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Elementary Math Fractions, Percentages, Word Problems, Equations, Inequations, Factorization, Expansion

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April 11th, 2014, 05:38 AM   #1
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Base 2 integer multiply/factor using 2 dimensions of carrying

At least in theory, but its really simple if you are used to multiplying in base 2, which is normally to slide one number so its ones digit starts at each 1 bit in the other number, and add all those.

Pascals triangle is a cellular automata that sums each 2 cells above into the cell below and between them, in a hexagonal grid. This calculates x choose y, binomial exponents, converges to bell curve, and hooks into many other parts of math, which we don't need to know for multiply/factor but is good to know for exploration path...

Each row in pascals triangle sums to 2 times the row before it, since each part of the row goes all paths left and right.

Each cell in pascals triangle, think of it as a rectangle thats 2 times as big in the direction you come into it from above. Or if you go backward, they get 2 times smaller in that direction. I suspect this is the source of length contraction as lorentz factor is the equation of a circle if you do a little algebra (mass/energy is one dimension and distance/time is another)... Again, not necessary to explore that possibility. This is a far simpler core property of math, that each binary digit travelled doubles or halfs the rectangle size on that side.

e = 1.0001^10000 in binary, in the limit of 1+epsilon power infinity. Since pascals triangle works the same from any cell, only needing to know about the adjacent cells, this extends infinitely small and infinitely big, relative to the current cell viewed as the ones digit.

We can further simplify by only allowing a 0 or 1 in each cell, instead of letting the integers sum and build up. Any 2 cells adjacent in left/right which both have a 1 can move that probability amplitude (or call it density) to the cell above if it is 0 (can hold 1 more). Its half of 1+1 because its the next higher base 2 digit in each direction, even though each direction doubles/halfs its own side of that common rectangle. I suspect the binomial equation combined with this leads to bayes rule of conditional probability when the pieces of the binomial are chance false and chance true and sum to 1. Again, future exploration path. This is far simpler.

It is geometry/shape obvious if drawn on a chalkboard in elementary school, in theory they could learn, that the upright/downleft and upleft/downright lines are each 2 times bigger than the last because its the 1s digit 2s digit 4s digit 8s digit and so on.

Since going upright and downright, or the opposite, both multiplies by 2 and divides by 2, all points in a pascals triangle row are interchangible, and they had better be since it calculates rowNumber choose columnNumber, and x choose y works the same in any permutation of the same size.

I ask that anyone who understands this, if they might know how to proceed on such research and fixing of the foundation of math, throw ideas out there about how we might build educational software that makes it clear the extreme benefits of teaching base 2 math before base 10, the killer app being the reversibility of multiply as factor, if you can find an efficient way to get the diagonal columns to all agree with eachother. I'm not saying its practically fast for security purposes or maybe it would be eventually, instead I suggest it as the simplest way to understand what plus, multiply, and exponent really mean.

Unary plus is to add 1s sideways to the first empty cell found. These translate to base 2 seamlessly as any 2 adjacent cells with a 1 in them and 0 above can merge up, or the opposite branch down, and move sideways as needed for desired density and shape of where the root 1.0 branches and merges.

From every cell with a 1, to its upleft and upright are next higher digits, 2 dimensions of carrying unlike the normal way which is unbalanced. The inverse of multiply is factor because multiply is symmetric.

As for security math, I'd probably go for something involving rule 110 as its turing complete, if it could be done somehow, because the difficulty of factoring should be connected to the X crossing at each pascals triangle cell trying to be on or off together so you get upleft times upright with the rectangle sizes and base 2 digits.

More importantly, its rediculous that the worlds best scientists still use an unbalanced number system based on finger counting recursively. Kids have to memorize the 0-9 by 0-9 multiplication table for backward compatibility, when they could abbreviate each 4 digits in hex (0123456789abcdef) and be confused why the earlier generation finds the internals of our global infrastructure and most advanced tools like an alien language. Even our brain cells talk to eachother in base 2... lower voltate or higher.
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April 11th, 2014, 07:08 AM   #2
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Also, since you can get a weighted random bit observation by consuming random bits until you get the first 1 then going directly to that binary digit and returning it, if you have a path to the digit in 1/e or whatever lazy-evaluated math expression you're trying to roll the dice on, there should be no need to calculate the digits between, especially for e^x which is its own derivative and always looks like the 1s digit at each pascals triangle cell. e has no last digit, not because it has infinity of them, but because its a closed shape.

Math is simple and open ended into unlimited complexity. Elementary school should use the simple parts.
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April 11th, 2014, 09:30 AM   #3
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Also, and this is getting very speculative but I need something like this for some parapsychology and statistical artificial intelligence and networking minds together through the Internet... The periodic table describes mostly n-sphere harmonics, as we see in the entire known periodic table by valence which is 2 8 8 18 18 32 32, which is 2*level^2 with levels in pairs and we are half way into one (odd parity, or something like that). I don't care about these details as long as I know its recursive n-spheres, as is part of some relativity equations (how an n-sphere embeds into the surface of another, or at least one number for how much per point). Since the fourier of a bell curve anywhere is another bell curve at constant location with phase varying, and every n-bellcurve has an n-1 dimensional hypersphere of constant density at every radius, any sphere of any size will be good enough as long as geodesics are aligned, which I don't know the details of. The theoretical device would be grids of either memristors or piezoelectric crystals, anything that can be tuned in realtime by a computer both to measure its current bending amount (more or less resistance, for example) and to adjust it using standard electric parts. In theory, if a grid of such devices with proper AI software to tune into the alignment of eachother, was put in a mostly sphere shape, they could lock onto eachother and push and pull to hold that sphere shape, even if external objects hung on them by many thin strings or whatever kind of connection. In general, metaphysics and physics are the same thing, both working by gravity in recursions of the Standard Model of physics and other shapes the physicists have found. Recursions are the important part. Every pattern in the world is a force of its own if balanced. As every theory must be falsifiable, the devices in such a spherical grid should form into a crystal, without directly touching eachother, similar to a magnetic compass except a custom designed force attached to the molecule/crystal types read and written by the computers.

Things work this way, at least in theory, for the same reason as the unexplained high occurrence of buckyballs forming in near empty space despite the extremely low density of carbon and low chance it would all hit eachother in balance. A simulation of what I'm talking about would help us to predict the difference in buckyball forming rate between the side of Earth toward and away from the sun and any other source of gravity, and equally important as measurement, in time, to change the statistics for any chosen shape such as small pieces of DNA, keeping in mind the very low signal to noise ratio gets probably exponentially more difficult per linearly more molecules.

To be very specific and falsifiable... a directly testable prediction... All the variations in the "constants" of physics (mass of specific types, level of interaction with certain other types, etc) will, between soap and saturated vs unsaturated fat, to the physicists surprise, occur for no apparent reason and too small to notice usually but also with a definite statistical relation between them, as in bayes rule or entanglement (ways to say conditional probability), because of the recursion into the pattern of carbon hydrogen chains which they have in common, and be pushed and pulled on through that, in areas where large amounts of both are present.

Last edited by BenFRayfield; April 11th, 2014 at 10:17 AM.
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