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(Cue the banjo music: It's "Duelin' Nosists" time.)SenaryThe12th wrote:how do we interepret any unmarked, isolated digit pair? Like, say, "10"?Kodegadulo wrote:No, I think I have to insist that any unmarked, isolated digit must default to its plain vanilla face value.
I'm running a computer program right now which can predict which stocks are going to go up today.
Well here's another hypothesis: Your debate opponent has gotten frustrated, because he's got a good point, too. But despite repeated attempts to get it across to you, you're either not acknowledging it, or not getting it.SenaryThe12th wrote: When somebody starts pulling out the ad homs, that's when I know I've persuaded them that I might have a good point.
Well, most of the time the annotation schemes I and others here have cooked up just involve two levels, and so far at most three. But let that go. To answer the substance of your question, Senary, it sounds like you want everybody here to just give up their favorite base and use senary instead, with your alternate-carry idea, and not try to work compatibly with any other base. And I'm just not for that.SenaryThe12th wrote:But why do we have to go three levels deep for this? If eventually we'll all have to agree on the meaning of C, why don't we just all agree on the meaning of A? Why do we have to disambiguate A wiith B, and disambiguate B with C? I'm getting a mental image of your Greek Grandmother reaching around behind her head again.
That possibility is not mutually exclusive with the possibility which I've mentioned :-) Lets explore your possibilities:Kodegadulo wrote:Well here's another hypothesis: Your debate opponent has gotten frustrated, because he's got a good point, too. But despite repeated attempts to get it across to you, you're either not acknowledging it, or not getting it.SenaryThe12th wrote: When somebody starts pulling out the ad homs, that's when I know I've persuaded them that I might have a good point.
Yes. For several months now, I've been doing just that. I've already mentioned that I've found it useful to be able to to on-the-fly calculations in meetings. I hardly ever do calculations in decimal anymore. Mostly just base conversions, to communicate with decimal-normals. Most of those, even, can be done visually and instantly using the techniques of this thread.you've decided to immerse yourself in your alternate-carry senary as your "primary base"? Really? You use it for everyday stuff, like going to the grocery? At work? Paying your taxes? You're putting a really high value on doing fast arithmetic, and negging on algebra as not being fast enough for you?
I really don't see why that should be the case. Let's compare some polynomials and numbers, where the polynomial evaluates to the number where we take x = 10:SenaryThe12th wrote:You are right, it is the same. Positional notation is essentially a compressed polynomial notation. And overbar notation is cool in the fact that it reveals this connection to us, giving us a better understanding of how it all fits together.
Nevertheless, positional notation has been compressed from polynomial notation in a way which really facilitates computations. That's what made them such an advance over, say Roman Numerals. Which, come to think of it, can *also* be viewed as a kind of compressed polynomial form. But its compressed in a way which *doesn't* facilitate computation. I suppose a snappy way of making my point was that I found arithmetic using overbar notation to be too much like trying to calculate using Roman Numerals.
Not really. I really do find it just as fast to deal with overlines as it is to deal with ordinary unsigned integers, and just as fast to deal with actual polynomials. In fact if anything polynomials seem easier, as you never have carries, and so division (while tedious to write down) is mostly mechanical and you don't have to keep trying or estimating multiples. Comparing notations of numbers, overlines also seem easier for me as cancellation always works the same way (2 always cancels 2̅, whereas in your notation it might cancel 4 in senary, 6 in octal, 8 in decimal, a in dozenal, and so on) and the way negatives work is really obvious (compare signed decimal 2 * 3 = 14̅, 2 * 3̅ = 1̅4, 2̅ * 3 = 1̅4, 2̅ * 3̅ = 14̅ in overline notation with 2 * 3 = 16, 2 * 7 = 94, 8 * 3 = 94, 8 * 7 = 16 in your notation). Just because this isn't the case for you doesn't mean it can't be the case for me. ^_^SenaryThe12th wrote:I found treating integers as poynomials in the calculations slowed me down too much. YMMV. If you enjoy overbars, bless your heart! You must have the patience of Job :)
{6} default senarySenaryThe12th wrote:Which means, that the only real way you have to recognize that two numbers are equal, is to reduce them to a normal form, and then compare them digit-by-digit. Illustrating the point with balanced senary, the only way you know that all these refer to the same number:
[code block containing 303, 13̅03, 313̅, 13̅13̅]
is if you pick one of them, say 303, as a normal form, and try to reduce the others to that form, and then compare them digit-by-digit.
In Frege's terminology, sure, they all have the same reference. But they all have a different sense. They all may be "pointing" to the same number in some platonic realm, as it were. But we can't access that realm directly. It is mediated through symbols. If you happen to be Greek Orthodox, you know what I'm talking about here. And if the symbols are different, and multiple, it causes our brains as much work as if they were ambiguous.
Sure. Mapping between the overbars and the polynomials is straightforward. For me, it was, as it were, *too* straightforward. Its so straightforward that my mind was always just going straight to the polynomials, i.e. I just couldn't make the overbar notation "chunk".Double sharp wrote:
I really don't see why that should be the case. Let's compare some polynomials and numbers, where the polynomial evaluates to the number where we take x = 10:
2x^2 + 4x + 1 corresponds to 241
2x^2 - 4x + 1 corresponds to 24̅1
-2x^2 + 4x + 1 corresponds to 2̅41
ust because this isn't the case for you doesn't mean it can't be the case for me. ^_^
Well, as you can probably tell by now, this whole thing was a journey for me. I did try to make overbars + visualization work for a long time. That was one stop on my journey.In fact, given that I find your visualisation scheme suggests the overlines much more readily than the complement-digits, I'd suggest that you try using your visualisation scheme to handle polynomials.
Ok, maybe it would help if you walked me through the process, step by step. Suppose I have this equation I'm trying to solve:Double sharp wrote:
Actually, that's not how I think about the equality of those numbers: I don't try to convert them all to some normal form.
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31303 + X = 130313
You know, the more I think about it, the offset probably should be negative. I had initially gone with a positive offset so that the digit in the annotation would be the same as the first unusual digit in the base, but I don't think that's the most important piece of information. After all, it's not quite as simple as just shifting where carries occur. If it were, then [d+6] 6+6 would equal 2, when it should be 92. For the purposes of arithmetic, it's important to remember that the digits 6 through 9 in that system really do work differently than usual. In particular, when two digits from that range are added to each other, a 9 is carried out. If the addition also reaches at least 6 in the modular ring (which happens with 8+8, 8+9, 9+8, and 9+9), then there is also a 1 carried out as usual, cancelling out the already carried 9 to create a net carry of 0.SenaryThe12th wrote: Kodegadulo:Amen. In that spirit, if you think that it makes more sense if the offset is negative, and should be written with an overbar, I'd be ok with that too :-)It's all about coexistence, baby.
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[d] : 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20
[d-4]: 0, 1, 2, 3, 4, 5, 16, 17, 18, 19, 10, 11, 12, 13, 14, 15, 26, 27, 28, 29, 20
Um....dude....the very title I chose for the thread is "Subitizing/Visualizing large numbers using Signed Bases". As in, "Bases" plural. If I really was a Senary-uber-alles type fascist, why would I spend so many hours typing in ascii cubes to make what was originally a senary-only technique applicable to all human scale bases? In what universe would that make any sense at all?Kodegadulo wrote:
To answer the substance of your question, Senary, it sounds like you want everybody here to just give up their favorite base and use senary instead, with your alternate-carry idea, and not try to work compatibly with any other base. And I'm just not for that.
Good points. And come to think of it, having an overbar over the digit representing the offset would be a handy reminder that this feature was introduced to support signed digit arithmetic. One point of clarification tho:arbiteroftruth wrote:
You know, the more I think about it, the offset probably should be negative
Yeah, but that's really the same carry rule.....when you roll over from 5 to 6 (in signed decimal) you carry a one. When you roll *back* from 6 to 5, you borrow (i.e. subtract) a 1, which would be 9.In particular, when two digits from that range are added to each other, a 9 is carried out.
By inspection, the coefficient on the right is obviously equivalent to the one on the right, just a complement form so x is zero.SenaryThe12th wrote: Ok, maybe it would help if you walked me through the process, step by step. Suppose I have this equation I'm trying to solve:solve for X. What steps do you take?Code: Select all
_ _ _ 31303 + X = 130313
So you just do it in one step? Impressive. The first number has 6 symbols in it, so I guess I could learn to chunk it. But the second one has 8 symbols in it. That's beyond my subitization limits. I can't just look at 8 objects and subitize the number 8. Similarly, I cant look at the second number above and just understand it at one glance. Anything more than 5 or 6 symbols, I have to break down into smaller chunks to understand. For me, its kind of like being able to look at 11111111 and 111111111 at one glance and seeing whether they are equal. More than five or six "1"s in a row, and I have to count them up to know how many are there.Kodegadulo wrote:By inspection, the coefficient on the right is obviously equivalent to the one on the right, just a complement form so x is zero.SenaryThe12th wrote: Ok, maybe it would help if you walked me through the process, step by step. Suppose I have this equation I'm trying to solve:solve for X. What steps do you take?Code: Select all
_ _ _ 31303 + X = 130313
I didn't say I did it in one step, at a mere glance. I said I did it by inspection. Isn't the name of the game here doing as much as possible in your head? I did see quite obvious chunks: I mean, ignoring the zeroes, the only absolute values in the digits were one and three. And the ones and negated threes were always together. I had practiced a little with doing the carrying/normalizing transformation in a few bases, so something like the equivalence \(3 = 1\overline{3}_6\), in other words \(3 = 6 - 3\), was already primed in my mind. Once I saw that it was all just \(3\)'s and \(1\overline{3}_6\)'s, then it was instantaneous to see that both numbers were \(30303_6\).SenaryThe12th wrote:So you just do it in one step? Impressive. The first number has 6 symbols in it, so I guess I could learn to chunk it. But the second one has 8 symbols in it. That's beyond my subitization limits. I can't just look at 8 objects and subitize the number 8. Similarly, I cant look at the second number above and just understand it at one glance. Anything more than 5 or 6 symbols, I have to break down into smaller chunks to understand. For me, its kind of like being able to look at 11111111 and 111111111 at one glance and seeing whether they are equal. More than five or six "1"s in a row, and I have to count them up to know how many are there.Kodegadulo wrote:By inspection, the coefficient on the right is obviously equivalent to the one on the right, just a complement form so x is zero.SenaryThe12th wrote: Ok, maybe it would help if you walked me through the process, step by step. Suppose I have this equation I'm trying to solve:solve for X. What steps do you take?Code: Select all
_ _ _ 31303 + X = 130313
Yeah, it's pretty much the same for me. I don't get why that minus has to "weigh" as much as a whole digit in Sen's mind.Double sharp wrote: However, I consider there to be 5 symbols in the number on the LHS and 6 in the one on the RHS, not 6 and 8 respectively. The overline modifies an existing symbol, but it doesn't add the load of an extra symbol for me.
Well, that is because standard positional notation doesn't really support negative numbers without a negative sign. That's why you look like you have a mirror symmetry around 0 rather than a translation symmetry like you have everywhere else: counting backwards in tens from 39, you get 29, 19, and 9, but then -1, -11, and -21; only at zero does the units digit change. So we are not representing negative numbers by building them up step by step; we are representing their additive inverses and then negating those as a whole. This is the same problem as that you get when your set of digits doesn't include 0 (e.g. decimal with digits from 1 to a): you now have no way to write zero without ad hoc tricks. Similarly, if all the digits are negative except zero (e.g. decimal with digits from -9 to 0), only negative numbers have a native representation; you need a negation sign to express the positives.SenaryThe12th wrote:Ooops, when the odometer rolls past 00, we get 99. Looks like we are using complement arithmetic here...
No wonder I misunderstood you--I was under the impression you actually answered my question :-) I asked for a step-by-step explanation. Since you gave me one line, why wouldn't I assume you did it in one step?Kodegadulo wrote: I didn't say I did it in one step,
That was a little eliptical, but is it fair to say that you did some easy computations which translated both numbers into a normal form (30303) and then since they had the same normal form you could instantaniously see that they were equal?then it was instantaneous to see that both numbers were \(30303_6\).
Well, tha't wasn't the only suggestion I made. I also recommend using unbalanced signed senary. If you do, you can only get the digits 30303 for that value. The problem never even arises.All of that would have been incrementally harder to see, if you applied a suggestion you made a few posts back,
Every dog has its day---that notation would make it harder to see that 35124 and 35124 were the same number.Now, a monotonous \(11111111_6\) is certainly hard for me to distinguish from a monotonous \(111111111_6\). On the other hand, I wouldn't have any trouble spotting the difference between these two numbers if they were expressed like so:
\[\left[\sum_{n=0}^{11} 10^n \ne \sum_{n=0}^{12} 10^n\right]_6\]
It doesn't count as a digit--it counts as minus sign, because that's what it is.Kodegadulo wrote:Yeah, it's pretty much the same for me. I don't get why that minus has to "weigh" as much as a whole digit in Sen's mind.Double sharp wrote: However, I consider there to be 5 symbols in the number on the LHS and 6 in the one on the RHS, not 6 and 8 respectively. The overline modifies an existing symbol, but it doesn't add the load of an extra symbol for me.
There are two things which are impressive to me about doing it in one step:Double sharp wrote: I do it in one step by inspection too. However, I consider there to be 5 symbols in the number on the LHS and 6 in the one on the RHS, not 6 and 8 respectively. The overline modifies an existing symbol, but it doesn't add the load of an extra symbol for me.
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30303 130303
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30313 130313
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31303 131303
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31313 131313
Oh, I have no trouble with complement arithmetic--I'm a computer programmer, and computers use complement arithmetic with positive integers (and not overbars lol) because it simplifies the hardware circuit design.Double sharp wrote:Well, that is because standard positional notation doesn't really support negative numbers without a negative sign.SenaryThe12th wrote:Ooops, when the odometer rolls past 00, we get 99. Looks like we are using complement arithmetic here...
Oh, and I just realised that the trouble with division remainders can be explained by first using "10" as the divisor: then everything works out with the standard digits, as the remainder is always the last digit of the number. Then you can explain that you always measure your distance, positive or negative, to the nearest multiple. Explaining this by looking at division as sharing takes little more inspiration to come up with a way to look at negative remainders the same way as we do positive remainders; I'd suggest thinking of the remainder as the minimum signed distance we need to add or subtract so that the division can be fair. I think it's better to look at the number line and think of possible negatives and vectors than to stick to approaches where only positives are visible, though.
{6} default senarySenaryThe12th wrote:Yet, unless you memorize that, I really don't see how you could--in one step---determine that any two of those strings were equal. Even if you *could* chunk both strings in one step, which I, alas, cannot.
How the heck do you do that??????
I agree that the overline is a minus sign that we put over a single digit to negate it. What I don't agree is that an overline over a digit counts as two symbols. I think of overline-1 as a single symbol (albeit compose of two symbols) just as I think that š (s-háček) is a single letter.SenaryThe12th wrote:It doesn't count as a digit--it counts as minus sign, because that's what it is.
If I don't use them but use digits 30303 instead, it doesn't count as anything, because its not there.