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# the world's smallest number?

Wanderer
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[James]: there is no smallest number greater than 0 and hence no greatest number lesser than 1.

Out of curiosity, when was the last time anyone here argued otherwise? Who? I'm sure there was someone, but this thread has gotten so mired in silly digressions about how unthinkable it is to have a repeating decimal, I've lost track of who you're arguing with now. Several of us here said from the beginning that .999(9) was equal to 1, and thus the difference between .999(9) and 1 was zero. We can keep quibbling about the .999(9), but as long as you're no longer making outrageous statements about all other repeating decimals, I'm not that motivated anymore. If we're going to keep talking about it, perhaps we could identify someone who thinks that there is a smallest number greater than zero? Someone who's still around, posting to this thread? Then we can disagree with that person. Or we can listen to them to get a better understanding what they mean. Until then, I'll be asleep in the corner.

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Originally posted by James Christian:
Hence I define Zero as being the mere abcense of something in a defined subsystem.

So you are trying to give a name/value to a non-existence thing in your system, which holds no value as it does not exist.

So when we divide a number of oranges between 0 people and the result is 0 what we are saying is that the number of oranges in a portion to be shared out is 0. i.e. we are not sharing out any oranges and thus the remainder is equal to original quantity.
And what I am saying that as there are no people, division never happens for a simple reason that there is nothing to divide from. I think you call that nothing as zero in your system.

Can you sell 0 oranges ??
I think answer is big no.
But yes I can sell 1 orange.

Can you divide number by 0?
I think answer is big NO within the limits of mathematics rules defined for operation called division.
But yes I can devide any number by 1.

if something does not exist in any systen then it does not exist. Non-existing things dont have values or property associated with them. rt ?

You cant say that person that does not exist, weights 60 kg.
I think I will say the person that does not exist, if ever exist will have undefined weight.

So 0 person has infinity weight.(note: not Zero wt.)

Its something like before you divide your 100 oranges to 0 people, may I know the name of people who are going to get oranges.

So even practically, in real life you cant divide 100 oranges to 0 people. Let us say your system requires a name of person who buys oranges. Now before you sell any orange to 0 people what name will you enter ??

You have 100 orannges even after dividing it by 0 because that division never happend. 100 is not remainder it

P.S. WE HAVE ALREADY UNANIMOUSLY AGREED THAT DIVISION BY 0 PRESENTS AN EXCEPTIONAL CIRCUMSTANCE FOR THE DIVISION ALGORITH IMPLEMENTED BY THE HARDWARE.
CAPS S*\$KS !!
I dont know who agrees that division of any number by 0 is possible.
(for the time being forget the DivideByZero exception).
I think simple rule of division says that quotient cant be greater than divider so I dont know how can any positive number be less than 0.
[ April 11, 2005: Message edited by: R K Singh ]

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It looks like a new datatype should be introduced to the Java and Python languages called fraction.

>> aFractionVeriable = Math.PI()
>> aFractionVeriable
355 / 113
>> aDoubleVeriable = Math.Double(aFractionVeriable)/* casting */
3.1415929203539825

lowercase baba
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James,

can you define for me what you mean by "0.000(0)1"??? What most people seem to think is that the (0) means "an infinite number of 0s". but then you say "put a 1 after that".

So does your definition of "infinte" actually mean "finite"? if so, how many 0's are there, since finite means you can say exactly how many there are?

Definition: Inifinite - "having no limits or boundaries in time or space or extent or magnitude" (emphasis mine)

here's the wiki on infinity.

if we can't agree on what the 0.(9) means, the rest of these areguments are pointless.

Bartender
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Originally posted by Jim Yingst:
[James]: there is no smallest number greater than 0 and hence no greatest number lesser than 1.

Out of curiosity, when was the last time anyone here argued otherwise? Who?

I think back on April 10 around 11:29 AM Gerald Davis did:

Something that is infinite cannot equal something that is finite like an integer or float number.

...but I may have mis-interpreted his comments.

Ryan

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Originally posted by James Christian:
And finally what about this? 7.1000(0)1 Is this the smallest number greater than 7.1?

Well, no, as 7.1000(0)1 is not a number at all...

Ilja Preuss
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Originally posted by fred rosenberger:
James,

can you define for me what you mean by "0.000(0)1"??? What most people seem to think is that the (0) means "an infinite number of 0s". but then you say "put a 1 after that".

Notice the smiley - I think James tried to be funny...

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Or we can listen to them to get a better understanding what they mean. Until then, I'll be asleep in the corner.

I totally agree. It seems we were all agreed that the smallest number doesn't exist all along.

So let's get back to my proposed behaviour of division by 0. As far as I'm concerned I still haven't been proven wrong. Although Jim gave a few example that almost qualified.

I'm sure if we all try harder someone can disprove my preferred implementation.

Anyway, it seems far more interesting than going round in circles when we already all agree that there is no smallest number greater than 0.

Everyone was really forthcoming when the topic was contraversial. And I had a whale of a time. So come on who thinks they can defeat my division of 0 theory. Who can demonstrate an example where the remainder should not be equal to the original quantity? And where the ** did the man who doesn't believe in 0 come from? That's sounds weird even to me!!!

Especially for you Mr. Singh:

Shall we throw errors for 1 + 0?
What about 7 - 0?
When you do the accounts at the end of the month and you haven't sold anything what will you enter in your accounts? Will you pretend you have sold 1 because you don't beleive in 0?
How many people do you know who are over 5 metres tall?
How many dolphins have you seen flying a hang glider?
If you have \$1000 in the bank and you take the \$1000 out how much will be left in the bank?
How much sense does your argument make?

I beleive you will find the answer to all the above questions to be a non existent figure by your own reckoning.

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if orangesPerEmployees = 0, code will likely assume:

"Oh, there weren't any oranges for the employees! We must need more oranges"

when in fact there was 6.022 x 10^23 oranges, just no employees.

P.S.
[JR] How many people do you know who are over 5 metres tall?
How many dolphins have you seen flying a hang glider?
If you have \$1000 in the bank and you take the \$1000 out how much will be left in the bank?
How much sense does your argument make?

How many people have a right to talk like [insults deleted, feel free to rephrase - Jim], in an argument in which his opinion is considered wrong by all knowledgable mathematic and programming communities?

I guess 1, but you're no Wittgenstein.

[ April 11, 2005: Message edited by: Nick George ]
[ April 11, 2005: Message edited by: Jim Yingst ]

Jim Yingst
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Now now, Nick. Haven't James' recent posts shows he needs our help more than our ire?

R K Singh
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Originally posted by James Christian:
Especially for you Mr. Singh:

Shall we throw errors for 1 + 0?
What about 7 - 0?

How much I know addition, subtraction has completely different rules and application.

AW good question...

why 1 + 0 = 1
and 1 * 0 = 0

why 1 + 0 = 1
and 1 * 0 = 0

why
1 + 0 = 1
1 - 0 = 1 (same result)
but
1 + 1 = 2
1 - 1 = 0 (different result)

Think of Zero as special number, if I remember correctly then natural number set does not contain Zero.

As it is a special number, it has special property and behaviour.[atleast in mathematics]

When you do the accounts at the end of the month and you haven't sold anything what will you enter in your accounts? Will you pretend you have sold 1 because you don't beleive in 0?
I will enter the stock I have

kidding apart, in your account will you write things/items which even you dont have as sold zero ?? Because they are not sold(reason is that you dont have them).

Do you enter in your house budget every month that you bought 0 plasma TV, 0 car, 0 jeep. And you spent 0 amount on tourism ??

If you have \$1000 in the bank and you take the \$1000 out how much will be left in the bank?
Now sense is being talk.

In my room, I have 0 helicopters... opsss helicopter is too big.
OK I have 0 TV, 0 radio, 0 music system, 0 cot. Does it make sense ??

In mathemical field we denote, nothing as Zero, the same way figures that cant be defined or may be continued to forever are denoted as infinity (00)[its not double zero].

Thats make sense, so we have Zero and Infinity figures in our life.
And n/0 = 00(infinity)

Now why infinity, becuase when you apply the mathematical rules of division, it can never be over.

And when you say you divide 10 oranges among 0 people in real life, actually no division happens and thats why you still have 10 oranges.

Lets say we dont know simple airthmetics.
Now we have to divide 10 oranges to 2 people.
I will give 1 orange to person X.
then I will give another to person Y and I will do this till I give all my oranges to persons X & Y.

Now can you explain how will you divide 10 oranges to 0 people ?? I am really intrested to know, how can one divide 10 oranges to 0 people.

Did anyone ever try to give 1 orange to 0 people(lets assume that s/he is 5 ft tall) ??
[ April 11, 2005: Message edited by: R K Singh ]

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Originally posted by R K Singh:

If you have \$1000 in the bank and you take the \$1000 out how much will be left in the bank?

If you have \$1000 in the bank and take out \$1000, the chances are that you'll be left with \$-30.25, because the unscrupulous corporate evil-doers at the bank charge a fee for withdrawal of funds and then charge another fee for going over-drawn!

In banking terms, you have to talk very nicely to you bank to find out what your final balance will be before making such a withdrawal so that you end up balancing your accounts with an equation more like:

\$1000 - \$998.75 = \$0

R K Singh
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Originally posted by Adrian Wallace:

\$1000 - \$998.75 = \$0

fred rosenberger
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James,

This might not be the BEST example, but it is an example where dividing by 0 giving a value does not work...

I'm going to write a program that takes an equation and graphs it, like many calculators will do. If I use your definition, when i graph something like

i would get a nice, neat, solid line equivilent to y = x+2. The program would never think anything was wrong.

But mathematically, my graph should have a hole in it at x=5. there is no y value at that point. With the exception being thrown, my program will say "AHA!!! something is wrong here. Don't graph a point at x=5 because it doesn't exist".

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Lets see if we can get the questions we are trying to answer laid out.

1) Is there a smallest number greater than 0?
I think we have agreed that there is not.

2) Is the correct mathmatical result of x/0, and therefore x%0, undefined (or infinity)?
I don't think there is any argument that this is true.

3) Knowing that the correct mathmatical result of x/0, and x%0, is undefined does it make sense to modify that behavior, within programming, to allow x/0=0 and x%0=x?

I think 3 breaks down into two basic questions.
3a) Does the behavior x/0=0 and x%0=x pose problems, within programming, that are too difficult to overcome.

3b) Are there benifits to the behavior x/0=0 and x%0=x that overcome any difficulty associated with their use.

In my opinion the answer to both 3a and 3b are no.

James Christian
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code:

void orangesPerEmployee(int oranges, int employees){
return oranges/employees;
}

if orangesPerEmployees = 0, code will likely assume:

"Oh, there weren't any oranges for the employees! We must need more oranges"

when in fact there was 6.022 x 10^23 oranges, just no employees.

Just out of interest, what kind of a plonker would base stock ordering on how many surplus oranges were divided out to employees. Most folk I know would use the stock count and maybe past sales patterns to predict future demand.

James Christian
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code:

x^2 - 3x -10
y = ----------------
x - 5

And as for this one, could you run that one by me again please, but slower this time? I have absolutely no idea what your point is here.

I gather that you expect there to be an unnoticabley small hole in your graph but I'm not alltogether sure I understand why you expect this behaviour.

Remember, you are speaking to an idiot who thinks you can divide by 0 and that 3*4=11, so please give a slow step by step explanation of why we should expect this behaviour.

And while your at it could you also tell me when in a program you would expect to plot a graph that fulfills such a behaviour.

Bartender
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Originally posted by fred rosenberger:
James,

This might not be the BEST example, but it is an example where dividing by 0 giving a value does not work...

I'm going to write a program that takes an equation and graphs it, like many calculators will do. If I use your definition, when i graph something like

i would get a nice, neat, solid line equivilent to y = x+2. The program would never think anything was wrong.

But mathematically, my graph should have a hole in it at x=5. there is no y value at that point. With the exception being thrown, my program will say "AHA!!! something is wrong here. Don't graph a point at x=5 because it doesn't exist".

Interestingly enough, my graphing calculator appears to reduce before graphing, as it has no problems showing a nice, neat line. And wheere x = 5, y = 7.

Joel McNary
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Originally posted by Steven Bell:
It seems to be you could use a similar argument to remove the NullPointerException from Java. Any code that would currently throw a null pointer exception should simply do nothing and if a return value is expected simply return null.

I'm tired of checking for null's all the time anyway.

That's what Objective-C does....

fred rosenberger
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When would i need to plot a graph like this? if i was writing, say, a graphing calculator program. a tool that Algebra students use to help them understand how graphs work, how to make them, how changing parameters effects a graph, etc.

Granted, not the most commonly used tool, but i think a valid one. when i taught algebra, we spent quite a while on different kinds of graphs...y=Ax^2 + Bx + C, for example. what happens as you change the values of A? the graph becomes "fatter" or "skinnier". As you change C? it moves up and down, etc.

there are dozens of functions you can graph, that have special properties. parabolas, hyperbolas, circles, ovals... and on and on...

my example was

Granted, this is a contrived example i created to illustrate what i consider a flaw in your approach. But i think it's a valid equation to try to graph.

traditionally, to graph this, you'd pick some values for x, do the calculation to find Y, and plot the points... you keep doing this until you find a pattern, and extrapolate the curve.

now, you are allowed to transform this equation into another equation that has the same "solution set. for example,

(1) y = 2x/2

can be transformed into

(2) y = x

these are NOT the SAME equations, but because of the rules of mathematics, it's easy to transform one into another without changing the solution set.

I guess i should also back up, and talk about Domains and Ranges. the Domain is the set of all possible values you can try as X. for equations (1) and (2), the domain is the set of all real numbers (let's forget about imaginary for now). There is no real value i can put in as X that would cause me problems. the same cannot be said for factorial...

(3) y = x!

factorial is generally defined as

x! = (x) * (x-1) * (x-2)... 3 * 2 * 1

ok, that's not REALLY a definition, but it illustrates how factorial works. Notice that if you put in a negative number, there is a problem. Same with fractions. by this definition, you can ONLY put in positive whole numbers.

the Range is the set of all values you can GET by putting in whatever values for x.

y = x^2

i can put ANY real value (again, ignoring imaginary) in for x. but i will ONLY get non-negative values for y.

Back to my original equation... if i wanted to graph it, i could start putting values in for x and see what i get for y. but that's an ugly equation, very complicated. if i could TRANSFORM it into an equation with the same solution set (an equation that gives me the same values for y as i substitute values in for x), it might make life easier.

Algebra to the rescue!!! i know from my algebra days that

x^2 - 3x -10 is the same as (x-5)*(x+2). now i have

GREAT!!! I can cancel a common factor out of the numerator and the denominator. in this case, (x-5), to get

this is MUCH easier for me to graph by hand. i can use the slope/intercept, or just plug in values

x|y
---
0|2
1|3
2|4
3|5
4|6

hey - it's a straight line!!! easy peasy!!!

BUT... i have to be careful. my Domain for my transformed equation MUST be the same as the Domain for my original equation. in my original equation, i had an (x-5) in the denominator. According to the rules of arithmatic (remember, i don't have a computer yet), i am not allowed to divide by 0. so, my domain must be all real numbers EXCEPT 5.

and yes, i would exactly expect there to be a hole in my graph, as if someone took a paper-punch and punched out that point, (5,7). it would not be unnoticably small. it would be exactly one point big. if i were plotting it on paper, i would see a solid line, with a circle drawn at the point (5,7). My drawing of the circle would be MUCH larger than the actual point, but that's a limitation of my pencil/eyes, not the mathematics.

now, most calculators wouldn't work this way. they wouldn't bother trying to simplify the equation - it would be too complicated/take too long. Calculators would use 'brute force', calculating hundreds if not thousands of values of y for different values of x.

Ideally, the calculator should return the same graph I got by hand. Should it happen to try and substitute x=5, it should let me know that there was an issue... something isn't qutie right here.

More than likely, though, the calculator will not actually hit that value. i just tried it on my TI-85. when my graphing range was -2 through 10, it didn't hit 5 exactly (it hit 4, though). but when i changed to 0 through 10, it did. and, it left a hole in the line.

Now, if you're asking me when a graphing program would need to graph an equation like this... i'd say anytime a user input an equation like this. also note that sometimes it's not quite as simple... maybe i want to play with the quadratic formula, and see what happens as the a parameter changes, while b and c are held constant

could be graphed as
, substitu.ting in some values for b and c

this can't be simplified. and this equation doesn't work when a (or x) = 0

Again, this is a contrived example. but if i have some 8-variable function (performance of an engine? turbulance over an airfoil??? i don't know...) that i want to graph in 8-dimensional space, there very well might be some values i can't allow to happen in the denominator.

fred rosenberger
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Joel,

what kind of calculator do you have? I'm genuinly curious... plus, i wouldn't want to buy it.

are you SURE it's plotting the point (5,7)? mine goes from
(4.9523809524,6.9523809524) to (5.0476190476,7.0476190476).

only when i set up the display range with 5 in the middle does it exactly hit that point. and there is a hole in my display there.

i have a TI-85, that's at least 10 years old.

Joel McNary
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I've got the TI-86 (Which is simply the newer version of the TI-85). When I trace the graph, I can get it to read x=5 y=7 at the bottom, and I don't seem to have a hole in my graph.

Joel McNary
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The key point here is that dividing by zero can mean whatever you want it to mean. Somebody earlier stated that in not quite those terms. You could take it to mean 0 remainder x. You could take it to mean 7. You could take it to mean 21324597865413465754321657498. There is no constant meaning for division by zero. It's not that it can't be 0 remainder x, it's that that isn't always the semantic intrepretation we want. Just like the difference between dividing a cake among 0 people and dividing a cake into 0 peices. In the former, you get 0 people with cake and 1 cake left over. In the latter, you get 1 peice (the cake). You can't get fewer than 1 peice without eating the cake, but that's subtraction, not division. It's for this reason that mathematics has defined division by zero to be undefined.

Take fred's example and add another equation:

Now, from simplification, we can see that y = x+2 in the first case and z=x+3 in the second. So, if x = 5, then y = 7 and z = 8. But plug 5 into the original equations -- in both cases, you get zero divided by zero. But we know that when x = 5, y = 7 and z = 8. Since we know these stetements to be equals, we can say that 0/0 = 7 and 0/0 = 8. And we can say that with certaintude and validity, because dividing by 0 is undefined -- all and any answer is possible.

A computer cannot read minds, however. It doesn't know what answer you are looking for. So, it gives the only answer that it can -- the "I don't know what to do in this situation" example. It's up to you, the programmer, to tell it what you want. (I recently worked on a project that reported metrics. In some cases, dividing by 0 indicated 0% of the work was done, in others is was 100% of the work was done. It all depended on semantic context. Semantics is something that computers are not yet good at.)
[ April 19, 2005: Message edited by: Joel McNary ]

fred rosenberger
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Originally posted by Joel McNary:
I've got the TI-86 (Which is simply the newer version of the TI-85). When I trace the graph, I can get it to read x=5 y=7 at the bottom, and I don't seem to have a hole in my graph.

Strange. mine at the bottom says x=5 y=<just blank space>

James Christian
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Thanks guys for the easier to follow explanation of the graphs.

However, now it has become cleare to me several questions come to mind. Seeing as you explained that the equation that looks complicated is in actual fact a simple linear graph does it seem normal to you for a linear graph to have a hole in it?

Where would you place the hole in y = x + 2 or 2y = x ?

And this hole which you defined as being exactly a point in size. How big is a point in mm? Would it not be our infamous smallest number greater than 0 which we have already agreed does not exist? Would you not agree that your calculators are producing a flawed graph by inserting a hole in a linear graph?

I'm no mathematical expert but I never remember my teacher saying that linear graphs have a hole in them. Wouldn't you find it strange and inconsistent that a calculator produces a graph without a hole when given y = x + 2, yet produces a graph with a hole from a more complex input which mathematically simplifies to the same thing.

All things considered, wouldn't my proposed implementation of division by 0 give a more consistent output?

And furthermore seeing as before it was argued that since multiplication and division are inverse operations shouldn't they cancel one another out?

just as 2*5/2 cancels to produce 5 where is the problem with cancelling
for all values of x even when x happens to equal 0?

James Christian
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With regard to our 0.111(1), 0.333(3) etc. I wanted to add a final comment.

The reason we propose with this notation to append an infinte number 3's to 0.333 is because no matter how many 3's we append the actual amount is always slightly greater, and the second we append a 4 the actual amount is always slightly less. Therefore when we say that 1/3 = 0.333(3) what we are actually saying is that due to an inadequacy of expression in base 10 we denote 1/3 as 0.333(3) because no matter how many 3's we append the actual amount is always slightly greater. So we hypothesize that 1/3 is equal to 0.333 with an infinite number of 3's appended. However, as has already been agreed since infinity is a number which does not exist we cannot say that 1/3 is EXACTLY equal to 0.333 with an infinite number of 3's appended as such a number cannot and does not exist.

Hence when we say that 0.333(3) (which is merely an imperfect representation of 1/3) * 3 = 0.999(9) what we are actually saying is that the result is always slightly greater than 0.999 no matter how many 9's we append. However, contrary to 1/3 we have no difficulty in exactly representing this number in base 10 as it is precisely equal to 1.

fred rosenberger
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Seeing as you explained that the equation that looks complicated is in actual fact a simple linear graph does it seem normal to you for a linear graph to have a hole in it?

i did not say "it is in fact a simple linear graph". what i said was, using algebra, we can convert the 'complicated' equation into a 'simpler' one, with the same solution set (i.e. for every x in the the domain, i get the same y in the range). However, since my original equation does NOT allow for me to have an x value of 5, i can't use an x value of 5 in my 'simplified' version of the equation. it's simply not allowed.

now, if i were JUST graphing y = x+2, then there would be no hole. but i am not. i am graphing that more complicated equation. So i am bound by ITS domain. HOW i go about graphing that is irrelavent to the actual solution.

Where would you place the hole in y = x + 2 or 2y = x ?

if this were the original equation i'm given to graph, there would be no hole, because there is never a value of x that would make a denominator of 0. HOWEVER, if i were given a more complicated equation that i could simplify to this, there may or may not be a hole, depending on that original one.

And this hole which you defined as being exactly a point in size. How big is a point in mm? Would it not be our infamous smallest number greater than 0 which we have already agreed does not exist? Would you not agree that your calculators are producing a flawed graph by inserting a hole in a linear graph?

a point, by definition, has no size. but that doesn't mean it's not there. Do you have a problem with me graphing the single point (5,7)? If so, how do you graph anything? and if it's ok to graph a single point, why is it not ok to NOT graph a single point? No, this is not the same thing as a smallest number that does not exist. the point (5,7) DOES exist. it simply has no size - as in it has a width of 0, and a height of 0.

I would not say my calculator is 'inserting' a hole in the graph. the hole is already there. Any tool i have is limited by the physics of reality. Yes, it draws a hole much bigger than the actual hole, but I do the same thing when i plot a point with pencil and paper.

Also, remember, i am NOT plotting a linear graph. i am graphing a somewhat complicated equation, that can be TRANSFORMED into something similar. it's like converting 1000 pennied into a \$10 bill. they are NOT the same thing, but they have the same value. i can convert back and forth as needed, without losing anything. and in many cases they are equivilent. but sometime, they are not. if you owed me ten dollars, i wouldn't take payment in 1000 pennies. i'd wait until you had a \$10 bill, or two \$5s, etc.

I'm no mathematical expert but I never remember my teacher saying that linear graphs have a hole in them. Wouldn't you find it strange and inconsistent that a calculator produces a graph without a hole when given y = x + 2, yet produces a graph with a hole from a more complex input which mathematically simplifies to the same thing.

Your forgetting that when i transform an equation, I can't change the Domain. the domain of the original equation said "x can be anything but 5". so, after the transformation, I MUST STILL USE THE SAME DOMAIN.

if the input was y = x + 2, then no, i would not expect a hole. but that was NOT my input.

Find a high school Algebra II textbook (maybe Algebra I). I'm not an expert either, but i distinctly remember teaching this very concept to my students when i taught high school.

All things considered, wouldn't my proposed implementation of division by 0 give a more consistent output?

no. if you allow division by 0, all mathematics breaks down. allowing it lets me prove that 1=2, 1=3, 1=4. your idea that 5/0 = 5 allows for me to also say with just as much validity that 5/0 = 11. Plus, before division could be done by the hardware, it would have to check to see if the denom is 0. if it were, it would have to use one rule, and if not, use another.

And furthermore seeing as before it was argued that since multiplication and division are inverse operations shouldn't they cancel one another out?

just as 2*5/2 cancels to produce 5 where is the problem with cancelling
for all values of x even when x happens to equal 0?

because, by the very definition of division, you CANNOT divide by 0. you can cancel the (x-5)s in ALL CASES EXCEPT WHEN x = 5. in that case, you can't cancel it out, so your stuck with the original equation. which gives you a 0 in the denominator. which is, by definition, undefined. there IS no solution at that point.

Joel McNary
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Originally posted by James Christian:
With regard to our 0.111(1), 0.333(3) etc. I wanted to add a final comment.

The reason we propose with this notation to append an infinte number 3's to 0.333 is because no matter how many 3's we append the actual amount is always slightly greater, and the second we append a 4 the actual amount is always slightly less. Therefore when we say that 1/3 = 0.333(3) what we are actually saying is that due to an inadequacy of expression in base 10 we denote 1/3 as 0.333(3) because no matter how many 3's we append the actual amount is always slightly greater. So we hypothesize that 1/3 is equal to 0.333 with an infinite number of 3's appended. However, as has already been agreed since infinity is a number which does not exist we cannot say that 1/3 is EXACTLY equal to 0.333 with an infinite number of 3's appended as such a number cannot and does not exist.

Hence when we say that 0.333(3) (which is merely an imperfect representation of 1/3) * 3 = 0.999(9) what we are actually saying is that the result is always slightly greater than 0.999 no matter how many 9's we append. However, contrary to 1/3 we have no difficulty in exactly representing this number in base 10 as it is precisely equal to 1.

I think that about sums it up as best as it can be said.

Jim Yingst
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[Joel]: I think that about sums it up as best as it can be said.

Well, aside from a continued inability to accept the idea of an infinite number of terms which add up exactly to a finite result. Plus the erroneous use of the term "hypothesize". But yeah, that's about as close as we're likely to get in this conversation.

James Christian
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it simply has no size - as in it has a width of 0, and a height of 0

Right, so what your so saying is, is that your graph has a hole with width and height equal to 0.

Errrm, isn't that the same as there being no hole?

James Christian
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i did not say "it is in fact a simple linear graph". what i said was, using algebra, we can convert the 'complicated' equation into a 'simpler' one, with the same solution set (i.e. for every x in the the domain, i get the same y in the range). However, since my original equation does NOT allow for me to have an x value of 5, i can't use an x value of 5 in my 'simplified' version of the equation. it's simply not allowed.

now, if i were JUST graphing y = x+2, then there would be no hole. but i am not. i am graphing that more complicated equation. So i am bound by ITS domain. HOW i go about graphing that is irrelavent to the actual solution.

One question as I have evidently not understood. Do both graphs produce the same line or not (apart from the "hole")?

James Christian
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erroneous use of the term "hypothesize".

Erroneous eh??? Well, if 0.333(3) is the shorthand, I'd be interested to see you produce the longhand.

Jim Yingst
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0.333(3) = limn -> ∞ Σ k = 1n [ 3 / 10k]

I've distorted the notation a bit because HTML doesn't support mathematical notation that well - but the idea should be clear to anyone who's worked with limits and series. The idea of a limit as some variable approaches infinity is actually rather rigorously defined in most good first-semester calculus courses. The variable may never reach infinity, but the limit of an expression containing that variable can still have a well-defined exact value - such as 1/3 in this case.

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Right, so what your so saying is, is that your graph has a hole with width and height equal to 0.

Errrm, isn't that the same as there being no hole?

Nope. i don't think it is. You seem ok with placing a bunch of 0 width and height objects together to get something with an infinite size (i.e. a line), or even a finite size (a segment from, say, 0-1) but leaving one of them out doesn't leave a hole. I find that interesting. and, i disagree with it.

Do both graphs produce the same line?

i guess the simple answer is yes. the more complicated answer depends on what you mean by "both graphs". If someone said (case 1) "graph Function 1. Now graph function 2." one would have a hole, and one wouldn't. If they said (case 2) "Graph function one by reducing it to the other", both would have a hole.

Again, it all goes back to what your Domain is. In case 1, i have two separate problems. each graph has its own domain independant of all others. so, funtion 1 had Domain = (all Reals where x != 5). Function 2 has a domain of (all Real numbers).

in case 2, i have one function to graph. that initial graph requires the domain to be (Reals != 5). so no matter how i transform the equation, i still have a domain of (Reals != 5).
[ April 22, 2005: Message edited by: fred rosenberger ]

James Christian
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Is 2y = 2x equal to y = x or not? I rather feel that it is. That's the beauty of algebra. We can simplify it but it is still equal to the original expression. I have to say that when we have to resort to saying these two lines are exactly equal except for the non-existant hole which we have to add to satisfy out generally accepted and time-honoured traditonal view of division by 0 it doesn't convince me very much. All things being equal would or would not my proposed implementation of division by 0 produce a more consistent rendering of the two graphs? I think we all know that the answer is yes and there is no practical usage of inserting the non-existant hole of 0 dimensions other than to convince our unquestioning students that our current definition of division by 0 is in fact the correct one.

Just out of interest if the graph represented what we should pay employee y for efforts x how much should pay employee y when he puts in effort x where the hole exists? Should we send him home after a months hard work saying "sorry we're not paying you this month because our graph was derived from the more complex equation which doesn't allow for division by 0. Had we used the simple and otherwise equivalent linear version of the graph your family could have eaten this month and you could have paid the mortgage but unfortunately your going to have to risk repossession this month and let that be lesson to you for over-performing. You're making the other employees look bad!!!"???

I'm sorry but although your example was good I just can't see the practical usage other than persuading teenage algebra students that there is a non-existant hole in the equivalent graph.

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The smallest number is defined more by our lack of understanding or limitations rather than our understanding of the same.

A 4 digit calculator will have 0.001 number as the smallest number while number cruncher can ideally go uptill thousands of digits before it runs out of memory or crashes ....on the same line a human can keep on it and try to say that this is the smallest number till the next one comes and adds some extra 0's after the decimal and come up with the smallest number.

0 and infinity are terms which can not be calibrated, we say absence of something is zero ie if something dosen't exist it is zero hence it is something which cannot be calibrated... something like null in java.

0 or infinity are not numbers in the true sense of logic leaving aside mathematical equations or sum theories which have rules governing them whic cannot be set aside.

If we consider 0 time as the start of the universe then 0.(0)1 [the number of recurring zeros is again something which cannot be quantified by the prevelant number system] is the smallest number.

if in the above case we say 0 is the smallest number then 0-0 is zero itself which means there has been no event and everything is still where it was as at the start.

similarly 0- 0.(0)1 gives a definite difference which points to the occurance of an event which can be quantified and calibrated accordingly.

There is a concept of shunya and kshana in vedic mathematics. shunya means zero also meaning the absence of something while a kshana means the smallest measurable quantity which i cannot describe in decimals more due to my lack of knowledge than anyting. The two numbers are distinct and have different weightage in the numeric system.

shunya being 0 is the start of the universe while kshana being what i take to be 0.0(0)1 the point on time scale just after the universe is created,the smallest calibration the humans mind can possibly comprehend.

fred rosenberger
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2y = 2x and y = x are different. the resulting graphs end up being the same. the rules of algebra allow us to transform one equation into the other. But we must follow ALL the rules, not just the ones that make life easier, or simpler, or that we feel are wrong. Either you accept the rules of mathematics, or you don't. It's fine if you don't, but then you have to come up with a stable set of rules that DO work.

anyway, one of these rules says, basically, "as long as you follow the rules of division, you can divide both sides of an equation by the same thing. Further, you can also cancel out a common factor in a numerator and a denominator." One of these rules of division, which you don't seem to like, is you cannot divide by 0.

so, in my example, i would say that the complex equation can be transformed to the simpler one for all values of x, EXCEPT when x=5. the trasformation doesn't work at this point. when x=5, i cannot reduce to y=x+2 (or whatever it was), because that would require me to divide by 0.

in your example, everything is fine, becase... wait for it... you are NOT dividing by 0. so there are no holes, one graph doesn't look any different from the other.

if you want a better example, how about some basic phyics equations?

Capacitance = charge / voltage.

if i have 0 voltage, i don't think i can have a capacitance. it just doesn't make sense. However, by your definition, i could have a charge with no voltage, WITH a capacitance.

Jim Yingst
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[Fred]: Capacitance = charge / voltage.

if i have 0 voltage, i don't think i can have a capacitance. it just doesn't make sense.

You can have a capacitance that has 0 voltage and 0 charge, sure. Capacitance can be caluculated from other physical factors - two parallel metal plates have a capacitance between them which is based on their area and separation distance, as well as the material between them. This capacitance is fixed, even if there's no charge on the plates. But if you do put a charge on the plates, the above formula allows you to determine the voltage. Or vice versa.

So according to James' preferred arithmetic, if the charge is 0 and voltage is 0, 0/0 = 0, remainder 0. Which is very much in error for any real capacitor. According to standard arithmetic, 0/0 is underfined, which simply means that we don't know the capacitance unless we calculate it another way. I would say that in most cases (including this one) it's better to know that we don't know something, rather than think that we know something which is actually incorrect (i.e. thinking the capacitance is 0, which is wrong).

[James]: All things being equal would or would not my proposed implementation of division by 0 produce a more consistent rendering of the two graphs?

For x = y and 2x = 2y you'd be fine. But for an equation like

if you evaluate this at x = 5 you'd get 0/0 which (according to you) is 0 with remainder 0. If you plot this point, you'll see it doesn't fit into the rest of the graph at all. The rest of the graph is a line that goes through (0, 7) - except there's a tiny hole there. It, is, as you would say, a zero-width hole, which you may wish to fill in. OK. Except that your division technique fills in the hole with a point at (0, 0), while the hole itself is at (0, 7). And it's extrememly possible you wouldn't even notice the problem unless you actually drew the graph. With standard division rules, you'd notice a problem when you got 0/0 as undefined. This would then signal that you should try analyzing the equation other ways - e.g. by factoring the quadratic expression and canceling the two 7-5 terms. That's easy once you notice the problem. But your proposed division rules don't tell you there's a problem - they just give you the wrong answer.

If that equation looks too contrived (since you know it's pretty much equivalent to y = x + 2 aside from the hole - how about this one?

y = sin x / x

What's the value of y when x = 0?

fred rosenberger
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You can have a capacitance that has 0 voltage and 0 charge, sure.

thanks. I really don't know physics or electronics very well. i just went to a physics web page and grabbed a formula that would (IMHO) help substantiate my point.

James Christian
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thanks. I really don't know physics or electronics very well. i just went to a physics web page and grabbed a formula that would (IMHO) help substantiate my point.

I thought so. Not that I know much about physics either. I'm starting to feel old now. It wasn't that long ago that I studied it at college. Maybe it's the Alzeimer's getting the better of me.

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