The big break though between Roman numerals and Arabic math is 0 as a number. In case you haven't noticed, 100 has two zeros in it, and this denotes there are no numbers in the 1's and 10's place. In other words, zero is a valid number, and is a base mainstay in our math system.

∞ is a bit trickier, and I will disagree that there are not differences between greater and lesser infinities. This I imprinted on in a math class many decades ago, not to mention Nova's discussion on the subject.

 

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.

It may be commonsense that there is one of X in X, no matter what X is, maths does not always follow commonsense.

The trouble with zero is that it annihilates anything you multiply it by:
0X = 0 and there is no remnant of X left, so there is no meaningful way to divide by zero. Allowing your definintion leads to problems like:

0*2 = 0*1

dividing by zero:
2 * 0/0 = 1

and if we accept your definition then

2 = 1

and as Bertrand Russell famously proved, given 2=1 I can prove that I am the pope (although these days it may be better to be Bill Gates).

 

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

Hi BillO

Would you please tell me which LaTex editor you used to compose the quoted text? Thank you.

 

Hi BillO

Would you please tell me which LaTex editor you used to compose the quoted text? Thank you.

Just used the LaTex reference here at AAC (click the little Ʃ symbol at the right end of the 2nd menu line at the top of the message editor) to insert tags and did the editing manually. It gets 2nd nature after a while.

 

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.

While I agree that infinity can never be physically realized, zero can, just like any other number. Sure, zero has some unique properties that other numbers do not, but it does exist and is realizable. If you think having zero Ferraris in your driveway is difficult to comprehend, imagine having -2 Ferraris in your driveway. It may just be you're not thinking about this in the correct fashion.

This sounds like an argument that was settled a few hundred years ago. Now, most agree that Natural numbers and the Whole numbers are the same set and they include zero in the Natural numbers, otherwise it does not satisfy the requirements of a commutative semi-ring (no additive identity). This limits its usefulness as a set.

Edit: Besides, when you get right down to it, all numbers are conceptual. Just a human construct to allow us to keep track of things. (Also, corrected my algebraic reference.)

2nd Edit: Actually, even the inclusion of zero does not give us a ring, or even a group, but a semi-ring. There are no inverses. The first number set capable of being a commutative ring is the integers. for a formal definition of the Natural numbers, see here http://en.wikipedia.org/wiki/Natural_number#Formal_definitions.

 

The big break though between Roman numerals and Arabic math is 0 as a number. In case you haven't noticed, 100 has two zeros in it, and this denotes there are no numbers in the 1's and 10's place. In other words, zero is a valid number, and is a base mainstay in our math system.
...

Agreed. Zero is real and provable. Since this is an electronics forum we could have a circuit with 2 amps, 1 amp, or zero amps. All values are equally real and valid.

...
Allowing your definintion leads to problems like:
0*2 = 0*1

What problem?
0*2 = 0
0*1 = 0
therefore 0*2 = 0*1

...
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.

Please feel free to climb up here with me and demonstrate something smaller than zero, or demonstrate a "nothing" that does not exist. But of course your proof must be valid in the real world and not just a concept limited to within the narrow scope of pure math (which as people have mentioned is not capable of working with zero very well).

To add to BillMarsden's example of a zero that exists, add this one; in a computer variable sized "unsigned char" there are 256 possible values. One of those values is zero. It is as real and valid and usable as the 255 other "real" numbers and exists in real working hardware.

...
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.

Well you may have missed that no knife is needed in dog/dog. But I admit dog/2 will get messy.

 

\(
Now,\ dividing\ animals\ by\ animals\ \cdots\ What\ is,\ \stack{lim}{\small{dog\ \rightarrow\ cat}}\ \ \frac{mouse}{dinner(dog)}\ ?
\)

AFAIK Fermat scribbled the entire proof to this little theorem 3 pages after the whole X\(^{N}\) + Y\(^{N}\) = Z\(^{N}\) dilemma.

By simple inspection, we get:

\({lim }{ \small{dog\ \rightarrow\ cat}}\ dinner(dog) = mouse
\)

And thus:

\(\frac{mouse}{mouse} = 1\)

 

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.

Hi Bill

I don't know if I should disagree or agree with you. I'm more of the opinion that mathematics is discovered which means mathematics is 'almost' perfect. If it was invented, then it would be more prone to error. I think mathematics is the only thing in this world which is faithful and flawless. I'm not saying you are wrong. I'm just a person of very limited knowledge. Please let me know your opinion. Thank you.

 

Math is a man made system to describe things around us. It started simple, I have 3 oranges for example. Or I had 5 sheep, but now there is four, so one is missing. Notice the relationship to fingers? The really important thing to keep in mind is math is a descriptive system, we make math systems to describe things around us. Roman numerals worked, but it was bulky and awkward to use. The Arabic system is much easier to use, and is based on the simple fact that zero is a valid number, a concept the Romans, who were very advanced engineers, missed. Imagine trying to do accounting with Roman numerals and you can see why we switched. You can record how much money you own, but how do you calculate interest?

We have invented new systems of math over several millennia, trigonometry was originally invented by the Egyptians to return land to the various farmers after a Nile river flooded and all the landmarks were erased. We have found new uses for old systems as they evolve.

The key term here is invented. Math is purely man made, the fact that it is so powerful a tool helps our civilization become what it has. Occasionally it has happened a new math system is invented, then a use for it is discovered. This was the case for Boolean algebra.

There is nothing that says a particular math is true, or useful. Straight Euclidean math works for Newton's Laws of Motion, but in the scheme of things Relativity turned out to be more correct as observed by practice. I've mentioned this before, but Newton's Laws were sufficient to get us to the Moon. If we go to other stellar systems we'll need a more powerful math, such as calculus and relativity.

The argument a while back about My Dear Aunt Sadie (Order of Precedence, Multiplication Division Addition Subtraction) is a case in point, this was a very firm rule that has been forgotten as we have let computers take over the bulk of the work for basic calculations. It was created to allow us to write in shorthand for equations, but with computers the need for shorthand has mostly gone away.

In other words, being an invention math evolves over time. Basics we take for granted may not be as true in the future.

If we come across something that doesn't fit the existing rules a new math to describe this system will be invented to describe it.

 

Hi Bill

I don't know if I should disagree or agree with you. I'm more of the opinion that mathematics is discovered which means mathematics is 'almost' perfect. If it was invented, then it would be more prone to error. I think mathematics is the only thing in this world which is faithful and flawless. I'm not saying you are wrong. I'm just a person of very limited knowledge. Please let me know your opinion. Thank you.

Actually, a mathematician proved the opposite. See Godel's incompleteness theorem.

You can get a very good sense of this theorem by reading "Gödel, Escher, Bach" by Douglas Hofstadter.

My opinion is mathematics exists all on its own separate from the physical world. In certain cases it seems to be an effective analog of physical processes, and thus has a certain utilitarian value.

Forget about infinities or imaginary numbers, just think of 1/2. If I have two apples, I can give you exactly half of them, but I cannot give you exactly half an apple.

Exactly half of a thing has no physical existence.

Want further confusion? Hand me half a piece of paper. No matter how you wiggle and slice and ponder, you will keep on handing me ONE piece of paper.

 

AFAIK Fermat scribbled the entire proof to this little theorem 3 pages after the whole X\(^{N}\) + Y\(^{N}\) = Z\(^{N}\) dilemma.

By simple inspection, we get:

\({lim }{ \small{dog\ \rightarrow\ cat}}\ dinner(dog) = mouse
\)

And thus:

\(\frac{mouse}{mouse} = 1\)

Well done!

 

Wow, great discussion, too bad I am stuck studying. I do think that math is man-made, but nature inspired (golden ratio?) or maybe we are just seeing things...

About 0/0, I wish I had the paper that I wrote on zero to reference from, but 0/0 was always a problem and I believe is still an undefined quantity. Some interesting things that I found:

http://www.friesian.com/zero.htm - a discussion with a short link to physics

p.s. while refreshing myself on this stumbled on "nullity" - solution to all of your zero problems...

 

Math is a man made system to describe things around us. It started simple, I have 3 oranges for example. Or I had 5 sheep, but now there is four, so one is missing. Notice the relationship to fingers? The really important thing to keep in mind is math is a descriptive system, we make math systems to describe things around us. Roman numerals worked, but it was bulky and awkward to use. The Arabic system is much easier to use, and is based on the simple fact that zero is a valid number, a concept the Romans, who were very advanced engineers, missed. Imagine trying to do accounting with Roman numerals and you can see why we switched. You can record how much money you own, but how do you calculate interest?

We have invented new systems of math over several millennia, trigonometry was originally invented by the Egyptians to return land to the various farmers after a Nile river flooded and all the landmarks were erased. We have found new uses for old systems as they evolve.

The key term here is invented. Math is purely man made, the fact that it is so powerful a tool helps our civilization become what it has. Occasionally it has happened a new math system is invented, then a use for it is discovered. This was the case for Boolean algebra.

There is nothing that says a particular math is true, or useful. Straight Euclidean math works for Newton's Laws of Motion, but in the scheme of things Relativity turned out to be more correct as observed by practice. I've mentioned this before, but Newton's Laws were sufficient to get us to the Moon. If we go to other stellar systems we'll need a more powerful math, such as calculus and relativity.

The argument a while back about My Dear Aunt Sadie (Order of Precedence, Multiplication Division Addition Subtraction) is a case in point, this was a very firm rule that has been forgotten as we have let computers take over the bulk of the work for basic calculations. It was created to allow us to write in shorthand for equations, but with computers the need for shorthand has mostly gone away.

In other words, being an invention math evolves over time. Basics we take for granted may not be as true in the future.

If we come across something that doesn't fit the existing rules a new math to describe this system will be invented to describe it.

Thank you, Bill. I have seen many people contend that mathematics is a universal language and the same people opines that mathematics is discovered. I remember I once read about a mathematicians who was an atheist but he still used to say something close to 'here is the proof from book written by some unseen hand'.

Some math systems are more efficient than others as are some tools than others. If I need more precise measurement I would use an equipment which can handle that precision. I can know the slop of straight line simply by using points on the line. But I can't use this method efficiently for a curve. Still, I can use two points on the curve which are not very far from each other to have a rough estimate of the slope. But to have an almost precise measurement of slope I need to use calculus tool. In this way, calculus more superior tool than the tool of 'using two points' because it can be used for any kind of curve including straight line.

Actually, a mathematician proved the opposite. See Godel's incompleteness theorem.

You can get a very good sense of this theorem by reading "Gödel, Escher, Bach" by Douglas Hofstadter.

My opinion is mathematics exists all on its own separate from the physical world. In certain cases it seems to be an effective analog of physical processes, and thus has a certain utilitarian value.

Forget about infinities or imaginary numbers, just think of 1/2. If I have two apples, I can give you exactly half of them, but I cannot give you exactly half an apple.

Exactly half of a thing has no physical existence.

Want further confusion? Hand me half a piece of paper. No matter how you wiggle and slice and ponder, you will keep on handing me ONE piece of paper.

Hi Ernie

Are you saying that Godel proved that mathematics isn't perfect and is human invention? Please let me know.

That depends how you interpret. If you ask me hand you over half a piece of paper. Then, I will simply try to divide it into two halves, and then hand you over an 'almost' half a piece of paper. Then, if you say that I have handed you over a piece or half a piece of paper that depends upon you. Mathematics can give us precise answers, we strive to reach that perfection as the need may be.

Best wishes
PG

 

Are you saying that Godel proved that mathematics isn't perfect and is human invention? Please let me know.

Obviously it's a human invention. I've yet to see any species other then human publish publish any papers in any field of mathematics. Otherwise follow the link (but GEB gives a better "feel" for a theorem I will never read).

That depends how you interpret. If you ask me hand you over half a piece of paper. Then, I will simply try to divide it into two halves, and then hand you over an 'almost' half a piece of paper. Then, if you say that I have handed you over a piece or half a piece of paper that depends upon you. Mathematics can give us precise answers, we strive to reach that perfection as the need may be.

You're proving my point. It is you who are applying arbitrary definitions to produce an approximate solution (that can be shown to be in error by 100%). If you are not giving me a "precise answer" you are not doing mathematics.

1/2 is as abstract a concept as \(\sqrt{-1}\)

 

Side comment, I think of j (

) as yet another sign, similar to + (plus) and - (minus). It follows rules that are very similar to signs. It is important to note it exists because it is required for the design of electronic filters (and maybe more). We couldn't get by without it, and it has some basis in reality. What and why I'm not sure, but it describes something real.