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That'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 thi
by jerf 1y ago
That'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