/x (x non-zero) the answer is always '1'. So one reasons that likewise, 0/0 should be '1'

But 0/x (with x non-zero) is always 0. So one alternatively (in relation to the previous statement) reasons that 0/0 should therefore be zero.

The result of 0/0 is therefore indeterminate.

You left out; "x/0 is always infinity"
And obviously non of those conflicting 3 pure math rules help you solve the problem.

...
What is a greater and lesser infinity? Think of ∞, then think of ∞+1. The latter is greater than the first, even though they are both ∞.

Greater and lesser infinities is one of the best proofs that infinity can't exist (other than in conceptual math). ie; infinity can't ever be real.

...
x/0, where x≠0 is undefined, not ∞

Wow! I really didn't expect that from a math guy! Isn't infinity widely accepted as the reciprocal of zero? Please explain!

...
Just to point out, both infinity and zero are conceptual numbers that can't exist in the real world.

That's bunkum! If I have a real eggcarton and remove 12 real eggs from it I have a very real zero eggs in the carton. Zero is real, infinity is not (unless you can demonstrate inifity eggs in the eggcarton). Zero amps is also very real.

 

Wow! I really didn't expect that from a math guy! Isn't infinity widely accepted as the reciprocal of zero? Please explain!

Not in real analysis. You simply cannot divide by 0. There are just to many problems that occur if this is allowed. There are literally thousands of examples in math texts and on the web that offer cases where some paradox arises if you allow division by 0. Feel free to look them up.

Now, if you want to get into complex analysis, then we can talk. But should be comfortable with complex manifolds if you want to proceed with a discussion. Look up Riemann sphere.

 

I am not an advanced math type. What I've learned in calculus I've mostly forgotten through lack of use and age. Certain concepts I've imprinted on for life, such as x/0=∞ and 0/∞=0, it may not be completely true, but it works for me.

I've never had a problem with ∞ myself, it is just a way of expressing Sagan's numerous concept, a short hand if you will. Again, it may not be completely true but close enough. While no concepts come to mind I have used ∞ to solve real world problems, though not recently (Trig tangent comes to mind). It is a necessary concept.

I'll leave the fine distinctions to real mathematicians.

 

Division comes about because one would like an inverse operation to multiplication. Thus, if x*b = a, then one might want to be able to solve equations where e.g. x represents an unknown and a and b represent given numbers. By dividing both sides of the equation by b, we get x = a/b.

Now, let's make it an equation with actual numbers: 4x = 3. OK, that means x = 3/4. Thus, 3/4 is the number that, when multiplied by 4, gives us 3.

Now let b = 0. We have x*0 = 3. Blindly following our solving method, we get x = 3/0. What is 3/0? It's not a number -- and mathematicians have declared this operation illegal because we are looking for a number x that, when multiplied by 0, gives 3. Now do you see the problem? Zero is defined in the axioms of a field so that the product is zero when it is multiplied by any number, including zero. Thus, there is no solution for x. Because of this problem, mathematicians simply declared division by zero to be undefined.

Thus, it is incorrect to write x/0 = ∞ because division by zero is undefined. That means you cannot do it within the rules of the arithmetic system you're working in.

Of course, anyone who has taken a calculus class learns about limits and limiting processes. This leads to the statements like "the limit of 1/x as x approaches infinity is zero". This is a completely different ball of wax, as it's often envisioned as a sequence of numbers that keep getting smaller and smaller. Still, it's one intuitive thing that leads to the common shorthand of x/0 = ∞.

Now, in the practical world, division by zero is supported in a python library I use a lot, numpy. And it's pretty convenient, as you wind up with a number that gets displayed as Inf or -Inf. You can also used it in subsequent calculations and it behaves as you expect. However, if you multiply Inf by zero, you get NaN, which means "not a number". But an expression like (number)/Inf will give you zero. This gives you calculational conveniences because it can mean that your complicated calculation won't stop after an hour if a division by zero occurs. But, at the same time, complete weirdness can propagate through your calculation when such things happen, so one needs to be cautious when allowing such things.

 

Division comes about because one would like an inverse operation to multiplication. Thus, if x*b = a, then one might want to be able to solve equations where e.g. x represents an unknown and a and b represent given numbers. By dividing both sides of the equation by b, we get x = a/b.

Now, let's make it an equation with actual numbers: 4x = 3. OK, that means x = 3/4. Thus, 3/4 is the number that, when multiplied by 4, gives us 3.

Now let b = 0. We have x*0 = 3. Blindly following our solving method, we get x = 3/0. What is 3/0? It's not a number -- and mathematicians have declared this operation illegal because we are looking for a number x that, when multiplied by 0, gives 3. Now do you see the problem? Zero is defined in the axioms of a field so that the product is zero when it is multiplied by any number, including zero. Thus, there is no solution for x. Because of this problem, mathematicians simply declared division by zero to be undefined.

Thus, it is incorrect to write x/0 = ∞ because division by zero is undefined. That means you cannot do it within the rules of the arithmetic system you're working in.

Of course, anyone who has taken a calculus class learns about limits and limiting processes. This leads to the statements like "the limit of 1/x as x approaches infinity is zero". This is a completely different ball of wax, as it's often envisioned as a sequence of numbers that keep getting smaller and smaller. Still, it's one intuitive thing that leads to the common shorthand of x/0 = ∞.

Now, in the practical world, division by zero is supported in a python library I use a lot, numpy. And it's pretty convenient, as you wind up with a number that gets displayed as Inf or -Inf. You can also used it in subsequent calculations and it behaves as you expect. However, if you multiply Inf by zero, you get NaN, which means "not a number". But an expression like (number)/Inf will give you zero. This gives you calculational conveniences because it can mean that your complicated calculation won't stop after an hour if a division by zero occurs. But, at the same time, complete weirdness can propagate through your calculation when such things happen, so one needs to be cautious when allowing such things.

I understand your point, and I also understand that X*0=3 can never exist. This was similar to my introduction to the calculus concept of limits. There are situations where conventional mathematics breaks down. Since very few things man made (such as math) is perfect, I can live with it.

There is also a fundamental logic error with your example, which is a basic tenet of GIGO (Garbage In, Garbage Out). Going through your example...

B*X=A

If B = 0 then X is solved, it is 0 no matter where the X is listed as. By making B=0 two, not one, variables have been eliminated. A is automatically solved, and can be no other number than 0.

There is a maxim that states you can prove anything given one false assumption, this is such a case.

 

You left out; "x/0 is always infinity"
And obviously non of those conflicting 3 pure math rules help you solve the problem.

And perhaps why the answer / solution is deemed "indeterminate".

 

Not altogether true. x/sin(x) is undefined at x=0

Sure;

\(\stackrel{lim}{\small{x\rightarrow 0}}\ \ \ \frac{x}{sin(x)}\ =\ 1\)

But;

\(\frac{x}{sin(x)}\ \neq\ 1, \ \ \ \ where\ x\ = \0\)

Spoken like a true mathematician Bill. <wide grin> I ran into the same thing in college, and while I agree with you view all my electrical engineering profs disagreed, taking the limit at that point such that it equals one. I forget what proof it was required for.

I'm reminded of some lecture I attended by an EE prof who was deep into the underpinnings of convolution. At the start he was asked did his work depended on the unit impulse function. When he replied "yes" all the mathematics profs left the room, as they thought that function was heresy (strike that) did not exist.

What is a greater and lesser infinity? Think of ∞, then think of ∞+1. The latter is greater than the first, even though they are both ∞.

Actually they are equal. Such are the ways or transfinite quantities. There are orders of infinities called the cardinality. The cardinality of natural numbers is less then the cardinality of the real numbers.

This fact and several dollars will purchase you a ride on the underground conveyance of your choosing.

 

Thank you someonesdad for the great explanation of why math fails to solve that problem.

As I see it the main issue is that people insist on using math and ONLY math to try to find an answer, as if math was the only tool in the toolbox. If you try to understand "torx head bolt" and the only tool you allow yourself is the standard "monkey wrench" then you would likely come to the conclusion that you can't undo "torx head bolt" because your standard method has proven it to be "undefined"...

Math guys make up very specific rules to make thier own work (pure math) go easier. In the same way carpenters have very specific rules "how you must build a wall" and stone masons have very different rules "how you must build a wall". But unlike carpenters and stonemasons who fully understand their rules are to suit their own specific work, math guys have this tendency to think they are "absolutely right".

Pure math is not the only tool that can be used to understand something or solve a problem. There are logic, common sense, prejudice (undertsanding the unknown thing based on similar known things) and of course intellect.

Since pure math struggles with the problem I previously posted a solution (OK it was tongue in cheek I admit).

Here goes again;
1. We can equate divison to "contains". In obvious example; a carton contains 12 eggs.
2. Nothing can be smaller than zero, this rules out all solutions where zero contains a number of items. Zero can't contain 2 (or more) items because there is nothing smaller.
3. Zero can't contain something larger than itself, this rules out all solutions where zero contains something larger than zero.
4. Zero contains itself, in the same logical way that dog/dog equals "dog contains 1 dog". Dog is a thing, that by it's own definition must contain 1 dog. Therefore; dog/dog = 1.
5. If you decree zero does not contain itself, then what does it contain? Nothing? Doesn't nothing = zero? Proving my point it MUST contain nothing, and nothing = 0, so it must contain itself, complying with the "definition rule" 4 that everything contains itself by definition.

So pure math may struggle to deal with working with zero but it's not that hard, the answer to 0/0 is 1. Zero contains 1 zero.

 

someonesdad: I'm not a Python user but I could see where that feature would be useful, where I could test a result for +Inf or -Inf instead of handling a "division by zero" error. However, all programming languages are computation, but mathematics. They are related but not identical.

Bill: Check that. BX=A for B=0 leads to A=0 and X undefined, as you may substitute any number for X you wish and still satisfy the equation.

RB: You've scaled a new height this time in not only ignoring negative numbers (that are by definition less then zero) but by further equating nothing with something.

Additional hilarity ensued when you started dividing dogs by dogs. As dogs are cute furry animals and not numbers I can only assume you used sharp knives to perform this operation.

 

Spoken like a true mathematician Bill. <wide grin> I ran into the same thing in college, and while I agree with you view all my electrical engineering profs disagreed, taking the limit at that point such that it equals one. I forget what proof it was required for.

\(Taking\ the\ limit\ is\ fine,\ just\ don't\ substitute\ zero\ in\ there\ and\ say\ it\ works.
\
\
At\ least\ \ \stack{lim}{\small{x\rightarrow 0}}\ \ \frac{x}{sin(x)}\ =\ 1\ is\ a\ valid\ limit,\ because\ it\ does\ behave\ well.
\
\
In\ other\ words\ \stack{lim}{\small{x\rightarrow 0^{+}}}\ \ \frac{x}{sin(x)}\ =\ \stack{lim}{\small{x\rightarrow 0^{-}}}\ \ \frac{x}{sin(x)}
\
\
In\ the\ case\ of\ \frac{1}{x},\ even\ writing\ the\ limit,\ \stack{lim}{\small{x\rightarrow 0}}\ \ \ \frac{1}{x}\ is\ undefined,
\
\
since,\ \ \stack{lim}{\small{x\rightarrow 0^{+}}}\ \ \frac{1}{x}\ =\ +\infty\ \ \ \ and,\ \ \stack{lim}{\small{x\rightarrow 0^{-}}}\ \ \frac{1}{x}\ =\ -\infty
\
\
Now,\ dividing\ animals\ by\ animals\ \cdots\ What\ is,\ \stack{lim}{\small{dog\ \rightarrow\ cat}}\ \ \frac{mouse}{dinner(dog)}\ ?
\)

 

Sheesh, this high-brow intellectual discussion is going to the dogs...

 

0 is real and exists very often. I have 0 Ferrari's in my driveway.

That's bunkum! If I have a real eggcarton and remove 12 real eggs from it I have a very real zero eggs in the carton. Zero is real, infinity is not (unless you can demonstrate inifity eggs in the eggcarton). Zero amps is also very real.

If there are zero, then there is not a physical value, simply the absence of a value.

Infinity and zero can both ever be reached. Infinity because we can't get anything that big, and zero because we can't get anything that small. If we do, it doesn't exist.