Feb 5, 2024
Recap of the stuff I learnt today
I learnt a cool algebra manuver and also the AM-GM inequality today
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Recap of the stuff I learnt today
I learnt a cool algebra manuver and also the AM-GM inequality today
Hello! for the longest time I've thought that I'm just "not good" at math, and that I'm just a creative type or something, but for the past few years I've been trying to brute-force my way through math drills and stuff to improve because I want to go into a STEM field. I've found that it doesn't work. I have, however, gained a kind of Stockholm syndrome for math, and I find myself wanting to know *more*. I feel like I'm missing out on something way beyond myself but idk where to even find that. So here i am. Asking a blog on tumblr for the secret math knowledge. If you have any resources that i could read about math that isn't just "here's how to do this" i would really appreciate it!!!
First of all, complete props to you for giving something that doesn't initially appeal to you, in this case, maths, a chance. Mathematics can be frustrating and even annoying, being an oftentimes nit-picky field. You can quickly realize that these are also the traits that grant the satisfaction and euphoria of mathematics. So don't forget to be proud of yourself for not disregarding mathematics as a whole and actually giving it a go.
Second of all, creativity is very necessary for maths. It is a shame that the educational system doesn't do maths justice (A Mathematician's Lament by Paul Lockhart is exactly about this). Mathematics is presented as something mechanical, force-feeding formulas and having students repeat them to find adequate results. That is not a fair portrayal of mathematics.
Different people will have different conceptions of what the learning of mathematics is or should be - I mean, for god's sake, mathematics hasn't even been defined properly and one expects people to know what to do with it. I can only tell you what I conceptualize mathematics, and its learning, to be. For me, mathematics is the boring work of examining multiple results and cases of the same formulas, it is the finding of patterns and attempting generalizations, it's the excited scribbles of formulas and the necessity of looking at a particular scenario from multiple perspectives. Most likely, the generalizations won't come easily, a minus sign will be forgotten making the following calculations obsolete, an approach to a problem will prove fruitless, and laborious work will be done only to find out there was a specific theorem that would have shortened the whole process. Patience is required for maths, and hopefully, you can now see that creativity is too, there is no shortcut to knowing where to look or what technique to utilize. As is the case with most worthwhile pursuits, and as you know from experience, mathematics is endlessly frustrating, but that also means it is endlessly fun.
Now, I do not have any secret math knowledge (or do I), what I can do is present you with some things that fascinate me and have led me to love mathematics as much as I do, outside of the conventional mathematics curriculum.
Maths really comes down to practice, practice, practice... as many other things. I heavily encourage you to start playing around with mathematics a little bit. Olympiad mathematics kind of do that, (I, very conveniently, have a lot of posts about that.) as do many other math competitions. Maybe try out some exercises from your national olympiad and, with no judgment, because this is just to have fun, play around a bit. Disregarding the conventional maths you know, test out your logic, laugh at mistakes and losses of time, and feel that happy rush when a conjecture you reach, or part of it, is correct.
In terms of the resources, I opted for a mix of funny math history moments and some actual mathematics, in no particular order. Prepare for confusion. ↓
Being autistic and raised "gifted" is a really weird kind of alienation because it is very rare for you to struggle with comprehension but it seems like you just can't properly communicate with educators and other students about the curriculum. Whenever you try to help another student it seems like everything you're saying isn't coming across, and whenever you ask an educator for help it takes a lot of time for them to properly understand your question.
The hardest part about lessons and tests isn't figuring out the answers, it's figuring out how to phrase it in a way that will communicate that or what you actually know.
This is why olympiads and competitions are an incredible privilege, since it funnels together a lot of people raised "gifted", including others who might've autism.
Without having participated in olympiads as a kid i wouldn't have met anyone who related and therefore understood the "gifted" part of me. I would either purposefully dissociate from that part of myself causing way worse academic performance or think that nobody cares about me, as happens a lot with people who assume that i do not know something which i struggle to communicate.
I say this while also keeping in mind that competing in some form of intelligence is a very prevalent driving force of the really toxic "naturally gifted (and therefore better than you)" mentality that a lot of smart people develop (which a lot of people close to me suffered from because i had developed that before any proper emotional maturing happened).
This all is about people who were raised "gifted" but i feel like the assumption that "anything you can't communicate efficiently is something you don't understand" might be a contributing factor to the more general widespread patronisation of people with ASD.
BMO1 2021 Question 6
This is another question from the 2021 BMO1 paper which is pretty neat.
Start with we know that n = 2^a + 2^b for some distinct integers a and b
Also, we know that n = 2^p - 1 + 2^q - 1 = 2^p + 2^q - 2 for some p and q such that 2^p - 1 and 2^q - 1 are primes
First, without loss of generality we can assume that a < b and p < q
For now let’s just focus on the Mersenne Prime, 2^p - 1
Notice that if you can write p = rs for some integers r and s, then without loss of generality if we assume that r > 1 and let x = 2^s then:
If r is even we can write 2^p - 1 = x^r - 1 = (x^(r/2) + 1)(x^(r/2) - 1)
If r is odd, we can write 2^p - 1 = x^r - 1 = (x - 1)(x^(r-1) + x^(r2) + ... + 1)
In both cases this violates the primality of 2^p - 1 whenever r < p or s > 1, and therefore we know that we must have r = p and s = 1. This is telling us that p is a prime number.
Now we may go back to 2^p + 2^q - 2 = n = 2^a + 2^b
If we consider both of these expression in binary:
- 2^a + 2^b can be written as (10)^a + (10)^b in binary, which tells us that the number has a 1 in the (a+1)th and (b+1)th digits and zeroes everywhere else. In particular, this tells us there are EXACTLY two 1 digits since a =/= b.
- 2^p + 2^q - 2 can be written as (10)^p + (10)^q - 10. When subtracting the 10, starting from the 2nd digit, any zeroes become a one until you reach the first 1. Because you know that p < q this means that you’ll always hit the 1 in the (p+1)th place before the 1 in the (q+1)th place. This is important because this tells us that q = b.
Additionally, since we know that q = b, we know that 2^p - 2 = 2^a and this can only be true if p = 2, a = 1. This is because the left hand side is divisible by 2 but not 4, so a = 1. If a = 1 then p could only be 2.
Now finally we have: n = 2^q - 2 + 2^2 = 2^q + 2
= 2^(q-1) + 1 + 2^(q-1) + 1
Notice that q-1 is even since q > 2 is prime and therefore odd. Let q-1 = 2k for some integer k. The we may write
n = [2^(2k) + 2*2^(k) + 1] + [2^(2k) - 2*2^(k) + 1]
-> n = (2^k + 1)^2 + (2^k - 1)^2
I liked using binary digits to deduce information about a, b, p and q. Ultimately it’s the same as considering various conditions of divisibility, but it was a nice visual aid.
The last step of adding in the terms 2*2^k - 2*2^k to get two squares was quite fun as well.
Solving An Incredibly Hard Problem For 15 Year Olds
Please help me mathblr :,)
I’ll be starting my degree in applied maths and computation in less than a month. Apparently, in my college, Linear Algebra and Integral and Differential Calculus are two of the hardest courses (I am unsure if it’s because of the complexity of the subjects or the lack of care by the teachers, although I’m willing to bet it’s both). Does anyone have any resources (Books, YouTube channels, online lectures, etc.) that could help me out? I’d really appreciate it!!
Content related to other subjects are also welcome, along with general advice about pursuing a (math) degree!!
Im terrified to start uni, can you tell?? Also excited tho
Abstract Algebra book recommendation?
Does anyone have any good abstract algebra books that you’d recommend for uni?