I was talking to my (12 year old) son about parts of math he finds boring. He said that he thinks absolute value is absurdly easy and extremely boring. I asked him if there was anything that might make it more interesting, he said "maybe complex numbers".
So I asked him "what would the absolute value of i+1 be?" he thinks for a little bit and says "square root of 2" and I ask him "what about the absolute value of 2i + 2?" "square root of 8"
I ask him "why?" and he said "absolute value is distance; in the complex plane the absolute value is the hypotenuse of the imaginary and real numbers."
So -- first of all, this was a little surprising to me that he'd thought about this sort of thing having mostly just watched youtube videos about math, and second, this sort of understanding is a result of some manner of understanding the underlying mechanisms and not a result of just having a huge dictionary of synonyms.
To what degree can these large language models arrive at these same conclusions, and by what process?
So I asked him "what would the absolute value of i+1 be?" he thinks for a little bit and says "square root of 2" and I ask him "what about the absolute value of 2i + 2?" "square root of 8"
I ask him "why?" and he said "absolute value is distance; in the complex plane the absolute value is the hypotenuse of the imaginary and real numbers."
So -- first of all, this was a little surprising to me that he'd thought about this sort of thing having mostly just watched youtube videos about math, and second, this sort of understanding is a result of some manner of understanding the underlying mechanisms and not a result of just having a huge dictionary of synonyms.
To what degree can these large language models arrive at these same conclusions, and by what process?