Overly Chatty Penguins
The Ready Room => Off Topic => Topic started by: Decimator on February 22, 2004, 03:56:28 PM
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Hmm, divide 2 by zero, i guess i'm doing pretty good. I have an infinite number of times their post...if that makes any sense... Oh well, if it doesn't make sense then it'll just crash your brains with a divide by zero error.
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Technically you can't devide anything by nothing. Dunno why, just one of those 'laws of math'.
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Technically you can't devide anything by nothing. Dunno why, just one of those 'laws of math'.
True Grimm. You can only divide zero by another number, which of course gives you zero. If you divide a number by zero, it becomes undefined. To prove this, get a calculator and divide by zero; you'll get an error. WEWT!
EDIT: Added a word so it made sense!
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Bah, you guys have too much error protection.
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To prove this, get a calculator and divide by zero; you'll get an error.
True, but the error message reads "Infinite Result."
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another way to explain it, how can you divide something into X number of '0-parts'...?
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If I have 1 apple, and I divide that 1 apple between 0 people...
Something like that was the example my pre-calculus teacher gave us.
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If I have 1 apple, and I divide that 1 apple between 0 people...
Something like that was the example my pre-calculus teacher gave us.
.. then you still have an apple, just no one gets it... so the answer would be.... i?
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No, i is the square root of a negative apple.
Or, possibly, applesauce. Higher maths make me hungry.
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No, i is the square root of a negative apple.
Or, possibly, applesauce. Higher maths make me hungry.
But would the square root of -2 be i times the square root of 2? So it wouldn't just be i for the square root of a negative apple.
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If the inability to explain division by zero is sufficient reason to deem it impossible, explain division by fractions, or multiplication by fractions. Explain, also, multiplication and division by negative numbers. Or are they, too, impossible?
And while we are on the topic of i, what is the square root of negative i?
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uuhh........some kind of negative apple sauce?
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No wonder people find maths boring. Gimme chemistry with its brightly coloured liquids, bunsen burners and strange odours any day.
"Chemie ist wenn es qualmt und stinkt
Fysik ist wenn es nicht gelingt."
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I like any science that I can get my hands on and experiment with, whch, unfortunately, hasn't been the kind of science at my school. Most of it is either loads of textbook work (biology) or loads of math work (chem and physics).
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If the inability to explain division by zero is sufficient reason to deem it impossible
I dunno people still believe in God...
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Is there a law somewhere that states that all forum discussions must eventually gravitate towards a religious flame war? I don't really see the point of arguing it, but why even bring it up. You're playing with FIRE, son. FIRE!!!!!
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It seems to me that the only basic mathematical operations that might result in an undefined number are multiplication of zero by infinity, division of zero by zero, and division of infinity by infinity. But perhaps even they result only in the simple number one.
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Zero times infinity would bebe zero, and infinity divided by infinity would be one.
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Zero times infinity would be zero
I can see your reasoning there, as normally any number times zero equals zero. But I am seeing things differently. I am seeing zero as the reciprocal of infinity. This makes perfect sense because the reciprocal of a number that is very close to zero (10^-100, for instance) is a number that is very close to infinity (10^100). Since any number multiplied by its reciprocal equals one, zero multiplied by infinity must equal one.
At first it seemed as if the result should be undefined, because it simply seemed logical that zero times infinity could conceivably equal any real number, but now it seems to make more sense when the result is one.
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civilian you are thinking it terms of limits, where on the graph of y=1/x, the greater x becomes (aka infinity), the closer y comes to 0.
You cannot merely take a limit and start plugging that value in. The limit as x approaches infinity of y=1/x is 0, but that in no way means that (1/infinity) is 0.
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Zero times infinity would be zero
I can see your reasoning there, as normally any number times zero equals zero. But I am seeing things differently. I am seeing zero as the reciprocal of infinity. This makes perfect sense because the reciprocal of a number that is very close to zero (10^-100, for instance) is a number that is very close to infinity (10^100). Since any number multiplied by its reciprocal equals one, zero multiplied by infinity must equal one.
At first it seemed as if the result should be undefined, because it simply seemed logical that zero times infinity could conceivably equal any real number, but now it seems to make more sense when the result is one.
but if o is the reciprocal of infinity
using
oo for infinity
0 = oo^-1
0 = 1/oo
but
0 != 1/oo
on a side note:
oo^-1 == 10E-oo == 0.000000[infinite amount of zeros]0000001
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I'd never see zero as the reciprocal of infinity. I can understand if its Zero = Nothing, Infinity = Everything, but in the end, anything multiplied by zero is still zero.
Also, technically infinity isn't a number, its just the idea that numbers continue on forever. So technically we can't use infinity in math, atlhough I know we do (sum of an infinite sequence, for example).
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I don't see why should infinity be regarded as any less of a number than zero. Both are equally immeasurable.
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Actually, zero seems pretty easy to measure. Take a measuring cup, for example, and... let it sit there. You have zero! Unless you count the air in it, which I am not counting for this example.
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But that is measuring zero in terms of zero; assuming its existence.
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Could you elaborate on that a little bit?
Infinity is not a tangible number. 0 is. Look at the graph of f(x) = 1/x. According to your reasoning (I believe) f(0) = oo/-oo. How does it make sense that for one x value the graph as two values? We all know that this is a function, so such a statement has to be false.
BTW you can't get close to infinity. We sometimes use high numbers to try to predict limits of different functions, but it is impossible to replace infinity with a high number.
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So much controversy over infinity....
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Infinity is not a tangible number. 0 is. Look at the graph of f(x) = 1/x. According to your reasoning (I believe) f(0) = oo/-oo. How does it make sense that for one x value the graph as two values? We all know that this is a function, so such a statement has to be false.
Then perhaps infinity and negative infinity are the same. I have thought about this too, and it makes perfect sense to me. Starting with 2/2 (1), and subtracting from the denominator, 2/2, 2/1, 2/0, 2/-1.
BTW you can't get close to infinity. We sometimes use high numbers to try to predict limits of different functions, but it is impossible to replace infinity with a high number.
The way I see it, infinity and zero are equally difficult to reach. Infinity may be impossible to reach by simply increasing a number, but that is akin to simply increasing the denominator of a fraction in hopes of reaching zero. To reach zero, you must subtract from the numerator. Similarly, to reach infinity, you must subtract from the denominator.
But perhaps this is all only my misguided desire for elegance. I find this possible symmetry between zero and infinity to be irresistible.
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Zero times infinity is zero. Infinity divided by infinity is still infinity, believe it or not. Or that's what I remember from calculus, anyway ;)
And the problem with your reasoning, civilian, is that zero is a well-defined number. The problem with infinity is that by definition it cannot be well-defined.
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I can see infinity divided by itself still being infinity. Put something infinite into something infinite, and it will go in an infinite number of times. However, infinity times zero is still infinity??? That makes no sense. Anything, zero times... is still zero. Was that a typo, or true? And if true, how the HELL...?
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I have no idea what you're talking about (http://www.ews.uiuc.edu/~hunsley/shiftyeyes.gif)
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OMG MOD HAX. You could have at least fixed your other typo:
believe it or not
Quoting that a lot so that when you go back and re-edit your post you will then have to re-edit mine several times over to make yourself look good again. But I didn't reckon on your keen intelligence making a fool of me
So hah!
<3
[size=8]Last edited by BobTheJanitor, 6:30 PM February 25[/size]
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Infinity divided by infinity is a undefined term the same way that 0/0 is undefined. If you have to find the limit of such an equation like f(x) = (3x^2 + 5x)/(9x^2) as x approaches infinity, it would merely be 1/3.
You can't use infinity without limits because infinity is not a tangible number, and can't be substituted in normal algebra. 2/0 does not equal infinity. 2/0 has no solution because no matter how large a number you multiply 0 by, you will never reach 2, or any positive number.
To understand that you must realize that 0*oo does equal 0. Unless you've discovered a new branch of mathematics there is no way you can change that.
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To understand that you must realize that 0*oo does equal 0. Unless you've discovered a new branch of mathematics there is no way you can change that.
The product of zero and infinity being an undefined number is indeed the premise for my arguments, but I do not see why that must be untrue. Perhaps I have, then, discovered a new branch of mathematics.
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Your logic defies basic reasoning. You have to understand that 1/oo is an infinitly small number. It does not equal 0, but it approaches 0. You can't say infinity equals anything because it is not a number.
This can be demonstrated by plugging in increasing numbers. You'll obviously see the trend that that the result gets smaller and smaller.
However, according to you 0 * oo approaches an undefined number. This is not true.
If you plug in ever increasing numbers, you get the same result each time, and the trend shows that this function approaches 0. No matter how infinitly high you make x, y will always be 0.
Granted, when you say 1/oo is 0, I assume you talk about limits, because it is impossible to pull real numbers out of any function with infinity.
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You mention plugging in increasingly large numbers to simulate infinity. But this does not work. I would liken it to plugging in increasingly small fractions in an attempt to simulate zero. Infinity is an infinitely large number, and cannot be reached by simply counting upwards.
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I have never actually seen infinity as a number. I've always seen infinity as something continuing on and on and on and on forever, basically.
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So why can't you plug smaller and smaller numbers to simulate 0? How does that not work?
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Take division of zero by zero. Replace the zeros with x, and plug in any number other than zero or infinity for x. The result is one. But is that the same [edit - or, to word it better, indicative of the] result that would be achieved by dividing zero by zero? It is not; as you stated previously, zero divided by zero is undefined.
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That is the one oddball example where nothing really holds true, just because of the fact 0 divided by 0 is undefined.
In pretty much every other example this holds true. If f(0) = 3, the closer and closer your x values come to 0, the closer and closer your f(x) values will come to 3. Unless you have a jump discontinuity, the value of a function will always be the same as the limit.
My whole point is you can never find values for functions involving 0 in the denominator or oo. You can however find limits. Saying that 1/oo is 0 is absurd, because oo by definition is an undefined term.
I find your treating of infinity like a number is quite odd. Infinity is not a number and cannot be used in algebra in any way shape or form. Infinity is a term used to describe an infinitly increasing number, which can be used in calculus.
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oo is infinitely big
1/oo is infinitely small
0 is the absence of anything
only one can be measured
only one can be used as a mathematical value
but, some calculaors use oo to go around the 'error: overflow' bit
what does teh calculatar say?
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WTH!?!? i leave yONE post alone, i don't read it. JUST ONE!! and you guys go CRAZAY ON ME!!! sheeesh!
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*ahem*
http://mathforum.org/library/drmath/view/62486.html (http://mathforum.org/library/drmath/view/62486.html)
This guy explains it well. Infinity is a concept used when exploring calculus, not a number.
(BTW I'm not Kaiser)
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That is the one oddball example where nothing really holds true, just because of the fact 0 divided by 0 is undefined.
In pretty much every other example this holds true. If f(0) = 3, the closer and closer your x values come to 0, the closer and closer your f(x) values will come to 3. Unless you have a jump discontinuity, the value of a function will always be the same as the limit.
Simulating zero with increasingly small numbers also fails to work when multiplying it by infinity (assuming infinity is always simulated with a number that is equally large as the number that is used to simulate zero is small, or assuming infinity is truly infinite). Likewise, simulating infinity with increasingly large numbers works in all situations except multiplication of it by zero and division of it by itself.
Infinity may not be already defined as a number, but if that is the case then I am defining infinity as a number.
And Black Mage, your calculator tells me exactly what I want to see (oo * 0 = undefined).
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My god you people go so off-topic with this crap... I think we need to make meandering threads, then seal the doors and gas everyone that posted in it :p
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I think we need to make meandering threads, then seal the doors and gas everyone that posted in it :p
That group would seem to include you.
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Ok, after rethinking my calculus a bit I have figured that there are examples where infinity times 0 can be 1, 0, or infinity.
The only examples I could come up with were when it equals 0.
lim (x -> 0) [ e^x / x! ] = 0
or
lim (x -> 0) [ x * ln(x) ] = 0
finally I looked at lim (x -> oo) [ x * 1/x ] = 1
So yes, infinity times zero is in fact undefined.
But 0 is still not the reciprocal of infinity, and 1/0 is a null set.
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Infinity may not be already defined as a number, but if that is the case then I am defining infinity as a number.
To clarify on this, I do not see why it cannot be regarded as a number. Undefinable? Just as you could say one represents a single object and zero represents no objects, you could say infinity represents an infinite number of objects. Intangible? Zero is no more tangible than infinity.
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proof that infinity is not a number:
i have an infinitum of apples
i split it among all lm slot holders
civ, how many apples do you have?
<a civilian> errr ... an infinite amount?
so if i started with oo apples and i have all the slotholders an equal amount of apples (say there are 20) and they each have oo
1 * oo = 20 * oo
divide by oo (since oo is, for this experiment an algebraic quantity)
1 = 20
true
and if you were to say that:
1 * oo = 20 * oo
simplify
oo = oo
true
that proves that oo can not be an algebraic number, as 20 * oo can not equal oo because 20 is not the number of multiplicative identity
as far as algebra goes, oo is a concept and i'm going to find the ascii keystroke-sequence-thing for it so we can actually have the character
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[edit]This is not a reply to Black Mage.[/edit]
In 1/x, plugging in increasingly low numbers to simulate zero will result in increasingly high numbers for the quotient. This indicates that when zero is plugged in, the operation will yield an infinite result. Is this not so?
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And in reply to Black Mage, that is not sufficient proof; I could use the exact same method to prove that zero is not a number.
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there appears to be an infinity chatachter in times new roman, but not an alt-keycode for it
this should be it: ∞ (copy + paste)
and in reply to civ
but i never gave any apples to any regs
1 * 0 = 20 * 0
through the multiplicative zero property as:
1 * 0 = 0
20 * 0 = 0
infinity is not an algabraic quantity as it cannot be quantified or counted
calculus deals with infinitums but most real world math is algebra
1 + 1 = x, etc
<-- begin somewhat large algebraic proof -->
#1, real values
∞ > everything
20 * ∞ = ∞
1 * ∞ = ∞
20 > 1
(20 * ∞) > (1 * ∞)
by defenition ∞ is:
for any real x
∞ = x + 1
therefore anything less than infinity is x
x is real
20 > 1
(20 * ∞) > (1 * ∞)
20 * ∞ = ∞
1 * ∞ is less than ∞
1 * ∞ is therefore real
false
#2 simple algebra
20 * ∞ = 1 * ∞
for any a*b = c*d
a/d = c/b
therefore
20/∞ = 1/∞
reduce
20 = 1
<-- begin somewhat large algebraic proof -->
the second is easier to understand
oh and on topic: as of this posting we have 7,883 posts
well on our way to yet another discussion about finite math!
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∞ > everything
20 * ∞ = ∞
1 * ∞ = ∞
20 > 1
(20 * ∞) > (1 * ∞)
by defenition ∞ is:
for any real x
∞ = x + 1
therefore anything less than infinity is x
x is real
20 > 1
(20 * ∞) > (1 * ∞)
20 * ∞ = ∞
1 * ∞ is less than ∞
1 * ∞ is therefore real
false
Allow me to reword:
0 < all positive real numbers
20 * 0 = 0
1 * 0 = 0
20 > 1
(20 * 0) > (1 * 0)
by definition 0 is:
for any positive real number x
0 = 1/(x + 1)
therefore anything greater than zero is x
x is nonzero
20 > 1
(20 * 0) > (1 * 0)
20 * 0 = 0
1 * 0 = 0
0 > 0
0 is therefore nonzero
false
20 * ∞ = 1 * ∞
for any a*b = c*d
a/d = c/b
therefore
20/∞ = 1/∞
reduce
20 = 1
20 * 0 = 1 * 0
for any a*b = c*d
a/d = c/b
therefore
20/0 = 1/0
reduce
20 = 1
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Allow me to reword:
0 < all positive real numbers
20 * 0 = 0
1 * 0 = 0
20 > 1
(20 * 0) > (1 * 0)
by definition 0 is:
for any positive real number x
0 = 1/(x + 1)
Wrong. For any real number x, 1/(x+1) is nonzero. Therefore the rest of this proof is useless
therefore anything greater than zero is x
x is nonzero
20 > 1
(20 * 0) > (1 * 0)
False. This inequality is only valid for positive numbers. Zero is not a positive number. Therefore the rest of this proof is useless.
20 * 0 = 0
1 * 0 = 0
0 > 0
0 is therefore nonzero
false
20 * ∞ = 1 * ∞
for any a*b = c*d
a/d = c/b
therefore
20/∞ = 1/∞
reduce
20 = 1
20 * 0 = 1 * 0
for any a*b = c*d
a/d = c/b
therefore
20/0 = 1/0
reduce
20 = 1
Unfortunately, this doesn't hold up either. Any rule breaks down at certain points. This rule breaks down when both sides of the equation are undefined as in your example. It can't really be used to prove this. Of course, BlackMage's proof in this case also has some problems, because the "reduce" step is not as simple s he is making it out to be when one is dealing with infinity
see comments^
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Wow. You guys are going a lot more in depth with this than I had thought you would.
I think I'm just gonna keep on thinkin infinity isn't a number, and zero is, and I'll leave you all to your proofs and hypothesis.
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Wrong. For any real number x, 1/(x+1) is nonzero. Therefore the rest of this proof is useless
That is true; but also true is that for any real number x, x + 1 is finite.
False. This inequality is only valid for positive numbers. Zero is not a positive number. Therefore the rest of this proof is useless.
Nor is infinity a positive number.
Unfortunately, this doesn't hold up either. Any rule breaks down at certain points. This rule breaks down when both sides of the equation are undefined as in your example. It can't really be used to prove this. Of course, BlackMage's proof in this case also has some problems, because the "reduce" step is not as simple as he is making it out to be when one is dealing with infinity
That division by zero is not undefined, but rather infinite is one of the things I have been arguing all along. Without it as a premise I do not claim infinity to be a number.
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Maybe I should have said positive values. Infinity is, by nature, positive.
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Maybe I should have said positive values. Infinity is, by nature, positive.
Infinity is as positive as zero. Exceed infinity (from the positive standpoint) and negativity results, as can be seen here: 2/2, 2/1, 2/0, 2/-1. (I am, again, using the premise that division by zero is infinite.)
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Infinity is the largest value possible. Therefore, you cannot exceed infinity by definition, nor can it be anything but positive.
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In a way you can, as I just did it in my example. But regardless, infinity can be approached by either addition or subtraction, which shows that it is not positive.
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Civilian usually you denote positive or negative infinity to show continuously increasing or negatively increasing numbers.
There is a well established difference.
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i'm not going to waste my keystrokes over this...
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Hey guys. 0.999... = 1
Argue that now. >:D
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i'm not going to waste my keystrokes over this...
You just did Black Mage...
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Hey guys. 0.999... = 1
Argue that now. >:D
1/3 = 0.3333333
thus 3/3 is discussably 0.999999999, but it is 1...
Edit: this is caused by our 'change of "unit" system'(lack of better word/expression) per tens, if we had another system a 1/3 wouldn't be a problem, however other fractions would.
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Civilian usually you denote positive or negative infinity to show continuously increasing or negatively increasing numbers.
There is a well established difference.
But that is only when infinity is viewed as a mere concept, designed only to serve such fuctions as that which you mention. When infinity is viewed as the quotient of any number and zero, it becomes apparent that infinity is neither positive nor negative.
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Civilian usually you denote positive or negative infinity to show continuously increasing or negatively increasing numbers.
There is a well established difference.
But that is only when infinity is viewed as a mere concept, designed only to serve such fuctions as that which you mention. When infinity is viewed as the quotient of any number and zero, it becomes apparent that infinity is neither positive nor negative.
Actually, in your case if a limit of a function as x approaches 0 of 1/x, then yes the function approaches both positive infinite and negative infinity.
If the function was 1/(x^2), then it would only opproach positive infinity.
See the difference?
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Yes, but that is similar to how as x approaches infinity of 1/x, the function approaches zero from both the positive and the negative sides, while with 1/(x^2), it approaches zero only from the positive side. And yet zero is not said to have both a positive and a negative value.