sethoflagos

My first wild stab in the dark would be their shared initial value and symmetry about zero. = common end point?

I'm sure I missed most of the subtlety of @studiot's point, but if as he seems to be saying, (-1)*(-1) only has a value of +1 by convention, then reading it backwards, -1 is simply a label for the 'other' root of +1, isn't it?. This instantly reminded me of my picture of +/-i being axiomatic labels for the roots of -1. As someone who habitually conceptualises ideas in geometrical terms, the paralellism is certainly apparent when the two relationships are expressed as rotations of a unit vector in the complex plane: two rotations of 0 or pi radians restore the unit vector; two rotations of +/-i*pi/2 give the negative unit vector. ... or to preserve the state of the unit vector under zero rotations. NB this is far from the OP objective: just trying to better explain what seems to have been seen as a daft idea.

So does -1 stand to 1 as i does to -1, no more no less?

Sounds like the one I scanned through on Jstor recently, and yes of course, what you say is unarguably so. Your own point regarding natural gastric alcohol production suggests that the concept of 'total abstinence' is illusory. We all consume alcohol to a certain extent. Gauging the effects of one extra milligram here or there would be challenging in the extreme.

There have been a multitude of studies on mortality vs alcohol consumption. High on the search-engine listings is Alcohol Consumption and All-Cause Mortality: A Systematic Review The balance of the evidence is perhaps surprisingly in favour of limited consumption of alcohol being a healthier choice over total abstinence. The argument isn't proven, but it suggests your fears are indeed difficult to justify.

Your input combination was one of the 1,300 other possible solutions also given on my spreadsheet. There is no unique 'correct' solution. If D is volume, and d is specific volume, then yes. If any of the various q values share a common factor, there will likely be multiple solutions. This is readily apparent in the coins problem as the common factors are of a similar order of magnitude to q. The problem does not go away even with irrational numbers. Express in terms of specific volume rather than density and the form is identical to your stated rule of mixtures.

Pretty much it. Main point is kids understand stairs so there's a clear physical and intuitive link to each operation that you can leverage. Give boys half a chance to compete at who can take stairs two or three at a time and they'll be doing it at playtime too.

All the Scotch Yoke actuators I've worked with have been quarter-turn per.piston stroke, though in principle this could be varied. More to the point, the slider moved parallel to the piston on (effectively) a variable throw crank. The OP design principle appears quite different.

If we've learnt the operations of ascending-by-one, ascending-by-two, etc.. What happens when we ascend-by-zero? What happens when we ascend-by-(-1)? What happens when we descend-by-(-1)? Do we see that ascending by 1, 2, 3... Is the same as descending by -1, -2, -3...?

Good question! Let me dwell on it a while! (Though my label was simply '-1' without committing to multiplication or subtraction)

Your OP is specific to the latter. I have clear memories of measuring and cutting strips of cardboard from old cereal packets, and arranging them as 'staircases' of varying gradients. Once we were familiar with ascending and descending the aboveground flights of stairs in various step sizes, we could have a look at what might call the first step down towards the cellar. Following my own train of thought, it seems to start as simply a label for a descent of one step from ground zero. Giving the result the same name as the operation is an idea at least.

To me, the very notion of negative numbers implies multiplication by -1. This suggests that prior introduction to multiplication of the natural numbers (not to mention the role of zero) is a more logical order of learning.

I didn't say it was. I'm trying to highlight an intermediate step that's possibly being overlooked.

How do you address the very different physical interpretations of: a) I take six oranges from the box b) I add minus six oranges to the box There's a strong scent here of two different operations being conflated into one without explanation.

As far as I remember: Learn multiplication tables by rote Apply multiplication tables to negative multiplicands Logically extend multiplication tables backwards for negative multipliers Hope little minds latch on to the underlying symmetries

Engineers and chemists generally have opposing views on whether thermodynamic work performed BY a system ON its surroundings is a positive or negative quantity. Engineers tend to the historic thermodynamics tradition of treating it as positive. Quoting from https://en.wikipedia.org/wiki/Work_(thermodynamics) It's a persistent source of confusion. Negative signs often appear and disappear in these calculations as the various equations are modified to fit a preferred convention.

I lost my harp in Sam Frank's disco

Of course, though $755.00 is a lot of coins, and it is a highly instructive question for a reason I didn't anticipate (or at least bother to check in advance). It turns out that I'd wrongly assumed that the standard coin masses have no large common denominator whereas your reference indicates that five dimes and two quarters actually have indentical mass and value. This guarantees that if there is one solution, there are many: just substitute five dimes for two quarters as many times as you like. Let us start by guessing there are no quarters. So nickels and dimes sum to $755.00 and 25.242 kg. Two variables; two equations which simply solve to 2,100 nickels: 6,500 dimes. Yay!! Integers!! So we have a solution!! Do the same for one quarter and we get 2,100 nickels again but a non-integer number of dimes (6,497.5) so we can reject this one. But for two quarters we get another solution 2,100 nickels: 6,495 dimes - ie we've taken the first solution and exchanged 5 dimes for 2 quarters. And so on... So we can do no better than state that the box contains $105 worth of nickels and $650 dollars of mixed dimes and quarters... ... because: Will you please elaborate? The above example illustrates this very well. If instead of 5.67g we set the mass of the quarter to 5.669g, we retain the previous solution of 2,100:6,500:0 however, the slight deviation from a large common denominator introduces increasingly large deviations from integer values which invalidates all other potential candidates. This is easily demonstrated with a simple Excel spreadsheet (I've omitted lines 16 - 3,200 for sanity's sake) Your methodology requires significant common denominantors in alloy composition figures to keep the number of permutations of composition down to a manageable finite number (to facilitate a brute force computational sieve), but component densities should ideally be irrational numbers (which in actuality we'd expect them to be) to prevent the existence of multiple integer solutions. For purposes of my argument, any equal incremental step process is essentially based on stepping through integers. I'd often wondered in the past why banks etc. went to the trouble of counting coins individually rather than just weighing them in batches and exploiting the limited possible combinations to compute the value. Now I've a clearer picture. Thank you for that.

On the contrary, your point is abundantly clear. But I think that bird has flown a while back. The mystery is not in the mineralogy, it's in the mathematical method. Please refrain from tabling dead cats. The topic is metallurgy. What we have here is effectively a closed box containing material of a precise given weight and monetary value. Given no further information I have no idea what the contents are. However, if I'm told the contents consist solely of nickles, quarters and dimes, I think I might be able to work it out. Restricting a search solely to integer roots of equations provides a massive simplification.

Still to this day a Tolkien fan. Similarly, I can find the rather odd physics of the Minecraft universe fully engaging. It can be every bit as satisfying to design and build automated processes in that reality as in this one. Absolutely! And now that the kids are all grown up, why restrict oneself to a single path anyway?

It can be IFF: - You know all the components beforehand - The components have significantly different elemental densities - Elemental densities have no significantly large common denominator - The alloying process does not induce a volume change - Compositions are precise integer percentages by mass (or moles, volume etc) - Alloy lattice is flawless - Density measurements are accurate to ~ 4 significant figures +

If integer values are used for component mass percentages in calculating this 'Archimedean' density function, then performing a brute force scan of the inverse function for integer solutions can recover those input integer values.

Different vein. Science may help provide some understanding of what paths are at least feasible, whether meaningful or absurd. Religion may be one such path. But one of restricted choice largely of someone else's choosing. Utterly absurd in my view but others may differ. Personality, I'm not averse to a little absurdity now and then. But preferably absurdities of my own choosing.

+1 The referenced discussion document makes no allowance that I can find for alloy density being a function of a lattice structure specific to that alloy. Rather, as you seem to suspect, alloy densities are simply assumed to be mass weighted averages of elemental densities. As far as I can tell, such a weighted average is calculated from an alloy composition constructed from integer component percentages, and of the infinite potential compositions that match that density, the composition that most closely yields integer percentage values is picked as the 'Most Probable Composition'. It's a few years since I studied statistical analysis techniques and I think I must have missed the lecture on the Hogwarts Sorting Hat. Best guess:- deadcatting.

Because almost all of the assumptions are only approximations of real world gas behaviour under similar atmospheric conditions to those we are all routinely familiar with at earth's surface. For simple, non-critical cases these approximations are often close enough for the differences to be unimportant. However, for more complicated, more critical situations, it is necessary to replace the less valid assumptions with empirically grounded correlations specific to the use-case. And to do this you need a thorough working understanding of what those assumptions are in the first place. O, yes and in passing - if you don't have that thorough working understanding, you don't even get to judge what counts as simple and non-critical.