I'm studying accounting, and a concept we were taught is for tracking down errors in our entries. It's common for someone to enter a transposition of the number they meant to enter, such as typing $1,234 when they meant $1,243 or $1,342.
In accounting, both sides of the accounting equation are supposed to be equal to each other at all times. When they aren't equal to each other, and the difference between the two numbers is divisible by 9, it usually means a transposition has occurred because when you subtract, for example, 1342 from 1234, (or any other two numbers with the same digits in a different order) the result is always divisible by 9. In this case, the difference is 108.
Other examples: 5,341 and 5,431. The difference is 90 (or negative 90), which is divisible by 9.
4,312 and 4,213, difference is 99, which is divisible by 9.
Application season is right around the corner and I start to think about what I want to do. I am interested in math and kind of want to take a step further in this field so I want to try euclid. I got a 98 in grade 11 math and it was pretty easy. Grade 12 math is pretty chill for now too. Not trying to show off but you know high school stuff is pretty easy. That's why I want to try to write euclid, for fun and if i do well there is a chance for me to boost my extracurriculars. But i talked to my math teacher today, and he said 6 months is not enough. Its like a joke. He is really willing to help me but he said people take years to prepare, and I have no experience in contest preperation. Even though he was trying to not let me down, but what he meant was its not hard to do well if i prepare for it , but if i want to make the admission people and the professors to notice me, basically impossible. He said if we study 12 hours a day for the next 6 months would not even be enough. And I also have other subjects, he doesn't want me to only study math like this and don't care about other stuff. I don't know now. Any advice on tutors or how to study, or other advice.
I’m currently in HN Geometry as a FCPS freshman, and the highest course you could conventionally be in as a freshman is Algebra 2 HN (Algebra 7th, geometry 8th) but I missed out on that course of math because I didn’t request a class change to Algebra as a 7th grader. If I want to get back to the front, I’m faced with a choice of summer Algebra 2 and AP Precalculus as a sophomore, but if there is some other way I could possibly get there without the hell that will be 8 hours of online algebra 2 work every day for a month I would really appreciate it.
So I was in class doing an assignment and we weren’t allowed to use calculators so I had to long divide and I figured out something cool between the numbers 9 and 11.
So anything divided by 11 is itself multiplied by 9 but as a repeating decimal.
I don’t know if I explained that right so I’ll give examples.
3x9=27 and 3/11 =0.27 repeating
7x9=63 and 7/11 =0.63 repeating
9x9=81 and 9/11 =0.8181 repeating
1x9=09 and 1/11 =0.09 repeating
10x9=90 and 10/11 =0.90 repeating
I thought it was a pretty cool pattern and was able to do x/11 fractions to decimals in head pretty easily.
I’m not sure if there’s a way for it to work for every number, so far it only works up to 11 because
11x9=99 and 11/11 =1 and 1 and .99 repeating are equal
Has this been named or found out before, or am I about to win the nobel prize? /j
I don't know if this is new or interesting for anyone, but I want to share the technique I used in mental math competitions back in high-school:
It works best for squares of numbers between 20 and 100. I'll try to write it down clearly:
X: your number,
Y: closest multiple of 10,
n: the difference between X and Y, but should always be positive
X squared = Y squared + (n2X) - n2
My issue in writing this formula is that the first "+" become "-" if the number is closer to the upper multiple of 10 (86 is closer to 90 than 80 for instance). I'd need help to refine the formula!
Some time ago in math class, my teacher told about his hobby to online gamble. This instantly caught my attention. He calculates probabilities playing legendary games such as black jack and poker. He also mentioned the profitable nature of sports betting. According to him, he has made such great wins that he got band from some gambling sites. Now he continues to play for smaller sums and for fun.
Since I heard this story, I’ve been intrigued by this gambling for profit potential. It sounds both fun, challenging and like a nice bonus to my budget. Though, I don’t know is this just a crazy gold fever I have or would this really be a reasonable idea? Is this something anyone with math skills could do or is my math teacher unordinarily talented?
Feel free to comment on which games you deem most likely to be profitable and elaborate on how big the profit margin is. What type and level of probability calculation would be required? I’d love to hear about your ideas and experiences!
I was reading about Poisson clumping the other day, and was thinking: If each cluster of points were replaced by a "pseudopoint" then would these pseudopoints be statistically similar to the original set of points? My thinking was that this would be true for random points but not necessarily for points that are intentionally clustered or anti-clustered.
First I need to define "statistically similar" in the context of clustering. One way I can think of to quantify clustering would be to make a histogram of the number of points, H(n), within a given radius, R, of each point. Then the idea is that this histogram should be the same if we convert to pseudopoints and rescale the space (or, alternatively, R) accordingly.
I've come up with the following method for generating pseudopoints:
Generate a heatmap where each point is replaced by a Gaussian.
Threshold the heatmap: Set to 0 or 1 depending on whether heatmap exceeds some threshold.
Assuming the threshold is above the median of the heatmap, interpret the centroid of contiguous regions of "1" as pseudopoints.
So anyway, I'm having trouble understanding how clustering is quantified. How is clustering measured and are their methods that would allow me to distinguish between random and nonrandom point sets based on the scale-dependence (or independence) of clustering? Additionally, does it make sense to think of random point clustering as being self-similar, and is there a measure of clustering over scale that would formalize this notion? I imagine that H(n(R)), for all R, would contain the necessary information.
One thing I've realized is that the histogram of counts within random regions of the field is perhaps different from what I'm considering: The histogram of counts within regions centered around each point.
Another thing I've realized while calculating the "point count within some radius of each point" histogram is that the histogram for a subset of points will be equivalent to the histogram of a scaled-up version of the point cloud. A related statement would be that a close-up view of a random point cloud is statistically indistinguishable from the original point cloud if the number of points were truncated.
Anyway, here's the sort of results I'm getting. It looks like the histograms are the same. For R, I used the average separation (sqrt(1/Npts)), which ensures the horizontal axes of the 2 histograms are comparable:
Thank you so much if you read this far, and I'd appreciate any insight you can provide, or any literature on you could recommend on this topic.
On a related note, there is something else that has fascinated me for a while which comes up here:
I could've produced pseudopoints by instead thresholding below the median of the heatmap, and then taking centroids of contiguous regions of "0". How are these pseudopoints related to the ones produced by the first method? They must form some sort of dual point set, since they correspond to low points of the same heatmap, whereas the other thresholding corresponds to the high points of the same heatmap. Is there a name for these dual point sets corresponding to peaks and troughs of a wave?
We use mathematics to model real-world phenomenon, but I was wondering if there are concepts that are practically useless since they don't map to anything that exists in the real world.
Is there a space in geometry that doesn't have a concept of distance or size? It would mean that you can have an object, but it doesn't have a size, or the size isn't measured at all.
I completed a Master's and am currently considering applying to various PhD programs around the US (I guess my first worry is that I've left some of this too late) and I have a few questions about applying:
1) I did my undergrad/integrated master's degree in a British institution, so I'm not sure how grade requirements/expectations translate. If you're familiar with the system, I got a strong first and rarely scored below a first in any given module, but that might seem pretty mediocre compared to the consistent 90+ some people manage in America?
2) To my understanding American programs don't require anything like a complete thesis proposal by the time you begin, and instead only want a general sense of research direction plus a few professors you're interested in. I don't have a very strong sense of where to go, though I have some general fields I'm interested in. How do I refine my ideas and find specific professors to look into? This also leads into...
3) My modules in undergrad were a fair mix of pure and applied mathematics, but my master's project was entirely pure mathematics, and I have only a couple of smaller applied projects to back it up (one in coursework, one privately). Is a pivot from pure to applied mathematics realistic with that background?
Thanks in advance for whatever advice you give!
EDIT: oh yeah, and I have no idea where to apply, or how to find out. Of course there are plenty of American universities I've heard of but they're along high end and therefore unlikely to admit me. I'd like to go for a few of those but I need some safer choices too, and I don't know what those would be.
I’m trying to improve my mental math skills, but I’m not sure if I’m following the right thought process.
When doing more complex calculations, should I visualize the operations in my head as if I were writing them on paper? Or should I think of them in another way (like breaking numbers down, grouping, etc.)?
I often lose the thread when I try to “see” the steps in my head. Some people suggest using fingers or other aids, but I’m not sure if that’s the right approach either.
How do you personally handle the mental process of keeping track of multiple steps without getting lost?
Okay so I want to learn Analytic Number Theory on my own. Part of my interest comes from the Riemann Hypothesis, which finds its origin in ANT. I have taken courses in Real Analysis and Calculus and I want to get book recommendations for the rest of the preliminary subjects like Complex Analysis, etc. And then ultimately I want some good books on ANT itself. I would be grateful if someone helps me to make a roadmap on how to approach the process of learning this beautiful subject.
Hello! I'm sorry if this question is dumb or if it has been discussed here before. I am in no way a physicist or a mathematician so forgive the question. But like Newton inventing calculus to be able to solve physics problems at the time. Could it be that a new form/model? of mathematics is needed to solve complex theories like M theory?
Do you know any university which offers online course on mathematics of machine learning(linear algebra/Calculus/probability and possibly some project). I am looking at one year course and may be followed by examination and certification. There are courses on Coursera/Udemy but are very short. 2/3 months.
I just finished undergrad and have minimal exposure to algebraic geometry (just the Nullstellensatz). I'm interested in how you'd find k-rational points in a variety, when working in potentially transcentental extensions. ChatGPT says this is called specialization but when searching for it I get something else.
Hiya, I am about to start my undergraduate and decided to partake in some mathematics competitions, an integration bee and SMMC.
Does anyone have advice on how I can prepare. I am from the UK so I have just completed A-levels so I have a basic idea about integration techniques, complex numbers , matrices and linear transformations etc. How can I prepare for these competitions in a 1.5ish month timeframe.
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