4 ms·
The jumps between neighbors are sometimes so drastic—are we sure our numbers are in the right order?
by gavmor 1y ago
The jumps between neighbors are sometimes so drastic—are we sure our numbers are in the right order?
- jhanschoo 1y agoI don't know what you mean by that, but for an example, 16=2^4 and is therefore arranged as a grid, whereas 17 is prime, and must therefore be arranged as 17 dots on a circle.
- gavmor 1y agoThe primes are some of the worst offenders, eg the transition from 647 (prime) to 648 (3×3×3×3×2×2×2), but as we approach infinity, the visualizations increasingly appear circular, and it's the least-primey (?) that break from the trend. eg 854-856, & 857 (prime) are all virtually or perfectly circular. Or maybe I'm observing not mathematical but neuro-visual principles.
- coolcase 1y agoNeuro-visual I think. Hard to see 100s of dots in a circle as not a circle. The same principle is why this looks like a circle/ellipse but it isn't: O
- gavmor 1y agoStill, there's something to be said for the variance in the number of prime factors. https://quickchart.io/sandbox/#%7B%22chart%22%3A%22%7B%5Cn%20%20type%3A%20'bar'%2C%5Cn%20%20data%3A%20%7B%5Cn%20%20%20%20labels%3A%20Array.from(%7Blength%3A%20250%7D%2C%20(_%2C%20i)%20%3D%3E%20(i%20%2B%201).toString())%2C%5Cn%20%20%20%20datasets%3A%20%5B%7B%5Cn%20%20%20%20%20%20label%3A%20'Number%20of%20Prime%20Factors'%2C%5Cn%20%20%20%20%20%20data%3A%20Array.from(%7Blength%3A%20250%7D%2C%20(_%2C%20i)%20%3D%3E%20%7B%5Cn%20%20%20%20%20%20%20%20let%20n%20%3D%20i%20%2B%201%2C%20count%20%3D%200%2C%20d%20%3D%202%3B%5Cn%20%20%20%20%20%20%20%20while%20(n%20%3E%201)%20%7B%5Cn%20%20%20%20%20%20%20%20%20%20while%20(n%20%25%20d%20%3D%3D%3D%200)%20%7B%5Cn%20%20%20%20%20%20%20%20%20%20%20%20count%2B%2B%3B%5Cn%20%20%20%20%20%20%20%20%20%20%20%20n%20%2F%3D%20d%3B%5Cn%20%20%20%20%20%20%20%20%20%20%7D%5Cn%20%20%20%20%20%20%20%20%20%20d%2B%2B%3B%5Cn%20%20%20%20%20%20%20%20%20%20if%20(d%20*%20d%20%3E%20n%20%26%26%20n%20%3E%201)%20%7B%5Cn%20%20%20%20%20%20%20%20%20%20%20%20count%2B%2B%3B%5Cn%20%20%20%20%20%20%20%20%20%20%20%20break%3B%5Cn%20%20%20%20%20%20%20%20%20%20%7D%5Cn%20%20%20%20%20%20%20%20%7D%5Cn%20%20%20%20%20%20%20%20return%20count%3B%5Cn%20%20%20%20%20%20%7D)%2C%5Cn%20%20%20%20%20%20backgroundColor%3A%20'rgba(54%2C%20162%2C%20235%2C%200.7)'%5Cn%20%20%20%20%7D%5D%5Cn%20%20%7D%2C%5Cn%20%20options%3A%20%7B%5Cn%20%20%20%20responsive%3A%20true%2C%5Cn%20%20%20%20plugins%3A%20%7B%5Cn%20%20%20%20%20%20legend%3A%20%7B%20display%3A%20false%20%7D%5Cn%20%20%20%20%7D%2C%5Cn%20%20%20%20scales%3A%20%7B%5Cn%20%20%20%20%20%20x%3A%20%7B%5Cn%20%20%20%20%20%20%20%20display%3A%20false%2C%5Cn%20%20%20%20%20%20%20%20title%3A%20%7B%20display%3A%20true%2C%20text%3A%20'n'%20%7D%5Cn%20%20%20%20%20%20%7D%2C%5Cn%20%20%20%20%20%20y%3A%20%7B%5Cn%20%20%20%20%20%20%20%20beginAtZero%3A%20true%2C%5Cn%20%20%20%20%20%20%20%20title%3A%20%7B%20display%3A%20true%2C%20text%3A%20'Number%20of%20Prime%20Factors'%20%7D%5Cn%20%20%20%20%20%20%7D%5Cn%20%20%20%20%7D%5Cn%20%20%7D%5Cn%7D%5Cn%22%2C%22width%22%3A500%2C%22height%22%3A300%2C%22version%22%3A%222%22%2C%22backgroundColor%22%3A%22%23fff%22%7D https://quickchart.io/sandbox/#%7B%22chart%22%3A%22%7B%5Cn%2...
- jhanschoo 1y agoMost large numbers are divisible by some large prime; so suppose that a number is divisible by some large number 11. Then what appears on your screen is some pattern repeated at least 11 times arranged into a circle, because the "largest" pattern is that of the largest prime in the factorization. The numbers that can be factored into primes at most, say, 5 are comparatively very rare.
- jerf 1y agoThat's the difference between the additive view of the world and the multiplicative one. A lot of number theory involves trying to bridge that gap, and even this simplest view of numbers can rapidly fling you into the mathematical unknown. The "simplest hard problem", the Collatz conjecture, can be viewed as coming from this space. You either take a step in multiplicative space, or a step in multiplicative space and then additive space, and ask a very simple question about where those steps can take you. And that's all it takes to end up at an unsolved problem in math. You can spend a lifetime on this simple observation that "the jumps between neighbors are so dramatic", just trying to reconcile the complex relationships between the additive view of the world and the multiplicative one. And we haven't even done anything like bring in complex numbers, or rationals, or introduce exponentiation!
- gavmor 1y agoHow can a layperson approach and develop correct intuitions for "the multiplicative view" of numbers? Is it practical?
- tocs3 1y agoI am a layperson but... You could try writing out an addition table and a multiplication table and see if you can find patterns and differences (you can). The Sieve of Eratosthenes is good. I asked ChatGPT what branch of mathematics high school algebra is and it suggested the field theory of real numbers. I have since been looking at groups and fields with some enjoyment. I made a video that I thought of as a play on Ulam's Spiral [2][3] a while back. Instead of marking primes I marked points of the (square) spiral that were x mod n 0. It is sort of silly and maybe a bit confusing. [1] https://en.wikipedia.org/wiki/Sieve_of_Eratosthenes https://en.wikipedia.org/wiki/Sieve_of_Eratosthenes [2]https://en.wikipedia.org/wiki/Ulam_spiral https://en.wikipedia.org/wiki/Ulam_spiral https://www.youtube.com/watch?v=fRzv1SXKCY8 https://www.youtube.com/watch?v=fRzv1SXKCY8 (my video play it slowed down maybe)
- ViscountPenguin 1y agoThe best way to get a multiplicitive view of numbers is to transform multiplication into a special form of addition. One of the more general forms of this is the logarithm. The defining equation of the logarithm is that log(a*b) = log(a) + log(b). This turns multiplication into a nice linear addition problem. A less general but more powerful transformation exists on the integers. You can factorize any integer into it's prime factors, then multiply by adding together the powers on matching primes. This may seem like more effort for a similar result, but the prime factors of an integer tell you a lot about that integers divisibility, so it's worth the effort. If you're talking about the interplay between multiplication and addition: Get a degree in pure maths, learn lots of number theory, and cry at how inelegant it is. Trust me, I did it in a past life