I think this is the kind of thing I could probably Google and get an answer but...could you describe in 100 words or so what "number theory" is? I'm asking because like a lifetime ago (actually the divorce was only finalized 10 years ago but I may be experiencing some weird emotional red/blue shift because I want it to be in the distant past) I was married to a guy who minored in mathematics as an undergraduate and he was incredibly dismissive of people who in his words confused "arithmetic" with "mathematics." And in his view, calculus and statistics was "arithmetic." Mathematics, apparently, was restricted to the kind of arguments that might have been put forth by an intellect like Srinivasa Ramanujan.
Your post was very cool and it got me thinking along some lines I had not previously thought about, but it's also the kind of reasoning my arrogant ex would have dismissed as "arithmetic" -- solving a problem through clever but ultimately iterative, exploratory, "find the number that fits" methods -- not a universal abstract solution expressed in the language Bertrand Russell would have used.
I'm really asking you to provide one more reason I should stop feeling inferior to a person who left my life a decade ago; if that's not your jam I perfectly understand. I really am interested in a Richard Feynman "explain it to a five year old" definition of number theory, though.
Your ex is of a type that I am confident I could describe pretty well. Narcissist and highly insecure, but overcompensates in dealing with that insecurity. Ironic AF that he only minored in mathematics. If he was half as smart as he pretended to be he'd have double majored in it, but if you major in mathematics you end up taking a broad enough array of courses in it that you'll have to take at least one or two in some area where you have no natural talent and that's a risk of proving you're not as smart as you think are. (Because all my natural talents are in things like writing and drawing I earned every bit of my competence through practice and effort, but I watched this happen a lot.) Our culture operates on the HIGHLY mistaken notion that "good at math" = a natural state for some people but not others and that this state is also equal to "intelligent," and it's complete and total horseshit. People who actually know what they're doing tend to only differentiate "arithmetic" from "mathematics" in a self-deprecating way that's clearly intended to be a joke. Like the time my advisor and one of his PhD students had spent hours stuck on something and I came by the office for paperwork reasons and spotted that they had divided 12 by 3 and gotten 3 near the bottom of one whiteboard, which got them unstuck. They laughed and said, "Never trust a mathematician's arithmetic." Which is both something they said while laughing at themselves, thanking me for pointing it out, and also something that, as a PhD and a PhD in training, they would've had the right to use in a way much closer to how your ex used it but clearly weren't. After all, they lost a huge chunk of an entire day--pretty damn important to not make those kinds of errors.
To answer your question, number theory problems have to do with the properties and relationships of types of numbers. There are many types of numbers, like the integers (comes from the same root word as 'integrity', i.e. whole, but includes negatives), natural numbers (sometimes also called counting numbers, and there's fierce debate over whether these start with 0 or 1), rational numbers (which can be written as fractions, which for example the square root of 2 cannot), etc. Number theory is about the types of problems that can be solved by using the properties and relationships of types of numbers, nearly always integers. For example, any even number can be represented as 2n for some value of n, and any odd number can be represented as (2n+1) for some value of n. Doing a proof of the fact that an odd number multiplied by an odd number always produces an odd product involves using two different representations -- two different stand-ins -- for odd numbers, like (2n+1) and (2m+1).
Your ex was a blowhard who wanted you to think he was a lot smarter than he actually was. Laugh at him if you can.
Hello Holly and thank you so much for this series. I found the eight values, but it doesn’t matter their order right? At least they all add up to 1001. However, in factoring, I came up with four copies of T and V and two copies each of YU, UX, XW AND WY. I don’t recall ever learning this kind of math before, or maybe I just glazed over when it was taught in school. Your explanations make lots of sense to me and even though I’m slow, I can follow along. It took me most of the morning and some afternoon to work through it while referencing your figures. The eight values I came up with are: 140, 120, 12, 14, 350, 300, 30 and 35. I’m not looking for a prize, just knowing that I solved the problem correctly is prize enough for me😀
Cool factoring method! I would not have thought to do that at first. It’s a technique I teach, but I’ve not really had occasion to use it.
I knew I'd end up there, but it was fun to show the thought processes, step by step.
I think this is the kind of thing I could probably Google and get an answer but...could you describe in 100 words or so what "number theory" is? I'm asking because like a lifetime ago (actually the divorce was only finalized 10 years ago but I may be experiencing some weird emotional red/blue shift because I want it to be in the distant past) I was married to a guy who minored in mathematics as an undergraduate and he was incredibly dismissive of people who in his words confused "arithmetic" with "mathematics." And in his view, calculus and statistics was "arithmetic." Mathematics, apparently, was restricted to the kind of arguments that might have been put forth by an intellect like Srinivasa Ramanujan.
Your post was very cool and it got me thinking along some lines I had not previously thought about, but it's also the kind of reasoning my arrogant ex would have dismissed as "arithmetic" -- solving a problem through clever but ultimately iterative, exploratory, "find the number that fits" methods -- not a universal abstract solution expressed in the language Bertrand Russell would have used.
I'm really asking you to provide one more reason I should stop feeling inferior to a person who left my life a decade ago; if that's not your jam I perfectly understand. I really am interested in a Richard Feynman "explain it to a five year old" definition of number theory, though.
Your ex is of a type that I am confident I could describe pretty well. Narcissist and highly insecure, but overcompensates in dealing with that insecurity. Ironic AF that he only minored in mathematics. If he was half as smart as he pretended to be he'd have double majored in it, but if you major in mathematics you end up taking a broad enough array of courses in it that you'll have to take at least one or two in some area where you have no natural talent and that's a risk of proving you're not as smart as you think are. (Because all my natural talents are in things like writing and drawing I earned every bit of my competence through practice and effort, but I watched this happen a lot.) Our culture operates on the HIGHLY mistaken notion that "good at math" = a natural state for some people but not others and that this state is also equal to "intelligent," and it's complete and total horseshit. People who actually know what they're doing tend to only differentiate "arithmetic" from "mathematics" in a self-deprecating way that's clearly intended to be a joke. Like the time my advisor and one of his PhD students had spent hours stuck on something and I came by the office for paperwork reasons and spotted that they had divided 12 by 3 and gotten 3 near the bottom of one whiteboard, which got them unstuck. They laughed and said, "Never trust a mathematician's arithmetic." Which is both something they said while laughing at themselves, thanking me for pointing it out, and also something that, as a PhD and a PhD in training, they would've had the right to use in a way much closer to how your ex used it but clearly weren't. After all, they lost a huge chunk of an entire day--pretty damn important to not make those kinds of errors.
To answer your question, number theory problems have to do with the properties and relationships of types of numbers. There are many types of numbers, like the integers (comes from the same root word as 'integrity', i.e. whole, but includes negatives), natural numbers (sometimes also called counting numbers, and there's fierce debate over whether these start with 0 or 1), rational numbers (which can be written as fractions, which for example the square root of 2 cannot), etc. Number theory is about the types of problems that can be solved by using the properties and relationships of types of numbers, nearly always integers. For example, any even number can be represented as 2n for some value of n, and any odd number can be represented as (2n+1) for some value of n. Doing a proof of the fact that an odd number multiplied by an odd number always produces an odd product involves using two different representations -- two different stand-ins -- for odd numbers, like (2n+1) and (2m+1).
Your ex was a blowhard who wanted you to think he was a lot smarter than he actually was. Laugh at him if you can.
Hello Holly and thank you so much for this series. I found the eight values, but it doesn’t matter their order right? At least they all add up to 1001. However, in factoring, I came up with four copies of T and V and two copies each of YU, UX, XW AND WY. I don’t recall ever learning this kind of math before, or maybe I just glazed over when it was taught in school. Your explanations make lots of sense to me and even though I’m slow, I can follow along. It took me most of the morning and some afternoon to work through it while referencing your figures. The eight values I came up with are: 140, 120, 12, 14, 350, 300, 30 and 35. I’m not looking for a prize, just knowing that I solved the problem correctly is prize enough for me😀
Work is kicking my ass the last few days, but I'm going to get back to you on this! Thank you!!