1, 2, 3 and to the 4: Getting Back into the Numbers Game
1, 2, 3 and to the 4: Getting Back into the Numbers Game
I've always liked math, but school ruined it for me. I took through calculus I in both HS and college. A few years ago I went back to school for economics, so I had to teach myself a bunch of math. This was challenging but fun. I studied in panic mode, learning what I needed to pass the classes. I'd like to keep working on mathematics, but it's difficult without direction. I'd like to learn stuff with real world applications for thinking about problems. So... modeling I guess? What else? I feel like I still need a comprehensive refresher in the basics too? If it could eventually lead to a job that'd be great, but it's not necessary.
I'm interested in learning more mathematical skills but also gaining (a) mathematical mindset(s). How would the more mathematically advanced here approach learning more math as an adult outside of school?
I'm interested in learning more mathematical skills but also gaining (a) mathematical mindset(s). How would the more mathematically advanced here approach learning more math as an adult outside of school?

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Re: 1, 2, 3 and to the 4: Getting Back into the Numbers Game
If you like games or solving puzzles, then look into number theory and higher geometry. They have practical applications and are useful for improving thought processes. The downside is that, for myself, it was something which required instruction to bring alive and does not lend itself to book learning.
Otherwise, I think applied statistics where you learn how to derive confidence bands is useful. An advantage is that spreadsheets really help and you can find books that teach you with spreadsheets.
Otherwise, I think applied statistics where you learn how to derive confidence bands is useful. An advantage is that spreadsheets really help and you can find books that teach you with spreadsheets.

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Re: 1, 2, 3 and to the 4: Getting Back into the Numbers Game
I'm on the side of grabbing a good textbook and just working through it. If you need a refresher on the basics that's not as dry, there's a series of books (targeted to children, but still useful for adults) called the Life of Fred which goes from basic numeracy for toddlers to calculus, statistics, and real analysis.
Once you've gotten that down, many universities post a bunch of free content online, particularly MIT.
Google search "{subject} textbook site:{university}.edu filetype:pdf" or search for lecture notes instead of textbook.
Some examples:
1. Learning the syntax: understanding what particular Greek letters and other symbols are supposed to mean, what someone is trying to convey when they write out an expression or equation. This just gets better with exposure and time.
2. Learn a concept well enough to take it for granted later: once you've walked through the proof for a concept, you need to internalize it well enough that if it's applied later you'll be comfortable trucking along without being like, "wait, why is that." For this, it's useful to create a running cheat sheet of useful equations that can be referenced when the textbook writer's shorthand skips a few things.
Once you've gotten that down, many universities post a bunch of free content online, particularly MIT.
Google search "{subject} textbook site:{university}.edu filetype:pdf" or search for lecture notes instead of textbook.
Some examples:
 Mathematics for computer science. Requirements: high school algebra and geometry, basic calculus. Fun stuff, my favorite.
 Computational biology. Requirements: discrete mathematics. Contains crash course in microbiology.
 Quantitative Finance: Requirements: calculus I and II. Very approachable.
 Econometrics: Requirements: linear algebra, multivariable calculus. What a slog.
 Differential equations: Requirements: multivariable calculus. Very easy to do DiffEq without knowing what you actually learned.
1. Learning the syntax: understanding what particular Greek letters and other symbols are supposed to mean, what someone is trying to convey when they write out an expression or equation. This just gets better with exposure and time.
2. Learn a concept well enough to take it for granted later: once you've walked through the proof for a concept, you need to internalize it well enough that if it's applied later you'll be comfortable trucking along without being like, "wait, why is that." For this, it's useful to create a running cheat sheet of useful equations that can be referenced when the textbook writer's shorthand skips a few things.
Re: 1, 2, 3 and to the 4: Getting Back into the Numbers Game
Basic: Design a shed (without using the metric system). Do it all on paper first rather than an app like Sketchup. There will be some mental math like converting between feet and inches (don't just make things an arbitrary height; design it for 8' tall lumber), some fractions as not everything will come down to an even inch, some geometry for a pitched roof, etc. Don't forget details like sheathing which may be an odd thickness like 5/16", trim boards, fasteners (how many pounds of nails or screws do you need for X surface area at Y spacing?).
For something more advanced, check out Project Euler. https://projecteuler.net/
There's a series of progressively harder problems that you'll need to learn how to program to solve. I don't remember much about the site from using it many years ago, but for the first few problems you can just program it in a straightforward brute force kind of way in whatever programming language you want. Then as the difficulty increases you have to think about doing things more efficiently or it could take your computer hours to solve  if it is even capable of solving it at all.
For example here is the first Project Euler problem.
For something more advanced, check out Project Euler. https://projecteuler.net/
There's a series of progressively harder problems that you'll need to learn how to program to solve. I don't remember much about the site from using it many years ago, but for the first few problems you can just program it in a straightforward brute force kind of way in whatever programming language you want. Then as the difficulty increases you have to think about doing things more efficiently or it could take your computer hours to solve  if it is even capable of solving it at all.
For example here is the first Project Euler problem.
And here is problem #500.If we list all the natural numbers below 10 that are multiples of 3 or 5, we get 3, 5, 6 and 9. The sum of these multiples is 23.
Find the sum of all the multiples of 3 or 5 below 1000.
The number of divisors of 120 is 16.
In fact 120 is the smallest number having 16 divisors.
Find the smallest number with 2^500500 divisors.
Give your answer modulo 500500507.
Re: 1, 2, 3 and to the 4: Getting Back into the Numbers Game
Thanks for the suggestions.
@BI: I like the idea of going back and starting from the beginning, I also like the idea of the Life of Fred stuff as a story. Any idea how to get started? I feel like I should start from the beginning, but there are 23 textbooks before algebra, which seems excessive, haha. I'm also wondering how much time I should spend mucking around with simple geometry and algebra, which I am kind of so so at, but which I know will be important later. The question is how much of my adult life do I actually want to spend factoring polynomials? Is there anyway to know what is likely to be useful later on vs. what is likely to be over emphasized in textbooks, because the audience is young and/ or schools need to have a years worth of work?
@BI: I like the idea of going back and starting from the beginning, I also like the idea of the Life of Fred stuff as a story. Any idea how to get started? I feel like I should start from the beginning, but there are 23 textbooks before algebra, which seems excessive, haha. I'm also wondering how much time I should spend mucking around with simple geometry and algebra, which I am kind of so so at, but which I know will be important later. The question is how much of my adult life do I actually want to spend factoring polynomials? Is there anyway to know what is likely to be useful later on vs. what is likely to be over emphasized in textbooks, because the audience is young and/ or schools need to have a years worth of work?

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Re: 1, 2, 3 and to the 4: Getting Back into the Numbers Game
What's overemphasized in textbooks are often the methods used to solve the kind of textbook problems that can be solved within an hour. This is becvause a lot of education is really just training or an IQ test to pass to get to the next level of training.
Differential equations are a great example. Most systems of differential equations can not be solved. Only a very small type of socalled ordinary differential equations admit pen and paper type solutions. Textbooks therefore dedicate an inordinate amount of time to ODEs despite this skill be almost useless in reality.
The same goes for integration (deducing the integral) albeit to a lesser degree.
If you want "applicable" stuff, read books with titles like "math for engineers", "math for biologists", or "data science". You're probably well aware of the need to consider the field specifics. For example, economists partake in some very strange way of describing rate of change usually formulated as "a 1% change in x relative to a 1% change in y equals ... " which a mathematician or physicist or any sane person really would write as "dlnx/dlny=..." That is only to say that the dialect you're adopting is also important. For all I know thinking in terms of 1% this or that makes more sense in economics, say.
Differential equations are a great example. Most systems of differential equations can not be solved. Only a very small type of socalled ordinary differential equations admit pen and paper type solutions. Textbooks therefore dedicate an inordinate amount of time to ODEs despite this skill be almost useless in reality.
The same goes for integration (deducing the integral) albeit to a lesser degree.
If you want "applicable" stuff, read books with titles like "math for engineers", "math for biologists", or "data science". You're probably well aware of the need to consider the field specifics. For example, economists partake in some very strange way of describing rate of change usually formulated as "a 1% change in x relative to a 1% change in y equals ... " which a mathematician or physicist or any sane person really would write as "dlnx/dlny=..." That is only to say that the dialect you're adopting is also important. For all I know thinking in terms of 1% this or that makes more sense in economics, say.

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Re: 1, 2, 3 and to the 4: Getting Back into the Numbers Game
@J+G I think it's more important that you can understand as someone steps through a proof "oh they factored the polynomial" than you being able to do it yourself. Conceptual understanding >>> rote technique. That said, you could probably take an online PSAT test like this one to see on what topics you might be due for a refresher.

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Re: 1, 2, 3 and to the 4: Getting Back into the Numbers Game
@bostonimproper  I think I disagree somewhat in that statement at least in how I'm reading it. I think rote stuff is important and even crucial. In the simplest terms, I think it's more useful to memorize the multiplication table than understand what multiplication is conceptually using group theory(*). It's more useful to be able to compute the asymptotic behavior of any function and do that a hundred times than to prove the formula for partial integration. The former makes it possible to make good guesses of what the underlying function for a dataset is or realize when one has a wrong answer. The latter ... I don't know.
(*) As such I am not a fan of https://en.wikipedia.org/wiki/New_Math (I'm old enough to have experienced new math in the 1st grade until it was dropped.)
Much of my math major (I'm ~1 semester short of a bachelor) was conceptual in nature. To give a more advanced example, I learned how to prove the spectral theorem, but we never used it to compute anything. There was some comment that this was a very powerful/useful result but no explanation of why. (In fact, a lot of these advanced analysis/algebra guys had a hard time coming up with applications beyond "the most efficiency way to divide a pizza.) It was only years (like 67) later that I saw it applied to quantum mechanical experiments that I understood it's "beauty".
One size might not fit all. Maybe this is just because I don't have an underlying Ti (where did @daylen go?) mode of thinking. The INTPs definitely enjoyed this way of learning math more. The INTJs mostly went into physics. The ENTJs and ISTJs into engineering. The approach to math in these quarters is more ghetto.
Or maybe it's just me. I had the same struggle when I was TA'ing physics in Switzerland. The curriculum focused on conceptual stuff and while these guys were brilliant in front of a blackboard with prepared notes, most of them couldn't calculate for shit when presented with a novel problem they hadn't seen before, e.g. a fountain has a throughput of 1000L/hr and a nozzle diameter of 10cm. What is the height the water reaches? The reaction when presented with such problems (also called "combat physics" or "physics kung fu") was one of They had learned a lot about physics (more than the system I came from) but they hadn't learned a lot of physics, because they couldn't use it for anything beyond talking about it.
(*) As such I am not a fan of https://en.wikipedia.org/wiki/New_Math (I'm old enough to have experienced new math in the 1st grade until it was dropped.)
Much of my math major (I'm ~1 semester short of a bachelor) was conceptual in nature. To give a more advanced example, I learned how to prove the spectral theorem, but we never used it to compute anything. There was some comment that this was a very powerful/useful result but no explanation of why. (In fact, a lot of these advanced analysis/algebra guys had a hard time coming up with applications beyond "the most efficiency way to divide a pizza.) It was only years (like 67) later that I saw it applied to quantum mechanical experiments that I understood it's "beauty".
One size might not fit all. Maybe this is just because I don't have an underlying Ti (where did @daylen go?) mode of thinking. The INTPs definitely enjoyed this way of learning math more. The INTJs mostly went into physics. The ENTJs and ISTJs into engineering. The approach to math in these quarters is more ghetto.
Or maybe it's just me. I had the same struggle when I was TA'ing physics in Switzerland. The curriculum focused on conceptual stuff and while these guys were brilliant in front of a blackboard with prepared notes, most of them couldn't calculate for shit when presented with a novel problem they hadn't seen before, e.g. a fountain has a throughput of 1000L/hr and a nozzle diameter of 10cm. What is the height the water reaches? The reaction when presented with such problems (also called "combat physics" or "physics kung fu") was one of They had learned a lot about physics (more than the system I came from) but they hadn't learned a lot of physics, because they couldn't use it for anything beyond talking about it.

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Re: 1, 2, 3 and to the 4: Getting Back into the Numbers Game
I've been wanting to relearn math almost from the algebra level forward for a while now. I had a tough time with math in school. Things really went off the rails in 8th grade. It was very hard for me to learn and stay focused on it. I'm one of the people who got effortless perfect scores on the verbal/reading/writing sections of SAT but maxed out at 520 on the math. I think I was incorrectly placed in "gifted" math classes at an early age and it really hurt me down the line even when I was moved back into normal classes when I got to HS. Math teachers would get frustrated, even quite mean to me, over my slow progress because they knew I was an accelerated learner in other subjects. This in turn increased my frustration levels with the subject and I later tried to avoid it at all costs in college.
Now as a 33 year old adult I feel like I have big hole in my brain that should be filled with this stuff.
It might just come down to laziness or attention deficiency. I'm an INTP and can read at a high level. Sometimes I feel like I can intuitively grasp things and use concepts in a way that surprises people with higher education levels. But understanding things through rote memorization can be difficult for me. A friend once suggested the book "Mathematics for the Million" to me. Anyone have any experience with that book or others that might be useful?
Now as a 33 year old adult I feel like I have big hole in my brain that should be filled with this stuff.
It might just come down to laziness or attention deficiency. I'm an INTP and can read at a high level. Sometimes I feel like I can intuitively grasp things and use concepts in a way that surprises people with higher education levels. But understanding things through rote memorization can be difficult for me. A friend once suggested the book "Mathematics for the Million" to me. Anyone have any experience with that book or others that might be useful?
Last edited by Papers of Indenture on Tue Apr 14, 2020 12:30 pm, edited 1 time in total.

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Re: 1, 2, 3 and to the 4: Getting Back into the Numbers Game
@jacob:
Just taking a stab at your physics problem because I could not resist:
1000L/min = 1 cubic meter of water per min
Now if forced into a cylinder of eater with 1 cm diameter:
1 = pi * (0.01/2)^2 x length of water column (or height in zero gravity)
Solving for h, h = 12,740m...12.7 km..wow!
Now since this happens in one minute..aka 60 seconds...the speed of the water column is 12740/60 = 212m/s. (sounds like a lot to me..like two thirds the speed of sound!)
So the water column will rise to a height 'h' where its weight * h (potential energy) will be equal to its kinetic energy.
So m*g*h = 0.5*m*v^2
Here for me it gets a bit tricky...since it is a solid column of water and not just a point particle and the speed at bottom is 212m/s and zero at top...I am inclined to modify above equation as follows:
m*g*(h/2) = 0.5*m*(v/2)^2
Solving for h gives me.. 1.1km...which sounds really high and makes me think I screwed up somewhere...or maybe not....does this sound right to you?
Just taking a stab at your physics problem because I could not resist:
1000L/min = 1 cubic meter of water per min
Now if forced into a cylinder of eater with 1 cm diameter:
1 = pi * (0.01/2)^2 x length of water column (or height in zero gravity)
Solving for h, h = 12,740m...12.7 km..wow!
Now since this happens in one minute..aka 60 seconds...the speed of the water column is 12740/60 = 212m/s. (sounds like a lot to me..like two thirds the speed of sound!)
So the water column will rise to a height 'h' where its weight * h (potential energy) will be equal to its kinetic energy.
So m*g*h = 0.5*m*v^2
Here for me it gets a bit tricky...since it is a solid column of water and not just a point particle and the speed at bottom is 212m/s and zero at top...I am inclined to modify above equation as follows:
m*g*(h/2) = 0.5*m*(v/2)^2
Solving for h gives me.. 1.1km...which sounds really high and makes me think I screwed up somewhere...or maybe not....does this sound right to you?

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Re: 1, 2, 3 and to the 4: Getting Back into the Numbers Game
I just pulled the fountain example numbers out of my ass. Then actually doing the problem made me realize that they would lead to an unrealistic result, so I revised them to something more realistic. Likely this happened while you were writing your post.
Otherwise, you solved it correctly.
Otherwise, you solved it correctly.

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Re: 1, 2, 3 and to the 4: Getting Back into the Numbers Game
I just reread the problem again...it is 1000L/hr not 1000L/min...so that totally changes magnitude.
So water velocity is 3.5m/s not 212m/s...and resultant height is more like 31 cms.!
Don't think I would have been selected for the team in movie Hidden Figures to calculate the area of splashdown!
So water velocity is 3.5m/s not 212m/s...and resultant height is more like 31 cms.!
Don't think I would have been selected for the team in movie Hidden Figures to calculate the area of splashdown!

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Re: 1, 2, 3 and to the 4: Getting Back into the Numbers Game
Ha ..ignore previous post...both rate of flow and nozzle size have changed.
Re: 1, 2, 3 and to the 4: Getting Back into the Numbers Game
Khan Academy: https://www.khanacademy.org/mathPapers of Indenture wrote: ↑Tue Apr 14, 2020 11:03 amI've been wanting to relearn math almost from the algebra level forward for a while now. ... Anyone have any experience with that book or others that might be useful?
Choose your math level, from Kindergarden, through Algebra, to Into to Stats and Multivariate Calculus. Enjoy very clear, very detailed and patient explanations. I would suggest these videos to anyone who has gaps they'd like to fill.
If you feel like it and you find the videos useful, consider making a donation. The site is nonprofit, and as I could gather from a Reddit post, the American school system basically runs off it in these times, creating a need for expanded server capacity.

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Re: 1, 2, 3 and to the 4: Getting Back into the Numbers Game
As someone who did badly at math in middle and high school, and am learning it now, I can second the Khan Academy recommendation. They're great.
Re: 1, 2, 3 and to the 4: Getting Back into the Numbers Game
I took a stab at Jacob's ass kung fu. Using Bernoulli's equation P + ½ ρ v^2 +ρ g h = constant on a streamline, it simplified to h = (v^2)/(2g), where v = (1 m^3/hr)/(pi*(0.005 m)^2)/3600 = 3.537 m/s = v. The water will reach a height of 0.638 m.

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Re: 1, 2, 3 and to the 4: Getting Back into the Numbers Game
@jacob Yeah, that's fair. I definitely approach math and physics in a more aesthetic, academic way. Like a isn't Gauss's Law beautiful sort. Once I know that a problem can be solved and roughly how, I'm less interested in the doing of it.
Re: 1, 2, 3 and to the 4: Getting Back into the Numbers Game
So I guess my preliminary question is how to go about reviewing elementary school through high school (so everything through algebra, algebra, geometry, stats, trig, calc) without wasting a bunch of time learning about stuff I already know, but with filling in the gaps from k12 math?
Once I've got the fundamentals down I'm very much interested in overcoming the domain dependency Jacob is talking about and actually being able to solve real world problems with math (or at least build models to analyze things that I find interesting ). I'm much more interested in applied mathematics than writing proofs (though obviously I recognize the important of both). Regardless, I think the first step is to master the "basics."
Once I've got the fundamentals down I'm very much interested in overcoming the domain dependency Jacob is talking about and actually being able to solve real world problems with math (or at least build models to analyze things that I find interesting ). I'm much more interested in applied mathematics than writing proofs (though obviously I recognize the important of both). Regardless, I think the first step is to master the "basics."
Re: 1, 2, 3 and to the 4: Getting Back into the Numbers Game
I actually know this! The answer is still khan academy. Let's look at algebra 2.Jin+Guice wrote: ↑Wed Apr 15, 2020 1:47 pmSo I guess my preliminary question is how to go about reviewing elementary school through high school (so everything through algebra, algebra, geometry, stats, trig, calc) without wasting a bunch of time learning about stuff I already know, but with filling in the gaps from k12 math?
Go here > https://www.khanacademy.org/math/algebra2
Then open the first chapter > https://www.khanacademy.org/math/algebr ... arithmetic
Go all the way down to the bottom and take the unit test. Questions are short, sweet, and multiple choice, and very, very comprehensive. When there's something you can't do, you can either look at the corresponding video, or you can look at their explanation of how they did it.
Countinue working through subsequent courses/chapters. I would start a little higher than where you think your proficiency is.
Might still be a bit too labor intensive for you, but if you do it, you will be totally and completely rock solid.