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aga
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So, because there is a 'reasonable' solution it is Correct ?
As you understand it, PLEASE explain !
Personally i do not get it at all.


NEMOChemistry
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It took an embarrassingly long time to get that! then i roared .
Because of the subject i was thinking altitude..........
Maths...Hmmm I leave that stuff to clever people.


clearly_not_atara
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Remember that mathematical truths are not facts in the sense of chemistry; rather, any set of mathematical rules might be considered to define its own
world, where the truth and falsity of any statement may freely differ from the truth of similar statements in another world.
The ruleset most students are taught simply happens to be more useful. Complex numbers are important for performing certain kinds of calculations
which are very unique (mostly because of something called holomorphicity), and really it is the simplicity of complex analysis, relative to vector and
matrix analysis, that motivates the idea that "i" can be considered a number and not just some strange object that someone made up in the middle ages.


mayko
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speaking of beautiful videos set in the complex plane:
https://www.youtube.com/watch?v=0z1fIsUNhO4
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"Chemicals, chemicals... I need chemicals!"  George Hayduke
"Wubbalubba dubdub!"  Rick Sanchez


careysub
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This is not true:
$$\frac00=0$$
$$\frac00$$
is simply undefined.
Division is the inverse operation of multiplication.
$$\frac nn=1$$ is simply restating that n x 1 = n, since 1 is the multiplication identity element (just as zero is the addition identity element). If
n=0, then sure enough 0 x 1 = 0.
But n x 0 = 0 for any and all n (not just n=1 or 0) so the inverse operation n/0 cannot be anything other than "undefined".
$$\frac nn=1$$ is not a mathematical (algebraic) law.
$$\frac nn=1$$ unless n=0
is the correct mathematical law, which would have been taught by any competent text or instructor when teaching about division, even at the lowest
grade level.
Mathematics is a humaninvented (or discovered) system of internally consistent rules that happens to map accurately on to the observable Universe,
indicating the the Universe itself follows these internally consistent rules.
The process of mapping human invented mathematics on to the Universe is an interesting process and began with the simplest operation of "counting"
discrete objects, that can be directly perceived.
But increasingly abstract mathematical constructs have been progressively found to map accurately on to the Universe in more complex ways. "Imaginary"
numbers accurately describe the behavior of electrical circuits, and the mathematics of waves generally.
There are many cases of mathematics thought to be too esoteric to have practical importance becoming very important in science.
About that which we cannot speak, we must remain silent.
Wittgenstein
Some things can never be spoken
Some things cannot be pronounced
That word does not exist in any language
It will never be uttered by a human mouth
 The Talking Heads


careysub
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Quote: Originally posted by aga  So, because there is a 'reasonable' solution it is Correct ?
As you understand it, PLEASE explain !
Personally i do not get it at all. 
I gather you are referrring to the BOGUS "proof" that infinity minus infinity equals Pi (or any other number at all).
This involves rearranging "conditionally convergent" series, ones with both positive and negative terms (often alternating). The ordering of the terms
is essential to their correct specification.
You can think of it fairly accurately by saying that it is groups of terms that are converging, and if you change the ordering, you are
changing the groups, and thus the series itself.
I remember as a child my parents made a chore schedule for us four siblings to rotate the chores. My older brother kept trying to trade the most
unpleasant task on his day for another one, saying he was just swapping days. But he kept trying to swap the chore so that he never did it. Reordering
a series can definitely alter its meaning.
It would be more accurate to say the CORRECT solution is, when you study it in detail, inherently reasonable.
[Edited on 26122016 by careysub]
About that which we cannot speak, we must remain silent.
Wittgenstein
Some things can never be spoken
Some things cannot be pronounced
That word does not exist in any language
It will never be uttered by a human mouth
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aga
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It's pretty obvious now that i got this Maths thing all wrong, the main error being trying to regard any system as 'the truth' or in some way
'absolute'.


Marvin
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Quote: Originally posted by careysub 
I gather you are referrring to the BOGUS "proof" that infinity minus infinity equals Pi (or any other number at all).
This involves rearranging "conditionally convergent" series, ones with both positive and negative terms (often alternating). The ordering of the terms
is essential to their correct specification.
[Edited on 26122016 by careysub] 
Fulmen said something similar and this isn't wrong, but I'd explain it differently.
I'd point out that infinity doesn't mean you can stop counting. For an infinite series it's like the terms come along on a conveyor belt. The
'paradox trick' happens by rearranging the terms, yes, but crucially some terms have be left aside while later terms are included. They are not
discarded, they'll be used if you don't stop, and you can tell yourself because no term is discarded that the sum is still valid. It isn't. This is
where the trick happens. As the terms come along the conveyor the pile of terms waiting to be used gets bigger and bigger and that is the error
building up between what the sum really is and what it's being forced to look like.


j_sum1
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IOW, for infinite sums, commutativity is not preserved.


SelfInflicted
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Quote: Originally posted by aga  Mind further boggled : Even Zero is now causing problems.
.....snipped.........
Forgetting 'infinity' as that's just Nuts anyway, if n/n=1 then 0/0 cannot equal 0.
......snipped......
. 
In my book infinity pi and 0 are special cases.
Infinity is not a number. It is abstract.
∞/∞ = ∞ because it would take an infinite amount of iterations to divide infinity an infinite amount of times.
∞/∞ = 1 because infinity equals infinity and an abstact number can be divided by the same abstract number exactly once.
0/0 = 0 because i divided zero pies zero times and still have zero pies.
0/0 = divide by zero error because you cannot divide a pie that does not exist.
0/0 = ∞ because you can divide nothing no times for infinity and still have nothing, except for wasted time which you still do not have so it is
still 0.


careysub
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0/0
∞/∞
are both indeterminate/undefined. They have no value at all.
I explained the reason for the first above ( X * 0 = 0, for all X so the inverse cannot have a defined value).
And infinity is not really a number, and does not have the properties of a number.
In fact there is no single "infinity", they are instead different infinities with different definitions.
There are infinite sets of different sizes. All of them infinite in that the number of elements of any of them is larger than any finite
number, yet some sets are larger than others.
This is actually quite easy to understand. The set of all integers is infinite. Yet for each integer there is an infinite set of real numbers between
it and the next integer.
The set of integers is said to be "countably infinite" and the set of real numbers is "uncountably infinite" because the integers "count" themselves,
but cannot count the real numbers.
This is can be extended into a hierarchy of infinite sets of different sizes. These sizes are called "transfinite numbers".
Although the elaborate apparatus of modern mathematics was beyond me in high school, I had no trouble understanding the basic principles of this
stuff, and the basic proofs that support it.
[Edited on 29122016 by careysub]
About that which we cannot speak, we must remain silent.
Wittgenstein
Some things can never be spoken
Some things cannot be pronounced
That word does not exist in any language
It will never be uttered by a human mouth
 The Talking Heads


SelfInflicted
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I like the way you say that.
Quote: Originally posted by careysub 
0/0
∞/∞
are both indeterminate/undefined. They have no value at all.
I explained the reason for the first above ( X * 0 = 0, for all X so the inverse cannot have a defined value).
And infinity is not really a number, and does not have the properties of a number.
In fact there is no single "infinity", they are instead different infinities with different definitions.
There are infinite sets of different sizes. All of them infinite in that the number of elements of any of them is larger than any finite
number, yet some sets are larger than others.
This is actually quite easy to understand. The set of all integers is infinite. Yet for each integer there is an infinite set of real numbers between
it and the next integer.
The set of integers is said to be "countably infinite" and the set of real numbers is "uncountably infinite" because the integers "count" themselves,
but cannot count the real numbers.
This is can be extended into a hierarchy of infinite sets of different sizes. These sizes are called "transfinite numbers".
Although the elaborate apparatus of modern mathematics was beyond me in high school, I had no trouble understanding the basic principles of this
stuff, and the basic proofs that support it.
[Edited on 29122016 by careysub] 


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