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Mathblr gets an exclusive sneak peek as to what video I’m uploading next Monday. Every Monday, I plan to upload some sort of math related video and Monday’s will now be dubbed “Math Monday.” You can check out my YouTube channel here! Please watch my latest video, like, and subscribe for more study/math content~

This video will be me in my PJ’s studying for abstract algebra on my whiteboard for 15 minutes. I tried to make the video 20 minutes but my phone fell down and ruined the recording LOL

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Why can we just make a hole in an equation by multiplying it by (x-n)/(x-n) with n being a number. It’s the same equation, and the hole can be canceled out, so why do we graph a hole there anyway? Couldn’t you just get rid of the whole graph by just replacing n by every x value that exists? Tldr: holes in equations???

(x-n)/(x-n) makes a hole at x=n because, for x=n, there is a zero in the denominator, and as everyone knows, you can’t divide by zero. To explain this fully, let’s graph the function y=(x-2)/(x-2). As you can see from the graph, (x-2)/(x-2) is equal to 1 everywhere except for x=2 where it is undefined (you can use the table function or check the value at x=2 by hovering over the point in Desmos to verify this). 

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So, when you multiply anything by 1, you get the same thing, and when you multiply something by ‘undefined,’ you get ‘undefined,’ which is represented on a graph as a hole because no value is returned by the function. This means that when you multiply a function by (x-n)/(x-n), you get the exact same function everywhere except x=n, where you get undefined, which again causes a hole in the graph. 

As to why you can’t ‘cancel’ the (x/n)s, the answer is complicated. The short answer is that it’s a function. Because there is not a single answer, every operation must be done in a way that does not hide or eliminate information, specifically zeroes in the denominator. Here’s why: canceling something in the numerator and denominator only works because it is the same as saying the two expressions are equal, ie 2/2*5=5. However, the same cannot be said for functions which cause zeroes in the denominator because (x-2)/(x-2) *is not equal to 5 when x=2 (because it’s undefined). I know this feels like circular logic, but it’s the best way to explain it.

– Admin Noah

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My favorite number is 2; therefore it is also my favorite single-digit (base 10. all of these are base 10. I will have to think more about my favorite primes in base 2) prime.

My favorite two-digit prime is 19.

My favorite three-digit prime is 101.

My favorite four-digit prime is 1093.

My favorite five-digit prime is 12289.

Please feel free to recommend good larger primes and argue in favor of your favorite primes less than 100,000.

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Normally wouldn’t post a personal caption (or share such scrawly notes) but just failed my NBME baseline and will have a lot (a LOT) of work to do over the next few weeks to get that score up as high as it will go! The apathy and burnout are real so trying to find a way to change that narrative in my head. May convert my diary partially into a CBT journal just to keep going.

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Otto Stern and Walter Gerlach performed a groundbreaking experiment in 1922 illustrating a so called two-state-problem. They used a collimated beam of silver atoms (in y-direction) and let it go through an inhomogeneous magnetic field so that the atoms did not only experience torque but also a force.

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For our purposes it is sufficient to look only at the z-component (x is going to be neglected in the further discussion. Still we should keep in mind that simplification is not quite correct because we need to fullfill Gauss’s law for magnetism). Assuming a constant magnetic momentum we can conclude:

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Thus Atoms with magnetic momentum larger than zero experience a downward force and vice versa.

Thinking in terms of classical physics one could assume that this experiment gives us a distribution of the distracted atoms looking like a gaussian curve. Why? Because the orientation of the atoms is random, thus the angular momentum (following from the electrons in the atom’s hull) and the magnetic momentum is random. The conclusion would be a somewhat continous gaussian looking distribution (in z-direction!)

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This is the moment when “strange” quantum mechanic stuff comes into the picture. Our assumption was wrong. What Stern and Gerlach measured was the following distribution:

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What does this mean? There is a quantization of the magnetic moment’s z-component (thus also of angular momentum)! It can be described mathematically as follows:

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This kind of  quantized “angular momentum” the electrons got is the spin we wanted to understand. In this two-state-problem the spin’s value is either +(½) hbar or -(½) hbar. Any analogies using rotations from everyday life are wrong! Spin is way more abstract. There is simply nothing in our everyday lifes corresponding to it - we can only accept that fact.

Actually this experiment does not really help to understand the nature of spin but at least we can understand why we need the concept of spin and that it really exists.

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I hope everyone has been enjoying Animal Crossing: New Horizons as much as I have! I’ve been playing ever since the Gamecube days, but only now has the series really given us the freedom to plan every last detail of our towns. The focus is now on taking an island wilderness and turning it into a functional town. So, let me talk about my experience planning my own beautiful island of Coraland:

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In short, the map is a big mess. Take a look at all the rivers and cliff edges. The island is broken up into lots of small pieces only accessible using a vaulting pole or a ladder. It’s really inconvenient!

I’d really like to make this place more accessible by foot, so I’m going to need to install some bridges and ramps. But those projects are expensive, and each one may take a few days to come to fruition. As a city planner, I’m faced with a question: What’s the most convenient way I can connect up all these pieces of land? Can I minimize the number of projects I’ll have to pay for?

To answer these questions, first we’re going to mess around with this map a little to make the problem easier to visualize. Then we’re going to turn to an area of math that deals with these types of problems: graph theory.

Keep reading

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Day 6

Math is generally such a hated subject and I literally don’t understand why. It’s like the whole world collectively decided that math is the symbol of everything wrong with the education system. It’s not! It’s so not. I feel that most teachers just don’t understand how to properly teach this subject.

If it isn’t obvious already, I absolutely adore mathematics. It is without a doubt my favorite subject. Not that arithmetic crap, I’m talking real math—calculus, trigonometry, coordinate geometry. I’ve always maintained that there’s something exciting about math that you can’t find in any other subject. There’s this pure satisfaction you get out of correctly solving a problem which you won’t find anywhere else. History, geography, biology, chemistry, etc. etc. are such memorization oriented subjects. I feel like ultimately they’re just a test of how much you can remember and then reproduce in the exam. Math, though, is about concepts and understanding and creativity. You can’t memorize the solution to a problem. You can memorize formulae, sure, but having a tonne of formulae stored in your brain doesn’t necessarily mean you would be able to find the solution to a tough question. That’s where you need to think, you need to apply logic, you need to be creative. People always find it ridiculous when I call math a creativity oriented subject. There’s so many rules and restrictions, they say. The thing is, I think of math like a puzzle. The problem placed before you is like a puzzle to solve. And what’s a puzzle without any restrictions? If the solution was plain as day, there would be no challenge to it. It wouldn’t be fun. It wouldn’t require creativity and lateral thinking to solve. Math is the same. If the solution was glaringly obvious the moment you looked at a question, then no, there’s nothing creative about that. But there is creativity both in creating a problem that’s difficult to solve and in solving it.

Math is straightforward. It’s black and white. Right or wrong. There’s no ambiguity to it, you don’t have partially correct answers. There’s no wrong way to solve a math problem, provided your final answer is correct. There can be countless approaches to solving the same problem.

I like things to be organised and arranged properly. I hate leaving things half finished or incomplete. I usually can’t stand ambiguity in any situation—if there’s a tense situation, I would be in a rush to resolve it as soon as possible. It doesn’t matter to me if the solution I come to isn’t a great one, as long as the situation is over (this tends to be a bad approach to a lot of things). That’s why I like math. You don’t leave things half finished. You don’t spend days debating the ins and outs of some hypothetical concept. It’s all clean cut and precise.

I don’t like leaving the glass half full or half empty. I’d rather empty it entirely or fill it up—anything is better than halfway.

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