Shwa "reverse" Notation

Shwa "reverse" Notation

Stella of the Sapphire
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Joined: Apr 22 2013, 08:08 PM

Sep 15 2013, 11:05 PM #1

(image links to webpage)


Honestly, two things disturb me about this notation. First of all, why is there a separate figure for the repeater? I can understand the utility of a repeater key, but how does the benefit of a repeater key carry on to a repeater figure? That doesn't make sense to me. Second, the first of three rules for using a positive or negative six is totally unnecessary; the second and third are sufficient to ensure that proper rounding (see note below) agrees with truncation. Why use the six that's opposite to the sign of the number? If the rule were changed for using the six of the same sign then numbers that are one-half or six times a power of twelve can be written with one fewer significant figure than with the rule for the opposite sign (e.g. 6 vs 16) But anyway, what is the benefit of always using the opposite sign for a least significant six, as opposed to using the sign that makes the next figure on the left even, or using whichever sign is more convenient for the given situation?

Furthermore, the symbols appear less distinct than the standard Western Hindu-Arabic numerals, at least as I see them. The symbols for three and four are almost horizontal mirror images, and that's more difficult for me to distinguish than horizontal and vertical flips as with "6" and "9" and not only that, many of the other symbols have similar shapes on the upper half. Perhaps someone can tell these symbols apart better than I...

Of course, this system has all the advantages of a balanced notation, that truncation is congruent with rounding, a minus sign is not necessary and the multiplication table is more succinct. That being said, I still get eyestrain from trying to read the figures. -meh-

NOTE: By proper rounding, I mean that a number should be rounded to whichever 'endpoint' number of a given precision is closer to the number to be rounded. That is, 56 should be rounded to zero instead of 100, 66 should be rounded to 100 instead of zero. However, a midpoint number can be rounded either way: 56 can be rounded to 50 or 60, whichever is more appropriate for the situation. An additional rule, such as round to even or round away from zero can guarantee an unique way to round a number to a given precision, but such a rule is beyond the definition of ''proper rounding'' that I use here.
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Zenarchist
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Zenarchist
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Joined: Nov 7 2012, 08:56 PM

Sep 15 2013, 11:22 PM #2

Balanced bases have always made my brain hurt, personally. I feel like it's too much added complexity for not enough added benefit.
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Stella of the Sapphire
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Joined: Apr 22 2013, 08:08 PM

Sep 17 2013, 04:15 AM #3

Reply to [url=http://z13.invisionfree.com/DozensOnline/index.php?showtopic=1035&view=findpost&p=22100117]this post[/url]: [quote="m1n1f1g @ Sep 16 2013, 12:46 PM"]Sorry, I messed up the 6-rules. I'll just quote them:[quote="Shwa"][/quote] Reviewing them, they seem quite difficult to apply. You have to start at the end of the number, then work back to the beginning (lest you end up in a situation where you have to change all previous 6/Ï‘s). I think there's a different way of reading Shwa numbers (distinct from the way of reading Shwa letters). The area of interest usually seems to be the horizontal edges of the vertical near-centre, plus the top. The letters are better, though.[/quote] The second and third rules ensure that truncation doesn't lead to improper rounding, that is, a number being rounded to an approximant number that isn't the closest possible to the number to be rounded, such as [color=blue]266[/color] being rounded to [color=blue]2[/color][color=purple]00[/color] instead of [color=blue]3[/color][color=purple]00[/color] which is infact closer to [color=blue]266[/color] than [color=blue]2[/color][color=purple]00[/color], so [color=blue]266[/color] should be represented as [color=blue]3[/color][color=red]6[/color][color=blue]6[/color] or equivalently [color=blue]3[/color][color=red]56[/color] to avoid this problem. The first rule, however, is not necessary, and I don't think it provides any useful benefit. [color=blue]3[/color][color=red]56[/color] is preferred by the first rule which says to use the least significan six of the opposite sign of the number, but the round-to-even rule suggests using [color=blue]3[/color][color=red]6[/color][color=blue]6[/color] instead. Does the author of the Shwa notation really not believe that sometimes [color=blue]3[/color][color=red]6[/color][color=blue]6[/color] may be a more useful representation than [color=blue]3[/color][color=red]56[/color] in some situations? What if one makes a measurement to three significant figures and can't pin down the fourth, but can at least see that it's closer to [color=blue]3[/color][color=red]6[/color] than to [color=blue]3[/color][color=red]5[/color]? [size=65][i]Let me know if you have any problems seeing the colors, because that's the best way for me to see the difference between positive and negative figures that I can easily input without much extra effort. I really don't want to use overlines because that's visually unappealing to me, and Greek letters are a pain to copy and paste or to constantly change my keyboard layout.[/i][/size]
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m1n1f1g
Dozens Disciple
m1n1f1g
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Joined: Feb 20 2011, 10:15 AM

Sep 18 2013, 10:04 PM #4

Stella of the Sapphire @ Sep 17 2013, 05:15 AM wrote: The second and third rules ensure that truncation doesn't lead to improper rounding, that is, a number being rounded to an approximant number that isn't the closest possible to the number to be rounded, such as 266 being rounded to 200 instead of 300 which is infact closer to 266 than 200, so 266 should be represented as 366 or equivalently 356 to avoid this problem. The first rule, however, is not necessary, and I don't think it provides any useful benefit. 356 is preferred by the first rule which says to use the least significan six of the opposite sign of the number, but the round-to-even rule suggests using 366 instead. Does the author of the Shwa notation really not believe that sometimes 366 may be a more useful representation than 356 in some situations? What if one makes a measurement to three significant figures and can't pin down the fourth, but can at least see that it's closer to 36 than to 35?

Let me know if you have any problems seeing the colors, because that's the best way for me to see the difference between positive and negative figures that I can easily input without much extra effort. I really don't want to use overlines because that's visually unappealing to me, and Greek letters are a pain to copy and paste or to constantly change my keyboard layout.
That's a good point. I think the idea was to ensure that all numbers have a unique representation, and one way to achieve that fits with the standard rounding operation. Also, I feel that rule #1 is an intuitive step from #2 and #3. We imagine that the number ends with infinite ‘0’s and comes back round to the start. The number to the right of the final digit, and after all those ‘0’s, is the first digit.
A few little conventions:
- Dozenal integers suffixed with prime (′). This is the uncial point.
- Decimal integers suffixed with middle dot (·). This is the decimal point.

You may see me use * prefix for messages before 11Ɛ7-03-1X, and a whole range of similar radix points. I will often use X and Ɛ for :A and :B.

Sometimes, I will imply that an integer is in dozenal, so I won't add any marks to it. You should be able to tell that "10 = 22 * 3" is in dozenal.
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Stella of the Sapphire
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Joined: Apr 22 2013, 08:08 PM

Sep 20 2013, 04:49 AM #5

m1n1f1g @ Sep 18 2013, 03:04 PM wrote:
Stella of the Sapphire @ Sep 17 2013, 05:15 AM wrote: The second and third rules ensure that truncation doesn't lead to improper rounding, that is, a number being rounded to an approximant number that isn't the closest possible to the number to be rounded, such as 266 being rounded to 200 instead of 300 which is infact closer to 266 than 200, so 266 should be represented as 366 or equivalently 356 to avoid this problem. The first rule, however, is not necessary, and I don't think it provides any useful benefit. 356 is preferred by the first rule which says to use the least significan six of the opposite sign of the number, but the round-to-even rule suggests using 366 instead. Does the author of the Shwa notation really not believe that sometimes 366 may be a more useful representation than 356 in some situations? What if one makes a measurement to three significant figures and can't pin down the fourth, but can at least see that it's closer to 36 than to 35?

Let me know if you have any problems seeing the colors, because that's the best way for me to see the difference between positive and negative figures that I can easily input without much extra effort. I really don't want to use overlines because that's visually unappealing to me, and Greek letters are a pain to copy and paste or to constantly change my keyboard layout.
That's a good point. I think the idea was to ensure that all numbers have a unique representation, and one way to achieve that fits with the standard rounding operation. Also, I feel that rule #1 is an intuitive step from #2 and #3. We imagine that the number ends with infinite ‘0’s and comes back round to the start. The number to the right of the final digit, and after all those ‘0’s, is the first digit.
I understand your thinking, but I don't agree with this ''loop around'' idea because it assumes an infinite set of figures of zero, which isn't acceptable to me. I mean, I just don't see any good reason for imagining that the number ends with "infinite '0's" as you say.
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m1n1f1g
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m1n1f1g
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Joined: Feb 20 2011, 10:15 AM

Sep 22 2013, 07:30 PM #6

Stella of the Sapphire @ Sep 20 2013, 05:49 AM wrote:
m1n1f1g @ Sep 18 2013, 03:04 PM wrote:
Stella of the Sapphire @ Sep 17 2013, 05:15 AM wrote: The second and third rules ensure that truncation doesn't lead to improper rounding, that is, a number being rounded to an approximant number that isn't the closest possible to the number to be rounded, such as 266 being rounded to 200 instead of 300 which is infact closer to 266 than 200, so 266 should be represented as 366 or equivalently 356 to avoid this problem. The first rule, however, is not necessary, and I don't think it provides any useful benefit. 356 is preferred by the first rule which says to use the least significan six of the opposite sign of the number, but the round-to-even rule suggests using 366 instead. Does the author of the Shwa notation really not believe that sometimes 366 may be a more useful representation than 356 in some situations? What if one makes a measurement to three significant figures and can't pin down the fourth, but can at least see that it's closer to 36 than to 35?

Let me know if you have any problems seeing the colors, because that's the best way for me to see the difference between positive and negative figures that I can easily input without much extra effort. I really don't want to use overlines because that's visually unappealing to me, and Greek letters are a pain to copy and paste or to constantly change my keyboard layout.
That's a good point. I think the idea was to ensure that all numbers have a unique representation, and one way to achieve that fits with the standard rounding operation. Also, I feel that rule #1 is an intuitive step from #2 and #3. We imagine that the number ends with infinite ‘0’s and comes back round to the start. The number to the right of the final digit, and after all those ‘0’s, is the first digit.
I understand your thinking, but I don't agree with this ''loop around'' idea because it assumes an infinite set of figures of zero, which isn't acceptable to me. I mean, I just don't see any good reason for imagining that the number ends with "infinite '0's" as you say.
I guess to be uniquely representable, it's best that numbers don't begin and end with infinite strings of ‘0’s. But the wrapping round still applies, in my opinion.
A few little conventions:
- Dozenal integers suffixed with prime (′). This is the uncial point.
- Decimal integers suffixed with middle dot (·). This is the decimal point.

You may see me use * prefix for messages before 11Ɛ7-03-1X, and a whole range of similar radix points. I will often use X and Ɛ for :A and :B.

Sometimes, I will imply that an integer is in dozenal, so I won't add any marks to it. You should be able to tell that "10 = 22 * 3" is in dozenal.
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Stella of the Sapphire
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Joined: Apr 22 2013, 08:08 PM

Sep 22 2013, 08:09 PM #7

If number strings don't begin and end with infinite copies of zero, then I don't see how wrapping around should apply at all.

Also, do we really need a repeater figure? Why not simply repeat the intended figure in writing and print, and have a separate key for the repeater if that helps?
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m1n1f1g
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m1n1f1g
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Sep 23 2013, 07:16 PM #8

Stella of the Sapphire @ Sep 22 2013, 09:09 PM wrote: If number strings don't begin and end with infinite copies of zero, then I don't see how wrapping around should apply at all.

Also, do we really need a repeater figure? Why not simply repeat the intended figure in writing and print, and have a separate key for the repeater if that helps?
Well, “the digit after the final digit” could be defined as either null or the first digit. Since we want it to always refer to something, we'd better have it refer to the first digit.

I agree with you completely on the repeater. It's useful in speech, where adjacent digits could blend into one, but it has no place in writing.
A few little conventions:
- Dozenal integers suffixed with prime (′). This is the uncial point.
- Decimal integers suffixed with middle dot (·). This is the decimal point.

You may see me use * prefix for messages before 11Ɛ7-03-1X, and a whole range of similar radix points. I will often use X and Ɛ for :A and :B.

Sometimes, I will imply that an integer is in dozenal, so I won't add any marks to it. You should be able to tell that "10 = 22 * 3" is in dozenal.
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pcyrus
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pcyrus
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Joined: Jun 19 2013, 04:02 PM

Apr 24 2018, 07:09 AM #9

The first rule has changed: Shwa now uses +6 for all positive numbers, and -6 for all negative numbers.  The original intent was to enable the two digits to share a key on a keyboard.  Without that rule, the numbers 6 and -6 would have been ambiguous.  But with the recent change to use vowel letters as digits, there's no benefit to sharing a key, and the new version of the first rule is clearer.  

I also added a negative zero, which nicely solved my remaining problems, for instance with percentages.  The same rule applies there: +0 with positive numbers (and zero), -0 with negative.
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SenaryThe12th
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SenaryThe12th
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Joined: Mar 1 2018, 02:03 PM

Apr 26 2018, 01:08 PM #10

pcyrus wrote:I also added a negative zero, which nicely solved my remaining problems, for instance with percentages.  The same rule applies there: +0 with positive numbers (and zero), -0 with negative.
Can you elaborate a bit about how the negative zero helped you?
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