So monzos are used in tuning theory to give a unique prime factorisations to intervals. An interval is just the distance between two notes. It's also common to keep the intervals between 1 and 2 because of octave equivalency—ie 2^(a)×r is viewed as the same as 2^(b)×r, for all integer values of a and b. So if we have the interval 3/2, we can write it as 2^(-1)×3^(1). We oftentimes put the powers inside [⟩ brackets so the monzo of 3/2 would be [-1 1⟩, [-2 0 1⟩ for 5/4, [1 1 -1⟩ for 6/5, [0 0 0 -1 1⟩ for 11/7, and [a b c... x, y⟩ for any arbitrary interval 2^(a)×3^(b)×5^(c)×...×pₙ₋₁^(x)×pₙ^(y). So yeah, it's pretty much just unique prime factorisations of all rational numbers and I wondered if something similar to this was used in contexts closer to pure maths.
When trying to find counterexamples for propositions akin to "E(XY) = E(X)E(Y) implies X and Y are independant" I always come up with solutions of the form X = f(Y,Z) where Y and Z are independant.
And now I'm wondering: if we have two real valued random variables X and Y, is there always a random variable Z independant of Y and a measurable function f such that X = f(Y,Z)?
My intuition is that it should be true. The Y part would represent the dependance of X upon Y and Z would be where the rest of the randomness lies.
In the extreme cases it is true. If X and Y are independant then take Z = X and f(Y,Z) = X. And if X is Y-measurable then we know there exists g measurable such that X = g(Y) we can then take any Z independant of Y and let f(Y,Z) = g(Y) = X.
For the general case I thought of considering E[X | Y] = g(Y), Z = X - E[X | Y] and f(Y,Z) = g(Y) + Z but it feels too simple to actually work. I'm really unsure as wether Y and Z would always be independant in that case.
Is this something well-known in the litterature?
P.-S. Probabilities are not my speciality so I may be missing something trivial, sorry in advance
Hi! I had a maths test today and everyone I've spoken to put different answers for this. I got -4(x+6)(x-6). I was hoping somebody could let me know what the correct answer is! Thanks :)
Can anyone help solving this question x ^ { 4 } - 6 x ^ { 2 } + 1
hey guys, second question in a day but this will be the last one. I understand part a and c, and have got the right answers for them. however, there wasn't an answer for b and I haven't heard this wording before. I don't really understand what to do.
thank you in advance!
I noticed that my answer was 2^6 × 7^14. It probably has an easy explanation, but I guess I'm still a bit sleepy....
All solutions I've seen use DP using summing but if there's a pattern, then we could exploit it and directly figure out the exponents in the prime factorisation, right?
EDIT:This works because:
- There's no gap of 2.
- The longest consecutive run is 5.
This means, that to get from one group of consecutive numbers to the next one you need to make a 3 jump, that is you need to jump between the edge numbers.
That is the solution is determined by how many ways you can get from the lowest number in a group to the highest in each of the groups.
This number of ways to traverse a group is only determined by its size and for sizes <= 5 its of the form 2^a × 7^b. The overall solution is just a product of the partial ones.
What’s more annoying is that I know the teacher won’t properly check my homework (I mean, math homework from 2 books, for around 40 students and adding their other classes I understand it’s a lot of work BUT DONT GIVE THAT MUCH HOMEWORK IF YOU CANT EVEN PROPERLY CHECK IT EHHH) Anyways, pay attention to factorisation while you can, it’s gonna be a pain in the ass later
I’m from the Netherlands and I’m looking for some stuff to make my lessons about fractions more interesting and/or more easy to grasp. I’m teaching children who are 11/12 years old.
The subjects that will pass are times tables but backwards, prime numbers, divisibility tricks, prime factorisation, simplifying fractions, adding and subtracting fractions and multiplying fractions.
I’m looking for a lot of stuff and maybe you can help me out:
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Activities or videos that make for a good introduction (the kids are Dutch though). Why do they need the subject? What’s fun and cool about the subject?
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Good ways to visualise what’s happening with the fractions such that the way they have to calculate things makes sense to them. I don’t want them to only learn tricks, I want them to understand it on a deeper level.
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Maybe you know of some interesting math olympiad like stuff that’s linked to this subject to keep the eager students busy.
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If you have other interesting stuff that’s linked to this subject, please let me know.
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Also, if there are better places on the internet to ask such questions, please let me know.
Create a function to find the prime factorisation of a natural number, n > 1. Your output should be a vector containing all prime factors, listing them each as many times as they are a factor. For example, MATLAB should output ans = 2 2 2 3 when asked to factorise 24.
You can't use fprintf and factor functions.
also can't use the sqrt function
How do I solve 4x²+4xy-15y² and why? I know you have to split the 4x, answer says into 10xy-6xy but why those numbers? Confused about these types of questions
How do I factorise the following;
A^(l)(x) = 2x - 8000x^(-2)
It's for an optimization question so it needs to be in a format where I can find the x-intercepts.
Thanks!
Dear community,
I am trying to factor x^4-x^3-x^2-x-2=0.
I want to get it into the form (x-2)(x+1)(x^2+1)=0.
I think we have to do it by grouping. It would be helpful if you can throw some pointers (preferably without solving it).
I'm really good at math and I have a decent grasp of computer science. I understand that multiplying two prime numbers to get a huge number is easy, but checking out if a huge number has only two prime factors is a monumental task for a computer. What I don't get is how this is used for encryption and coding and decoding messages. I keep reading about this in books and they keep talking about how one side is the key or whatever but they never really explained how it all works. Every book seems to love explaining the whole large-numbers-take-a-lot-of-time-to-factorise concept but not how it actually works in encryption. I understand basic message coding--switch around the alphabet, add steps that changes a message into a mess of letters; then the recipient has to do all those steps backwards to change it back. How do prime numbers and huge numbers fit into this? How does knowing a pair of factors enable me to code a message and how does knowing the product enable my recipient to decode it?
So I know this might be easy for a few of you, but I’m stuck on this one question
(A+B)(A+B) + 2A + 2B
If you know the answer, can you walk through how you got it? I’m really trying to understand it
How do I factorise indices for example 12*n+1
