The issue with odd and even and zero is that the concept of odd and even (male and female, good and evil) numbers predates the concept of zero being a number by centuries, let alone negative numbers. Numbers were conceived as some sort of spiritual entities, because they did have power to tell you how the world worked, and they definitely existed because four minus three will ALWAYs leave you with one. But what is 'One'? We've merely fitted the old mythology of the number entities to our extended numberline even though it doesn't really apply. The absence of a number, even a very large number, may be something that you can count, but it doens't make it an entity. Not even a spiritual one.
I hold that zero is not even. Neither is it odd. As for negatives, let's just not go there. Modern fabrications, merely!
I was actually about to post a question raised by my mum's maths teacher about proving that 0.99 reccuring equals 1...the three thirds one being the simplest of those discussed!
Now that's just our brains not dealing with infinity very well. If you were to start at the other end, adding 0.3_ three times would give you 1. But 0.9_ is not 1, tho it's a very close number 2.
For your next entry Ed, I highly recommend you rant about infinity for a while. You had me thorougly entertained at Ali's party. I just started you talking with a joke I'd heard and I think an hour later the beer was warm and I was still in stitches :)
:S but isn't the "last" digit of 0.3_ a three? so how does starting at the end of the number give you an answer of 1 as opposed to 0.9_?
also surely the difference between 0.9_ and 1 is only a theoretical one anyway? if you try to subtract 0.9_ from 1, you would get 0.{an infinite number of zeros}1 - more commonly known as zero. hmmm. but then maybe it's like i - it doesn't *actually* exist, but is a useful concept anyway? or like saying there's a difference between E flat and D sharp - there's not really a difference, but it makes some people feel good about themselves to say it and confuse everyone else! :P
The last digit isn't a 3 since where ever you point there is always another one to the right. From that point on the bit you leave out is exactly 1/3 of that point. Decimal just doesn't represent the concept accurately; it misses detail.
So I'd say 0.9_ is not 1, and 1-0.9_ is not 0, it's 0.0_ with a little 1 at the end, wherever you decide to stop representing in decimal. And that little 1 is the bit you leave out when you think about 0.3_ * 3 in terms of adding from left to right. At some point you stop putting in detail. It can't be represented accurately in decimal so doing math with it is going to be flawed. That's how I think of it anyway. *sits in armchair*
So yeah, I'd agree like i, with any recurring number.
E flat and D sharp, there's a calculable difference tho, it depends entirely on the key and temperament, or so the musos tell me :)
Well if you're pointing at a piano then in different chords the space between intervals is ever so slightly different, even though the D flat and E sharp is exactly the same note. So two major chords in different keys can have a very different colour -- in that case the note is the same but the chords are different ( apart having a different tonic of course! )
And if you pick two different signatures and tune perfectly for them then E flat and D sharp can be a different note.
I'm really late in posting this reply.eddie_ausJune 26 2006, 14:21:17 UTC
Anyway, 0.999... recurring = 1 as pointed out earlier because 1 - 0.9... = 0.0..(infinite num of 0)..01, which is equal to zero.
Strictly in maths terms, we define the sequence of numbers a_1 = 1-0.9 = 0.1 a_2 = 1-0.99 = 0.01 a_3 = 1-0.999 = 0.001
a_i = 1-0.9..99 = 0.0..01, where there are i-1 zeros between the decimal point and the 1.
Then, we have a convergent sequence, and the sequence of a_i s converge to 0. Hence 1 - 0.99.... = 0, and we have shown that 0.9 recurring is equal to 1.
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We've merely fitted the old mythology of the number entities to our extended numberline even though it doesn't really apply.
The absence of a number, even a very large number, may be something that you can count, but it doens't make it an entity. Not even a spiritual one.
I hold that zero is not even. Neither is it odd. As for negatives, let's just not go there. Modern fabrications, merely!
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For your next entry Ed, I highly recommend you rant about infinity for a while. You had me thorougly entertained at Ali's party. I just started you talking with a joke I'd heard and I think an hour later the beer was warm and I was still in stitches :)
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but isn't the "last" digit of 0.3_ a three? so how does starting at the end of the number give you an answer of 1 as opposed to 0.9_?
also surely the difference between 0.9_ and 1 is only a theoretical one anyway? if you try to subtract 0.9_ from 1, you would get 0.{an infinite number of zeros}1 - more commonly known as zero.
hmmm. but then maybe it's like i - it doesn't *actually* exist, but is a useful concept anyway? or like saying there's a difference between E flat and D sharp - there's not really a difference, but it makes some people feel good about themselves to say it and confuse everyone else! :P
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So I'd say 0.9_ is not 1, and 1-0.9_ is not 0, it's 0.0_ with a little 1 at the end, wherever you decide to stop representing in decimal. And that little 1 is the bit you leave out when you think about 0.3_ * 3 in terms of adding from left to right. At some point you stop putting in detail. It can't be represented accurately in decimal so doing math with it is going to be flawed. That's how I think of it anyway. *sits in armchair*
So yeah, I'd agree like i, with any recurring number.
E flat and D sharp, there's a calculable difference tho, it depends entirely on the key and temperament, or so the musos tell me :)
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Well if you're pointing at a piano then in different chords the space between intervals is ever so slightly different, even though the D flat and E sharp is exactly the same note. So two major chords in different keys can have a very different colour -- in that case the note is the same but the chords are different ( apart having a different tonic of course! )
And if you pick two different signatures and tune perfectly for them then E flat and D sharp can be a different note.
it's true.
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which is equal to zero.
Strictly in maths terms, we define the sequence of numbers
a_1 = 1-0.9 = 0.1
a_2 = 1-0.99 = 0.01
a_3 = 1-0.999 = 0.001
a_i = 1-0.9..99 = 0.0..01, where there are i-1 zeros between the decimal point and the 1.
Then, we have a convergent sequence, and the sequence of a_i s converge to 0. Hence 1 - 0.99.... = 0, and we have shown that 0.9 recurring is equal to 1.
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