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How can a layperson approach and develop correct intuitions for "the multiplicative view" of numbers? Is it practical?

by gavmor 1y ago

tocs3 1y ago
I 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 ago
The 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

How can a layperson approach and develop correct intuitions for "the multiplicative view" of numbers? Is it practical?

by gavmor 1y ago

How can a layperson approach and develop correct intuitions for "the multiplicative view" of numbers? Is it practical?

I 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 ago

The 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