# 1 is 0.9999999999999............

Tach to arfa brane;3138834... Let's try a simple exercise. Start with 0.n 1. Multiply be [tex said:
10^{-1}[/tex]. You get 0.0n, right? In what direction from the decimal point did zero propagate? (hint: you were taught the answer in 5-th grade)
2. Multiply again by $$10^{-1}$$. You get 0.00n, right? In what direction from the decimal point did zero propagate?

You miss the point that the decimal notation per se is represented as 00000.000000 to imply infinite PLACES extending in both directions from the transition zone where the DECIMAL POINT is included in the notation of the system PER SE.

However, when we actually USE the decimal system notation for ACTUAL STRINGS, then the infinity of zeros is NOT continuous because the actual significant digits OCCUPYING any one or more of those places ALWAYS INTERRUPTS the infinity-of-zeros continuity of the notation per se.

Hence the extension of zeros to infinity from the decimal point (for a FRACTIONAL string) is always interrupted by the 9 in .9 (9/10ths), and by the .09 (9/100ths) etc, no matter how many times you divide by ten.

Furthermore, in that FRACTIONAL string, the only effect on the 'value' of the fractional string (to change it from 9/10ths to 9/100ths) is brought by the LEADING ZEROS introduced into the string FROM the decimal point BUT in front of the 9 (or whatever string of significant digits involved in the FRACTIONAL value).

And once you see that LEADING zeros are important to the actual value of a fractional string, then the leading zeros extends to infinity UNINTERRUPTED and TO THE LEFT of the fractional string, and hence the zeros to the left extend past the decimal point notation symbol and CONTINUE UNINTERRUPTED to infinity to the LEFT of the decimal point (just as arfa brane is explaining to you, again).

Ok?

Based on your post, you missed it , since you introduced an unrelated subject
No, didn't miss your point, but it's clear you missed mine.

What does this have to do with the discussion?

No, didn't miss your point, but it's clear you missed mine.

I doubt it but I will give you the benefit of the doubt, what's your point?

5 , now what?

You miss the point that the decimal notation per se is represented as 00000.000000 to imply infinite PLACES extending in both directions from the transition zone where the DECIMAL POINT is included in the notation of the system PER SE.

However, when we actually USE the decimal system notation for ACTUAL STRINGS, then the infinity of zeros is NOT continuous because the actual significant digits OCCUPYING any one or more of those places ALWAYS INTERRUPTS the infinity-of-zeros continuity of the notation per se.

Hence the extension of zeros to infinity from the decimal point (for a FRACTIONAL string) is always interrupted by the 9 in .9 (9/10ths), and by the .09 (9/100ths) etc, no matter how many times you divide by ten.

So what? This is basic arithmetic, what does it have to do with my discussion with arfa? The extension of zeroes is to the right of the decimal point. Yes or No?

Tach said:
me said:
The 0 in 0.9 is still equivalent to an infinite string of zeros, and my corrected statement still stands.

This is a different issue, trivially correct but can you stick to the point and not resort to diversions?
This was the point I made initially:
me said:
the single zero on the left of the decimal point must really be like a string of zeros extending infinitely to the right
But it should say "extending infinitely to the left", as Undefined pointed out. The left of what? Of the decimal point (maybe I should have included that).

Or is that still confusing?

???

So what? This is basic arithmetic, what does it have to do with my discussion with arfa? The extension of zeroes is to the right of the decimal point. Yes or No?

No. The extension of zeros that matter are to the left of the fractional string generated at every division by ten; and these leading zeros extend to the left to infinity past the decimal point symbol. Any zero continuity right of the decimal point in a fractional string is INTERRUPTED by that string. And any trailing zeroes to the right of the zero-continuity-INTERRUPTING fractional string at any stage (of division process generating the actual string) do NOT affect the string value ITSELF at any stage. The only reason to include these trailing zeros would be as I already explained; ie, to convey further information to the reader regarding the level of precision/accuracy any particular measurement result string value was measured to. Please read again my earlier posts in context. Thanks.

I suppose.

But let's look a bit more closely at my list model: if you have a number like 0.999 and multiply it by 10[sup]-2[/sup], the result is 0.00999, the single zero on the left of the decimal point must really be like a string of zeros extending infinitely to the right (which is inductively true since we can multiply by 10[sup]-2[/sup] indefinitely). But this is true for any number, there is this infinite string of zeros on the right. We just write numbers like 000.9 as 0.9, because it's: "0.009 multplied by 100".

In case you are still confused about what is being discussed, here is your original post. In 5-th grade, they taught you that when you divided 0.999 by 100, as you did above, two zeroes extended to the right of the decimal point producing the correct answer, 0.00999.

5 , now what?

Actually it's two.
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The leading zeroes between the decimal point and the first non-zero number are not significant, but we write them anyway as a matter of convenience. We can rewrite the number as $$5.2\times10^{-4}$$ and it still has exactly the same accuracy.

Actually it's two.
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The leading zeroes between the decimal point and the first non-zero number are not significant, but we write them anyway as a matter of convenience. We can rewrite the number as $$5.2\times10^{-4}$$ and it still has exactly the same accuracy.

This Source disagrees with your answer, (and with the other two sources you are citing), according to it, the number of significant digits is 4, not 5. Did you realize that?

Personally, I prefer 5, keeps things clean, two doesn't tell you anything.

Tach said:
In 5-th grade, they taught you that when you divided 0.999 by 100, as you did above, two zeroes extended to the right of the decimal point producing the correct answer, 0.00999.
But when you don't divide 0.999 by anything, the zero on the left of the decimal point is like a string of zeros extending infinitely, and not to the right as I stated initially.
The direction they extend can only be relative to the decimal point, so the string of zeros cannot extend to the right (when not multiplying or dividing 0.999), so they must extend to the left, or the first zero is the rightmost and there is no leftmost. Didn't they teach you that in the 5th grade??

But when you don't divide 0.999 by anything,

Yet, we are talking about the case where you DO divide 0.9999 (by powers of 10)

the zero on the left of the decimal point is like a string of zeros extending infinitely, and not to the right as I stated initially.

Sure but this is NOT the case we are discussing. You need to stop this type of desperate diversion. Here is the post that is being debated:

arfa brane said:
I suppose.

But let's look a bit more closely at my list model: if you have a number like 0.999 and multiply it by 10-2, the result is 0.00999, the single zero on the left of the decimal point must really be like a string of zeros extending infinitely to the right (which is inductively true since we can multiply by 10-2 indefinitely).

I just noticed this bit of nonsense, at the end of the post:

arfa brane said:
But this is true for any number, there is this infinite string of zeros on the right. We just write numbers like 000.9 as 0.9, because it's: "0.009 multplied by 100".

Keep digging yourself.

I prefer 5, two doesn't tell you anything, you can have your convention as you like it. The number has 5 digits of precision.

It's not my convention, it's the convention being taught at tertiary institutions around the world, it's for the same reason that 52,000 is two significant figures, not five.

My point being that, as I first stated twenty or thirty posts ago that there is an undending string of leading zeroes that we agree not to write unless we absolutely have to - for example, they're moved to the right of the decimal point, and there is another unending string of trailing zeroes after the last significant figure, and the insertion or deletion of zeroes when multiplying by powers of ten is simply as a result of performing operations on the decimal powers of the significant figures. It has nothing to do with moving the decimal point or adding zeroes to the end of the string.

What's confusing me, right at this instant, is whether Undefined is now agreeing with this, or if he still, somehow, disagrees with this.

Tach said:
Yet, we are talking about the case where you DO divide 0.9999 (by powers of 10)
I see, so in your mind, that means we can't talk about a number that we don't divide. Glad we cleared that up.
me said:
But this is true for any number, there is this infinite string of zeros on the right
Note that this should say "on the left of the first nonzero digit". It's incorrect because I made the same mistake as before, when I referred to the zero on the left of the decimal point in a number like 0.9. Note that I can refer to this zero without having to refer to a multiplication/division operation on the number 0.9, whereas Tach insists that I can't (for reasons known only to him).

And what's wrong with
me said:
We just write numbers like 000.9 as 0.9, because it's: "0.009 multplied by 100".
?

Is it not true that 0.009 x 100 = 0.9, or something? Or are you trying to look stupid?

It's not my convention, it's the convention being taught at tertiary institutions around the world, it's for the same reason that 52,000 is two significant figures, not five.

My point being that, as I first stated twenty or thirty posts ago that there is an undending string of leading zeroes that we agree not to write unless we absolutely have to - for example, they're moved to the right of the decimal point, and there is another unending string of trailing zeroes after the last significant figure, and the insertion or deletion of zeroes when multiplying by powers of ten is simply as a result of performing operations on the decimal powers of the significant figures. It has nothing to do with moving the decimal point or adding zeroes to the end of the string.

What's confusing me, right at this instant, is whether Undefined is now agreeing with this, or if he still, somehow, disagrees with this.

Actually, Trippy, I do agree with you on that. The only quibble I had was that some tried to invoke the 'moving the decimal point' as some sort of actual mathematical 'action/function'. That was why I pointed out that the fundamental 'action' is the string behavior itself, not the 'formatting/shortcut' display convenience of 'move the decimal point'.

As to this latest to-do, In context, I merely asked arfa brane (and he graciously answered) for clarification about what he meant (left or right extending to infinity of the zeros) so that I could read his post more correctly and avoid my misunderstanding. Tach called my polite and courteous request for clarification "wrong' and 'bullying' (goodness knows where he got that from?). Since arfa graciously clarified, I then went on to point to two aspects. If you read my posts above, I point out that the actual fractional string interrupts any 'zero extending to infinity' view, except for the leading zeros to infinity to the left of the string or the decimal point (depending on how the fractional number string begins (ie, as .9 or .009 etc). These relate to the actual 'value' of the string and its 'behavior' under function/process number 'actions'.

Also, I tried to explain the non-value effect/information separate aspect/practice. Where the use of writing a certain zero-string trailing zeros in the fractional string representations because the writer wants to convey the furthest degree of precision/accuracy in the measurement construct used (hence they might write some fractional string like 0.90 to indicate the measurement was precise to the second decimal place of measurement precision/accuracy).

Anyhow, thanks again, Trippy, everyone, for all your polite and genuine positive contributions to the discourse so far. I appreciated them no end. Running out of time again. See ya round, Trippy, everyone.

This Source disagrees with your answer, (and with the other two sources you are citing), according to it, the number of significant digits is 4, not 5. Did you realize that?

Personally, I prefer 5, keeps things clean, two doesn't tell you anything.
Does it?

I presume you're referring to the their example in A.2 where they use 0.01023? I agree that that number is four significant figures, but that doesn't disagree with what I said.

Further, if you see their example on the last page:

The standard prefixes are listed in TableA.1 By using a prefix, the exponent used to report a measurement in scientific notation can be modified but the number of significant figures is unchanged. For example,

l = 0.0021m = 2.1 · 10[sup]−4[/sup]m = 2.1 · 10[sup]−1[/sup]mm = 2.1 · 10[sup]2[/sup]μm.

Which supports, rather than contradicts my assertion.

Could you be clearer about where you think it disagrees with me?

Does it?

I presume you're referring to the their example in A.2 where they use 0.01023? I agree that that number is four significant figures, but that doesn't disagree with what I said.

Yes, it agrees with what you said. Your point is, again?

I see, so in your mind, that means we can't talk about a number that we don't divide.

No, I am simply cutting off your repeated diversions and pointing you back to the post where you made your statements that started the whole debate:

arfa brane said:
I suppose.

But let's look a bit more closely at my list model: if you have a number like 0.999 and multiply it by 10-2, the result is 0.00999, the single zero on the left of the decimal point must really be like a string of zeros extending infinitely to the right (which is inductively true since we can multiply by 10-2 indefinitely).

You keep multiplying by powers of $$10^{-2}$$, you keep inserting pairs of zeroes at the right of the decimal point. Do you disagree with that?

And what's wrong with?

Is it not true that 0.009 x 100 = 0.9, or something? Or are you trying to look stupid?

But this is not what you write, you write the nonsense:

arfa brane said:
We just write numbers like 0.009 as 0.9, because it's: "0.009 multplied by 100".

How can a sane person write 0.009 as 0.9? They are two DIFFERENT numbers.

Or are you trying to look stupid?

Look in the mirror, again.

Yes, it agrees with what you said. Your point is, again?

My point has already been stated clearly and I have no desire to re-iterate it.

My point has already been stated clearly and I have no desire to re-iterate it.

OK, let's try again, what does your point (the convention on the number of significant digits) have to do with the way division by powers of ten is mangled in arfa brane's posts? This is what is being discussed.