Radio Times Wordfinder Week 51
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Divided by zero
If 0 is an infinite number (and I am having problems understanding this, because I think zero is absolute ie nothing:) then is 3 divided by zero not a larger number than 2 divided by zero (or is that the other way round!)? Not sure if this is maths, science, philosophy or fantasy.
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Have a look here in Wikipedia - fascinating!
http://en.wikipedia.org/wiki/Division_by_zero
In ordinary arithmetic division by zero is meaningless, so 2/0 and 3/0 are not different, because both are meaningless.
http://en.wikipedia.org/wiki/Division_by_zero
In ordinary arithmetic division by zero is meaningless, so 2/0 and 3/0 are not different, because both are meaningless.
(2-part post):
0 is a member of the set of integers and thus, by definition, finite rather than infinite.
Mathematicians don't recognise 'infinity' as a 'number'. The result of dividing anything by zero is simply referred to as 'not defined'. Alternatively, they state n�x tends to infinity as x tends to zero.
Here's an example of why one 'infinity' isn't (contrary to initial appearances)bigger than another one:
Start with the list of all the integers, i.e.
. . ., -6, -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, 6, . . .
Since the list goes on forever (in both directions), there are clearly an infinite number of integers in the set.
Now lets remove every other number from the list. So, if we remove the odd numbers we get this:
. . .-6, -4, -2, 0, 2, 4 6, . . .
0 is a member of the set of integers and thus, by definition, finite rather than infinite.
Mathematicians don't recognise 'infinity' as a 'number'. The result of dividing anything by zero is simply referred to as 'not defined'. Alternatively, they state n�x tends to infinity as x tends to zero.
Here's an example of why one 'infinity' isn't (contrary to initial appearances)bigger than another one:
Start with the list of all the integers, i.e.
. . ., -6, -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, 6, . . .
Since the list goes on forever (in both directions), there are clearly an infinite number of integers in the set.
Now lets remove every other number from the list. So, if we remove the odd numbers we get this:
. . .-6, -4, -2, 0, 2, 4 6, . . .
'Common sense' suggests that, although both lists are infinitely long, the first list must have twice as many numbers as the second one does. (i.e. it appears that we've got two infinities, where one is twice as big as the other).
However, let's now start with the first list again but simply double every number in it. (e.g. we replace 1 with 2, we replace 2 with 4, we replace 3 with 6, etc). The new list now looks like this:
. . ., -12, -10, -8, -4, -2, 0, 2, 4, 6, 8, 10, 12, . . .
Now that list definitely has the same number of elements as the first one, since we've done a 'one-for-one' replacement.
But, hang on a minute, that list turns out to be the same as the second list, which we thought was only half as big as the first one (because we'd deleted half of the elements from the first list).
So if we halve 'infinity', the result is no smaller than what we started with. i.e. one 'infinity' isn't bigger than another.
Chris
However, let's now start with the first list again but simply double every number in it. (e.g. we replace 1 with 2, we replace 2 with 4, we replace 3 with 6, etc). The new list now looks like this:
. . ., -12, -10, -8, -4, -2, 0, 2, 4, 6, 8, 10, 12, . . .
Now that list definitely has the same number of elements as the first one, since we've done a 'one-for-one' replacement.
But, hang on a minute, that list turns out to be the same as the second list, which we thought was only half as big as the first one (because we'd deleted half of the elements from the first list).
So if we halve 'infinity', the result is no smaller than what we started with. i.e. one 'infinity' isn't bigger than another.
Chris
OK sit back and fasten your seatbelt.
There are more than one infinity
In fact there are an infinite number of infinities and some are bigger than others although they are all infinite.
Is your brain hurting yet?
Georg Cantor who came up with this ended his life in an asylum
If you feel brave enough to tred in Cantor's footsteps start here:
http://www.mathacademy.com/pr/minitext/infinit y/index.asp
There are more than one infinity
In fact there are an infinite number of infinities and some are bigger than others although they are all infinite.
Is your brain hurting yet?
Georg Cantor who came up with this ended his life in an asylum
If you feel brave enough to tred in Cantor's footsteps start here:
http://www.mathacademy.com/pr/minitext/infinit y/index.asp
← here comes trouble . . .
At the end of the day there is/are no infinities in reality. Infinity is not a property, attribute or action of any entity. Infinity is nothing more than a mathematical abstraction. An infinite quantity of anything would crowd out the universe leaving no room for anything else, let alone an entity with a propensity to consider infinity.
However many times one divides a line into smaller and smaller segments at some point this process must come to an end leaving a finite number of segments whose length is greater than zero.
Zero has no life of its own. It is simply the absence of anything at all. Something, anything divided by nothing is meaningless and that is the only rational conclusion one can draw regarding a process that has no real application whatsoever.
In this context . . .
∞ = 0
by virtue of the fact that neither exist as real entities but only as mathematical abstractions.
At the end of the day there is/are no infinities in reality. Infinity is not a property, attribute or action of any entity. Infinity is nothing more than a mathematical abstraction. An infinite quantity of anything would crowd out the universe leaving no room for anything else, let alone an entity with a propensity to consider infinity.
However many times one divides a line into smaller and smaller segments at some point this process must come to an end leaving a finite number of segments whose length is greater than zero.
Zero has no life of its own. It is simply the absence of anything at all. Something, anything divided by nothing is meaningless and that is the only rational conclusion one can draw regarding a process that has no real application whatsoever.
In this context . . .
∞ = 0
by virtue of the fact that neither exist as real entities but only as mathematical abstractions.
At the risk of digging myself in a little deeper . . . who knows? A doorway leading into another universe (or in combination, the birth of our own) perhaps?
As I understand it, our concepts of time and space breakdown altogether at the point of a singularity (if not sooner) leaving us with no other options but to speculate about what might become of our reality beyond the visible horison of a black hole, although I see no reason to suspect that either zero or infinity would do it justice and far more fascinating to consider than either or both of the two.
At any rate, when confronted with zero or infinity, I believe the forging of new conceptual tools are required to refine the well worn tools that have served us so well so far. Admittedly a poor answer but thanks for asking all the same.
As I understand it, our concepts of time and space breakdown altogether at the point of a singularity (if not sooner) leaving us with no other options but to speculate about what might become of our reality beyond the visible horison of a black hole, although I see no reason to suspect that either zero or infinity would do it justice and far more fascinating to consider than either or both of the two.
At any rate, when confronted with zero or infinity, I believe the forging of new conceptual tools are required to refine the well worn tools that have served us so well so far. Admittedly a poor answer but thanks for asking all the same.
Hi folks. I left school at 16 with just O level maths so I have deep understanding of the subject. But with better teachers I may have - but that is another story.
However, I love the concept of maths and this discussion is fascinating me. Can anyone recommend a book on this subject which I may find entertaining and challenging but not too technical for someone at my standard?
However, I love the concept of maths and this discussion is fascinating me. Can anyone recommend a book on this subject which I may find entertaining and challenging but not too technical for someone at my standard?
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