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About LinkBacksWht is 0 * 0 and what results is giving to you all?
January 19th, 2008, 06:29 PM
Wht is 0 * 0 and what results is giving to you all?
January 19th, 2008, 09:45 PM
Zero times zero is zero. Multiplication can be looked at as an iteration of the addition operator. Zero times zero means you're adding zero to itself zero times, yielding zero. Why do you ask?Originally Posted by SolomonGrundy
And 0 : 0 is 0 i see ... How can 0 be addaed and multyply or put in an equation to become 1?
If by 0:0 you mean 0/0, i.e. 0 divided by 0, then no, 0:0 ≠ 0. Division by 0 doesn't make sense, in fact. Division is supposed to be the inverse of the multiplication operator. In mathematical terms, we have that:
a:b = c
means the same thing as:
a = bc
and we want b to be uniquely determined by a and c. The expression a:0 for a ≠ 0 makes no sense, as otherwise we would have a = 0c = 0. And the expression 0:0 makes no sense, as 0 = 0c for any value of c.
This may seem like a flaw of our number system; it may seem like we should be able to define 0:0 or 1:0. But we shouldn't. In fact, it's pretty essential to mathematics for these expressions to not make sense.
1 divided by zero to me means there is nothing to divide by therefore there simply is no answer, with nothing to divide by, no action can take place thus no result can bo obtained. Same for multiply.
You're suggesting multiplication by 0 shouldn't be meaningful?
A while ago I've tried to construct a kind of 'nulliair' mathematics.
My thought process was as follows:
There's a subtle difference between the divisions 0/0 and 1/0. The first is 'undetermined' and the latter is "undefined". You might reason that 0 x 0 = 0 and therefore 0 / 0=0, but there's clearly no definable solution like 1/0 = 0 since 0 x 0 is not equal to 1.
As an analogy to i^2 = -1, I then just tried to define 1/0 as follows:
j = 1 / 0 --> 0j = 1 --> 1/j = 0
And produced the following calculation rules:
Addition:Subtraction:
- aj + bj = (a + b) j for all a<>0 and b<>0
- aj + bj = 1 + bj for a=0, b <> 0
- aj + bj = aj + 1 for b=0, a <> 0
- aj + bj = 2 for a=0 and b=0
Division:
- aj – bj = (a - b) j for all a<>0 en b<>0 and a<>b
- aj – bj = 0j = 1 for all a=b
- aj – bj = 1 - bj for a=0, b <> 0
- aj – bj = aj - 1 for b=0, a <> 0
- aj – bj = 1 - 1 = 0 for a=0 en b=0
Multiplication:
- aj / bj = (aj * 1/j) * 1/b = (aj * 0) * 1/b = 1/b for all a<>0 and b<>0
- aj / bj = (aj * 1/j) * 1/b = (0j * 0) * 1/b = 0 for a=0, b <> 0
- aj / bj = (aj * 1/j) * 1/b = (aj * 0) * 1/0 = j for b=0, a <> 0
- aj / bj = (aj * 1/j) * 1/b = (0j * 0) * 1/0 = 1 for a=0 and b=0
Powers:
- aj * bj = (a*b)j^2 for all a<>0 and b<>0
- aj * bj = bj for all a=0, b <>0
- aj * bj = aj for all b=0, a <> 0
- aj * bj = 1 for all a=0 and b=0
(j^a):(a^j):
- j^a * j^b = j ^(a+b) for all (a+b)>0
- j^a * j^b = 1 for all (a+b) = 0 (a=-b or (a=0 and b=0))
- j^a *j^b = 1/j^(a+b) for all (a+b) < 0
- j^a / j^b = j ^(a-b) for all (a-b)>0
- j^a / j^b = 1 for all (a-b) = 0 (a=b or (a=0 and b=0))
- j^a / j^b = 1/j^(a-b) for all (a-b) < 0
Goniometric:
- a^bj * a^cj = a ^ ((b+c)j) for all (b+c) <>0
- a^bj * a^cj = a for all (b+c) = 0
- a^bj / a^cj = a ^((b-c)j) for all (b-c) <>0
- a^bj / a^cj = 1 for all (b-c) = 0
Commutative?
- tan (90 + n*180) = j (n = 0, 1, 2, ...)
- sin (90 + n * 180) = j * cos (90 + n * 180)
- cotan (90 + n*180) = 1/j = 0
For all a and b it's true that:
- aj + bj = bj + aj
- 2j + 0j = 0j + 2j (= 1 + 2j)
Distributive over addition? (hold your breath till the end Serpicojr! :wink: )
- aj * bj = bj * aj
- 1j * 2j = 2j * 1j (= 2j^2)
For all a, b and c it's true that:Associative?
- (aj + bj) * cj = aj * cj + bj * cj.
- (0j + 0j) * 0j = 1*1 + 1*1 (= 2)
- (1j + 1j) * 1j = 1j*1j + 1j*1j (= 2j^2)
- (0j + 1j) * 0j = 0j * 0j + 1j * 0j (= 1 + j)
For all a, b and b it's true that:
- aj + (bj + cj) = (aj + bj) + cj
- 0j * (1j * 2j) = (0j * 1j) * 2j (= 2j^2)
I've even plotted the first ever 'nulliair plane' (with the imaginairy axis cutting the real axis in 1).
- aj * (bj * cj) = (aj * bj) * cj
- 0j * (1j * 2j) = (0j * 1j) * 2j (= 2j^2)
- 2j * (3j * 4j) = (2j * 3j) * 4j (= 24j^3)
Of course the main problem is in the 'distribution over addition' where (0+0) j = 0j + 0j = 1 + 1 = 2 but also 0j = 1. So: 1=2 or any number you'd like...
I've never found a way to solve this problem other than making surreal assumptions like:
1. accepting that in the nulliair world it's just true that 1=2, 1=3 ... 1=n
2. assuming that (a + b) j is only "distributive over addition" for a<>0 and b <> 0. In the special case that a=0 and b=0 then (a +/- b) is always equal to 0j.
3. making all zero's unique in the nulliair world and assigning special properties to them.
4. dropping the whole idea...
Hmmm confusing your using 'j' as to engineers it is the square root of -1
I think you've hit the problem on the head. If you want lots of properties to hold true, you're going to get "nonsense" statements like 0 = 1. This is fine and dandy if you don't care about doing usual arithmetic in your new number system. But I have a feeling that you do. So... you have to drop some properties. Arithmetic with j can't satisfy all the properties you might hope it does. Evidently, the problem is with this argument:
1 = 0j = (0+0)j = 0j+0j = 1+1 = 2
What did you we here? Two things: 1 = 0j and distribution over multiplication by j. So you have to give up one of those guys. You want 1 = 0j. So you have to get rid of distribution over multiplication by j. I don't think this gets you off the hook, though--you've included a lot of properties, and so there's a lot of potential for inconsistency. In any case, things are getting kind of complicated, and arithmetic operations should aim to be simple (in my world anyway). At the end of the day, I would just give up on trying to divide by 0.
I recall looking at something recently (year or two ago?) where some guy made up rules for arithmetic that included division by 0, plus and minus infinity, and one extra number representing "undefined". Basically, any time he tried to define an operation that necessarily led to an inconsistency, he let the result be undefined, and any operation involving undefined resulted in undefined. But this is kind of a cop out and is no more useful than just saying that an expression is undefined.
Let me also add that a lot of mathematical subjects have very good ways of dealing with the notion of division by 0--complex function theory, for example. I could discuss this at greater length.
Think I found the guy that you mentioned.Recall looking at something recently (year or two ago?) where some guy made up rules for arithmetic that included division by 0, plus and minus infinity, and one extra number representing "undefined".
http://www.badscience.net/?p=335
His theory was as follows:
1/0 = inifinity
-1/0 = -infinity
0/0 = NaN (Not a Number) let's call it @
He then uses the following logic to solve a '1200 year old problem' of 0^0:
0^0
= 0^(1-1)
= 0^1 * 0^(-1)
= (0/1)^1 * (0/1)^(-1)
= 0/1 * 1/0
= 0/0
= @
I've read about the Riemann Spheres and also about 'wheels'. Is that what you're referring to?Let me also add that a lot of mathematical subjects have very good ways of dealing with the notion of division by 0--complex function theory, for example.
Yes, yes, and yes!
That's precisely the guy I was talking about. To be honest, the mathematics community isn't impressed by his work. I mean, it's all valid, and he clearly knows math, but there's a question about the importance of the work. It's a fine intellectual exercise to go through, but it's not going to cure cancer or prove the Riemann hypothesis.
The Riemann sphere is indeed an object which allows you to talk about functions which take on the value infinity (which is really the same thing as dividing by 0 in this scenario).
Actually, I wasn't really talking about wheels, but wheels are another solution to the problem of dividing by 0. I haven't really seen them used much, but that doesn't mean there's not some research group out there diligently working out wheel theory. If you want an algebraic/arithmetic/nonanalytic/nongeometric approach to dividing by 0, I'd say this is the way to go.
Just read his paper (it's in pdf on his site) and the idea of trying to avoid a 'division by zero' error in computersofware by constructing an elegant set of new rules, could have some practical use.I mean, it's all valid, and he clearly knows math, but there's a question about the importance of the work. It's a fine intellectual exercise to go through, but it's not going to cure cancer or prove the Riemann hypothesis.
Dr James Anderson, from the University of Reading’s computer science department, says his new theorem solves an extremely important problem - the problem of nothing. “Imagine you’re landing on an aeroplane and the automatic pilot’s working,” he suggests. “If it divides by zero and the computer stops working - you’re in big trouble. If your heart pacemaker divides by zero, you’re dead.”
This is a famous real life example:
On September 21, 1997, a divide by zero error in the USS Yorktown (CG48) Remote Data Base Manager brought down all the machines on the network, causing the ship's propulsion system to fail.
cosine of 0 is 1 right ?
sin 0 = 0: cos 0 = 1
why is that so?
True, avoiding these sorts of mistakes is worthwhile. I don't know and would be surprised if his solution was the first to this problem, though. And you don't need a new number system to accomplish this--you can just design your programs not to crash if division by zero is attempted.
True indeed.
so if cos 0 =1 then the solution to 0:0 is in the middle no?
You never get the ratio 0:0 when working with triangles unless all of the vertices are the same point, which is very degenerate. In particular, when working with right triangles, we always assume the hypotenuse has positive length, and often we assume that it is 1 (so that the other two side lengths can be described by sin and cos). So the fact that cos 0 = 1 says nothing about what the ratio 0:0 could possibly be. And I don't understand what "in the middle" refers to... in the middle of what?
"in the middle" is undefined and the undefined is all over the place in math
Can some of you define this undefined with a good ecuation?
so if 2 lines connect in point x and the point x is the 0 point named origin do we have 0 = 1 or just 0 = 0 meaning that the 2 lines do not connect ?
...huh?
in a function
I really have no idea what you're trying to say.
division and multiplication by 0 doesn't make sense, as there is no such thing as a void in the universe.
the void is a human construct.
Well, then, it's a good thing math is a human construct and is not tied to the physical universe!
Oh, and multiplication by 0 makes sense.
don't exactly understand this. The point of intersection is when the equations for two lines are equal to one another. Also isn't calculus designed to solve the problem of dividing by 0. Well, not necessarily solve, but determine the limit of an equation when x = 0.
To some degree, yes. Differentiation finds instantaneous rates of change. To calculate such a rate, say we're trying to find the velocity of an object, we can't just divide the change in position by the change in time, because instantaneous implies the change in time is 0. Calculus (limits, specifically) allows us to come up with a way of circumventing this problem by letting an instantaneous rate of change be approximated by average (usual) rates of change over smaller and smaller changes in time.
did you get it now?don't exactly understand this. The point of intersection is when the equations for two lines are equal to one another. Also isn't calculus designed to solve the problem of dividing by 0. Well, not necessarily solve, but determine the limit of an equation when x = 0.
I am talking about applying some rules and omiting other rules.
so 1=0 or 0=0 in that case ?
So Solomon, what would be the conjugate of that expression?
what answer do you want the short version or the long one?
either will do mate, lets have them both then!
if 2 lines connect in point x and the point x is the 0 point named origin do we have 0 = 1 or just 0 = 0 meaning that the 2 lines do not connect ?
this is the long one
and the short one
How can you make 0= any number or at least 1
it's easy to make something out of nothing as many of your excellent threads have shown!
i do not see your point
explain !
i guess you can add a number x to 0 and get any number ...
so there seems to be no natural value for 0at power 0 can be any nonnegative number, or infinity, or fail to exist .
Modern textbooks often define 0at power 0 = 1. For example, Ronald Graham, Donald Knuth and Oren Patashnik argue in their book Concrete mathematics.
What is the valid option?
0<sup>0</sup> is defined to be 1 by convention. This makes writing formulas more easily. For example, there is the power series for the exponential:
e<sup>x</sup> = <sub>n=0</sub>∑<sup>∞</sup> x<sup>n</sup>/n!
It'd be annoying to have to separate the first term from the rest, i.e.:
1+<sub>n=1</sub>∑<sup>∞</sup> x<sup>n</sup>/n!
So that mean that is not so if it is by convension ?
0 at power 0 is 0 x 0 no?
0<sup>0</sup> can be whatever you want it to be. It depends upon context. Most of the time, it makes sense to define it to be 1, or rather, to define x<sup>0</sup> to be 1 no matter what value x takes. But if we were in a situation where we wanted:
x<sup>0</sup> = lim<sub>r->0<sup>+</sup></sub> x<sup>0</sup>
then we should define 0<sup>0</sup> = 0.
i see so 0=1 whenever we like it ... not fine not fine at all ...
No, you're not getting the point.
Serious question, and I don't mean to insult. Is English your second language?
I get the point .
I am just saying that some parts of our math do not fit well is something missing.
No, you don't get the point, or you wouldn't have concluded that I was saying that we can set 0 = 1 if we want. I said nothing of the sort. I said that some mathematical expressions mean different things in different contexts.
ok i agree with you on that .
I did not concluded nothing just state that 1=0 as well as 2=1 can be true in some mathematical expressions
x = 1
Therefore: x² = x
x² - 1 = x -1
Factorising: (x - 1)(x + 1) = x - 1
Dividing through: x + 1 = 1
Substituting: 2 = 1
And this is one reason why we cannot divide by 0.
if 0/0 can be any number or infinty why it cannot be 1 then
0/0 is 0 you wish to divide nothing by nothing therefore no division takes place hence no answer.
did you read all this post?
0*0=0
one solution yes
so what do you think is another one then ?
If 0 can be any number why not?
0 cannot be any number 0 is just that, 0.
in some cases 0 can be any number
well show a proof then
start with nothing and see if you can make it into something then make a few 500 dollar bills!
0x1 = 0
0x2 = 0
so
0x1 = 0x2
and by division with 0
0/0 + 1 = 0/0 +2
so we have
1=2
or
0/0 - 0/0 =2 -1
and
0 = 1
:x
with no division by 0 is after or ...:x
Division by zero is not allowed. It's the 11th commandment.
i was just showing as in some cases 0 can be any number
I don't think I agree with that....
Cheers
why not?I don't think I agree with that....
Cheers
It's in the math you just gave...why not?I don't think I agree with that....
Cheers
dividing by zero gives nonsensical answers.
If you have two real numbers, they are either equal, or one is greater than the other.
0<1<2
Cheers
yes but if some say 0/0 = 0
then
0x1 = 0x2
so i put
0/0 =0 there
and
i have
0/0 x1 =0/0 x 2
0x1/0=0x2/0
0x0 x 0x2 = 0x0 x 0x1
so we have
0(0+2) =0(0+1)
that will mean
0(0+2)/0(0+1)=1
and there is
0+2/0+1 =1
so
0+2=0+1
0=2-1
0=1
I say 0/0 is undefined, as is x/0 where x = any number.
Check this out:
http://www.thescienceforum.com/1-%3D...rong-3629t.php
Cheers
that is what i am saying
undefined = infinty
so 0/0 can be any number
Seriously, Solomon, are you just trolling?
Okay...
more appropriately, 0/0 is an "indeterminate form."
Plus, x/0 = infinity is used for practical purposes, but it is not strictly true.
And...
there are different degrees of infinity!
Check this out;
http://www.thescienceforum.com/viewt...r=asc&start=30
Cheers
no
i just want to show the problems in your days math.
As opposed to the math of your days?
The only thing you're exposing is your inability to carry on meaningful discussion.
January 26th, 2008, 08:23 AM
It's clearly nil. (Ask me for the proof) But what is more interesting is the pattern that leads to infinity.
vis ...... 1/5 = 0.2
............1/4= 0.25
........... 1/0.5 = 2.0
............1/0.001 =1000.
The smaller the number divided into 1 the greater the answer. So does this mean that 1/0 = an ifinitly large number ~ infinity itself!
Is infinity a number? Is it part of the normal number sequence?
(Probably not 'cos you might then get infinity + 1 being the next logical value in the sequence).
So does infinity appear in any sequence of any description at all or not?
Not sure if this has been said, but 0<sup>0</sup>=1
Yeah, I said that earlier, although it's important to note that this is not so much fact as it is convention.
(a exp. 0)=1
0 is 0 and this cannot change.
You also cant divide with 0 and then consider the result as a number and then add or multiply with this, because division with 0 does not give you a number, it gives you infinity itself, as harryschneider said!
February 19th, 2008, 11:44 PM
It's interested that you said that because:It's clearly nil. (Ask me for the proof) But what is more interesting is the pattern that leads to infinity.
vis ...... 1/5 = 0.2
............1/4= 0.25
........... 1/0.5 = 2.0
............1/0.001 =1000.
The smaller the number divided into 1 the greater the answer. So does this mean that 1/0 = an ifinitly large number ~ infinity itself!
Is infinity a number? Is it part of the normal number sequence?
(Probably not 'cos you might then get infinity + 1 being the next logical value in the sequence).
So does infinity appear in any sequence of any description at all or not?
lim x-> 0 (x/x) = 1
limi x-> 0 (x/0) = undefined
lim x-> 0 (0/x) = 0
lim x-> 0 (1/x) = (+/-) infinity
Of course, limits are not the same as solving the equation
I can't believe this thread has made 6 pages.
why not?I can't believe this thread has made 6 pages.
All is here is the truth
No, all is here is stupid!
no it is not, if all here is stupid then all life is stupid and by default you are stupid too
If we define 1 := S(0), then b + 1 = b + S(0) = S(b + 0) = S(b). That is, b + 1 is simply the successor of b.
If we interpret the natural numbers as "excluding 0", and "starting at 1", the definitions of + and × are as above, except that we start with a + 1 = S(a) and a × 1 = a.
That's about the first sensible thing you've said in this discussion. Unfortunately, it's also a complete non sequitur, so I have no idea what point you're trying to make by saying it.
Umm... are you saying that you can cancel nothing and nothing...
as in saying
(1/0)/(0/1)=1...
what if you define 0 as 0/4546...
then the answer would be 1/4546...
it all really depends, since zero is such a wierd number, along with one...
Isn't that something...
the kindergarteners and preschoolers think that zero and one are the easiest to work with,
and the mathemeticians and scientists of the world find zero one of the most mysterious things in life...
Maybe, they should switch places...
No, mathematicians have a pretty good handle on zero.
It's quite weird to say that 0 is mysterious to many mathematicians.
Mathematics is defined by men so what do you mean should be the confusing about 0, the definition?
0 is weird indeed. Considered as the complex number 0+0i, it is (a) purely real because its imaginary part is 0, and (b) purely imaginary because its real part is 0.
0 is the only entity in the universe that is both real and imaginary!!
0 is divisible by every number!
It is the only real number that is neither positive nor negative. 8)
8/0 is a legitimate ratio. In the next cycle it could be 7/2 and the next cycle 6/1.
Sincerely,
William McCormick
You lost me there.
William McCormick’s posts make me think of Gratiano in Shakespeare’s The Merchant of Venice:
Bassanio: Gratiano speaks an infinite deal of nothing, more than any man in all Venice. His reasons are as two grains of wheat hid in two bushels of chaff: you shall seek all day ere you find them, and when you have them, they are not worth the search.
(Sorry William, but I feel I just have to say it.)
A machine is set to perform some x,y movement. It is done by threaded rod.William McCormick’s posts make me think of Gratiano in Shakespeare’s The Merchant of Venice:
Bassanio: Gratiano speaks an infinite deal of nothing, more than any man in all Venice. His reasons are as two grains of wheat hid in two bushels of chaff: you shall seek all day ere you find them, and when you have them, they are not worth the search.
(Sorry William, but I feel I just have to say it.
Screw, one turns eight times, screw two turns 0 times. The ratio would be 8:0, x:y movement. In this case the machine only moves on the x axis. But the ratio of screw turns to move the machine is correct. It is self explanatory. The next cycle the machine moves 7:2 then 6:1
You could also create a ratio like that for a ship and its screws. Right/Left screw 8:0 all ahead for 30 seconds. This would produce a preset left turn of the ship with rudders set straight, broken off, jammed or cut off from control.
It is just a ratio, it is eight to zero, it is not the same as eight to one. Power will be infinitely greater on the right side of the ship.
Sincerely,
William McCormick
Thanks for explaining, now I understand better (though still not completely). However, it seems you’ve completely missed the point of this thread. Some people here are trying to make sense of (or even define) “8/0” as “8 divided by 0”. You are merely using “8:0” as your own notation for something else totally unrelated.
I just reread through this thread, and I just have to say... oy vey.
In that second scenario with the ship and screws you need a time frame as well. Like 8 turns a second or 480 rpm, for 30 seconds.
To make use of any of those ratios you need to know what the rpm's are, that is a time frame. Then you could calculate how much of a turn or arc would be created, by any ratio over a given time.
But certainly 8/0 is a valid ratio. Eight is infinitely divided by zero. The ratio is eight right screw turns to zero left screw turns at what ever rpm. Right is Starboard and Left is Port.
When you are coming into a local channel from the ocean the rule is RRR, Red, Right Returning from sea. Meaning as you are heading back to port through inlets, channels and bays, you are to keep the red buoy on your right hand side. If there are only green buoys or poles with green arrows, you keep them on your left hand side.
But the ratio is the ratio. And it is infinity.
Sincerely,
William McCormick
No, I mean that 8/0 means that screw x, is turning eight times, while screw y is turning 0 times. That means that the ratio is infinite, screw x will move infinitely more then screw Y. It is totally correct and real. Nothing even out of the ordinary or strange about it.Thanks for explaining, now I understand better (though still not completely). However, it seems you’ve completely missed the point of this thread. Some people here are trying to make sense of (or even define) “8/0” as “8 divided by 0”. You are merely using “8:0” as your own notation for something else totally unrelated.
This scenario needs no time frame, it could happen at its own pace. The ratio will be the same. When x turns 8 times the ratio is complete.
No misunderstandings. This is real stuff.
Sincerely,
William McCormick
but when considering ratios we see that 1:2 = 2:4 = 256:512 however none of these are equal to 3:2. but with something like 8:0 we find that the ratio 8:0 = x:0 for any and all values of x, thus the ratio takes on no defined value or is indeterminant. this is far from saying that the ratio is infinite
William, Jane hit the nail on the head: you're talking about ratios when you write 8/0, other people were talking about the mathematical operation of division when they wrote 8/0. The former makes sense. The latter does not. We're not saying you're wrong--in fact, what you're saying makes sense. But your points aren't relevant to the discussion about division.
I am saying that with the Starboard engine or screw running at full or 8 turns a second, and the left engine at zero, that an infinite difference is occurring. Because it is an infinite ratio. It is a ratio but taken to the end of the ratios possible. To infinity. It is a one dimensional ratio. Even though it is a real ratio.
It would be like two twins one dies during birth. The one that lives is infinitely aging, compared to the one that did not live. You will see at any age the living twins ratio is an infinitely changing ratio, instead of a 1/1 ratio. As long as the one twin is alive, the actual ratio of aging will be infinitely changing, 1/0, 2/0, 3/0 4/0. But if the both twins were alive, it would go 1/1, 2/2, 3/3, 4/4.......
And if we got math in order we would live naturally to well over 135 years old. Less stress.
On the other end of the spectrum if the Port engine is running at full, that would be the opposite end of the spectrum. However the ratio of the two screws can be 8/0. It is just a very real ratio. That you would have to calculate into the total ratio of left and then right engine running at full and shut down. To measure the total capabilities of the ship.
How would you show the ratio of two variable speed screws when one is at a stand still?
8/1 is not the same as 8/0 in actual measuring of ratios. But both are real ratios, they are exact ratios. Everyday ratios. When the ration is 8:1 one screw will turn seven more times a second then the other. When the ratio is 8:0 one screw will turn 8 more times a second then the other.
If you do the math on 8/1 or 8/0 it comes out right. Over six seconds 8/1=48/6 and 8/0=48/0 it is just real ratios.
Sincerely,
William McCormick
A fraction is a ratio. We just often convert them into a number for use in a formula or to get a whole number they represent. To make it easier to see.
We used to do everything by hand and fractions are pretty easy and fun.
http://www.rockwelder.com/Flash/Frac...Fractions.html
Sincerely,
William McCormick
I see your point in our universe in with we all live a division by 0 is te basic of our existance.
Until this days we as human race we did not get the simple answer , so for us to make this simple 0/0 is an error, that is why computer have +0 and -0 for them to function ... is this ringing some bells for you
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