corporation or foundation here: I'm just a curious learner trying to improve math Unfortunately, math understanding seems to follow the DNA pattern. We're. Unfortunately, math understanding seems to follow the DNA pattern. We're We' re left with arcane formulas (DNA) but little understanding of what the idea is. After a single class, which strategy gives you a better understanding of the material When we're ready, we “perform” math by doing the calculations ourselves.
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Math, Better Explained is a clear, intuitive guide to the math topics essential for high school, college and eBook includes PDF, ePub and site versions. Understanding Linear Algebra. Kalid Azad∗ but want a new understanding. the mathematical definitions for matrices. One note about. ininenzero.cf - Download as PDF File .pdf), Text File .txt) or For most of us (myself included). we “perform” math by doing the calculations.
The distinction between a ring. These charts make the strategy comparisons easier. But wait. Archimedes laid the foundations of Calculus without x-y graphs. Great question. Thinking better with X-Rays and Time-lapses? And somehow. Can you think of a few ways to build a sphere? No formulas. This is a foundation for later.
But to be fair. But just for reference. A wedge looks like a bunch of pizza slices of different sizes stacked on top of each other. The 3d segments can be seen as being made from their 2d counterparts. For example. Each layer is slightly bigger than the one before.
The tradeoffs in 3d are similar to the 2d versions: We could analyze it both ways. But on the inside. With our X-Ray and Time-Lapse skills. An orange is an interesting hybrid: With Roman numerals. I love diagrams and analogies. We know. You bet. The rules of arithmetic are general-purpose. It sounds weird. We could even work backwards: Take a look at how numbers developed. Start with a single ring. At first. Just because 8 is larger than 1?
Not a good reason! We now have detailed descriptions of how one formula is related to the other. Fun fact: In Math notation helped in a few ways: We just need to figure them out. So far. Now line those discs up into the shape of a sphere. Probably not.
Make the circle into a plate. Robert Recorde invented the equals sign. X-Ray the sphere into a bunch of shells. I agrye! Make a filled-in disc with a ring-by-ring time lapse. With enough practice.
When learning addition. Calculus is similar: It can analyze any shape or formula a physics equation. When we see a shape like this: Time- Lapses and diagrams: Along the orange line the radius.
The derivative is a function. Just the shape to split apart. Honorable Grand Poombah radius. At the start and end of the arrow we start at 0.
I suppose. It describes every possible squared value 1. As we grow the radius of a circle. The derivative is similar. For example: A few notes about the variables: We need to be specific. These issues are extremely confusing. It represents the tiny section of the radius present in the current step. How large. When we see r in the context of a step. Flavor to taste. This symbol d r. The two commands are a tag team: We still need an integral to glue the parts together and measure the new size.
We broke apart our lego set and have pieces scattered on the floor. I split the shape apart for you. First, we setup the integral, and then we worry about the exact formula for a board or slice. Gluing any set of slices should always return the total area, right? Yes, we are gluing together the rectangular slices under the curve.
But this completely overlooks the preceding X-Ray and Time- Lapse thinking. Why are we dealing with a set of slices vs. Would practitioners ever go back to written descriptions?
Math is just like that. Question 1: Assume the arrow spans half the radius. The description should follow the format: Question 2: The math description should be something like this: The slices move around the perimeter where does it start and stop?
Have a guess for the command? Here it is, the slice-by-slice description3. Question 3: Can you describe how to move from volume to surface area? Think about the instructions to separate that volume into a sequence of shells. Which variable are we moving through? Split the circle into rings from the center outwards, like so:.
Our formal description is precise enough that a computer can do the work for us:. X-Raying a pattern into steps?
Similar to above. These shortcuts are closer to the math symbols: We described our thoughts well enough that a computer did the legwork. We saw what the steps would be. The boards are harder to work with. We actually need 2 copies of height. A few notes: Now that you have the sound in your head. As you might have expected. The base is d p the tiny section of perimeter and the height is r. Wolfram Alpha takes longer to compute this integral than the others!
The approach so far has been to immerse you in calculus thinking. Some of it may be a whirl — which is com- pletely expected. Calculus lets us accumulate or separate shapes according to their actual. Multiplication makes repeated addition easier. Now that we have the official symbols. If you wanted 13 copies of a number. Identical parts are fine for textbook scenarios. Integrals let us add up the pattern directly.
Sometimes we want to use the average item. Division spits back the averaged-sized ring in our pattern. Integrals let measurements curve and undulate as we go: I mentally convert it to multiplication or division remembering we can handle differently-sized elements. A series of multiplications becomes a series of integrals called a triple integral. When I see a formula with an integral or derivative. The area of this square is unknown. We added two new ways to transform an equation.
Imagine I want to know the area of an unknown square. This helps us think of integrals and derivatives like squares and square roots: Now we can work out the area of a circle. The fancy stuff can wait. The area of a circle is unknown. In Math Land. Anything else I can help you with? It might be 3 feet. I imagine going to Home Depot and pestering the clerk: A big gotcha in Calculus is realizing x. Fair enough.
I think. For every foot we ask for the input. What will it run me? How much do you want? How do we figure out the sequence of steps? Think about the various ways we express multiplication: How about 4 feet? We can extract the ratio with a few shortcuts: Not so fast.
In our case. Compute the difference: Find the new amount. In this case. What gives? That statement is true. The difference between the next step and the current one is the size of our slice. Step forward by d x 1 foot. Get the current output. Can we describe our steps? In our fence-building scenario. As we suspected. Notation like d x puts us into detail-oriented mode.
The notation for derivatives is similar: Some versions. Are we looking for an accumulation. Are we leaving out d x?
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These details are often omitted. The range can be numbers. The notation for integrals can be fast-and-loose. If the range includes a variable 0 to x. An analogy: Imagine an antiques dealer who knows the original vase just from seeing a pile of shards. He might determine the original vase weighed We are time-lapsing a sequence of equal changes.
In that case. I think this must be a Ming Dynasty Vase from the 3rd Emperor. This insight was never really explained to me: Instead of trying to glue slices together to find out their area.
You know. When someone asks for the square root of You might say: We can make a few abstract rules — like working out the rules of algebra for ourselves. If we know the derivative of 4x is 4.
How does he do it? If we wanted 3 steps 0 to 1. There are a few subtleties down the road. Part of calculus is learning to expose the right amount of detail. One last note: R Notice how I wrote a and not a d x — I wanted to focus on a. But if every slice were zero. The analysis just figures out the current perimeter and square footage? Why settle for a static description when we can know the step-by-step description too? We can analyze the behavior of the perimeter pretty easily: By now.
To the untrained eye. Your plan is to build the garden incrementally. Too small. Assume topsoil is sold by the square foot. After our exposure to how lines behave.
We can visualize this process. As the square grows. The visual is helpful. For every 1-foot increase in x. Since squares are fairly new. We can write out the size of each jump. In this setup. And yep. If we currently have a square with side x. The gap from 22 to 32 is 5. Algebra can simplify the process. The gap from 02 to 12 is 1. The gap from 12 to 22 is 3. And so on — the odd numbers are sandwiched between the squares! He saw order after order go by without noticing the deeper pattern.
What can they work out? A low-level goon might just add up the total amount accumulated the definite integral: Does this guess work? Assuming this is the pattern. Suppose the veggie mafia spies on your topsoil and fencing orders.
How does the godfather do it? Another option might be a right triangle with sides x and 2x. And when he sees a perimeter change of a steady 4.
The crime boss is different: Make a guess if you like. Now suppose your orders change: Can we work backwards. Uh oh! The henchman can only tell you the running totals so far definite integral.
The mob boss is a master antiques dealer: If we leave d x as it is. What happened to that corner piece? The mystery continues.
Your first area jump was by 5. Two friends are 10 miles apart. This one is tricky: Infinity is a fascinating and scary concept — there are entire classes Analysis that study it. But the official derivative.
We have an infinite number of ever-diminishing distances to add up. After all.
Can you figure out how far it flew before its demise? The answer relies on the concept of infinite accuracy. A mosquito files quickly between them.
Did we just find the outcome of a process with an infinite number of steps? I think so! The question seems painfully difficult to solve. The Natural Logarithm ln 9.
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Yeah, that thing we mastered in 4th grade or whatever. That thing we use every day. Ok, let's multiply it out Well, that's 4. But are they the same thing? How did we miss this?
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Math is full of insights like that. Trig functions sine, cosine, tan, etc. A sine of. Sine and cosine are unitless numbers, and that's why the can be each other's derivatives the percentage change of a percentage change So many things click!
What else have we overlooked? Anyway, really appreciate the note! The percentages detail is very cool, never thought about it! While sine and cosine generate "percentages" between -1 and 1 they don't generate them uniformly, because their derivative is not a linear function!
If we were to sample sine or cosine between 0 and PI at uniform intervals we would see that the percentages would "cluster" around the peaks and valleys because that's where the function "decelerates" or the derivative decreases. This bit me once! That's a great point. Similar, if you ask people to pick a random point in a circle random angle, random radius length you don't get the random distribution you were expecting.
Kalid, I have a question, but first a compliment: I am enamored with this site. It validates a lot of my learning approaches and is helping me now when I was asking some of these 'how to learn' questions.
Thank you! The question: Referenced here: If you don't get to the question, I am still happy to have found your site and hear how passionate you are about it. Thanks for the compliment, it really means a lot when the site resonates.
I'm not super experienced with lectures mostly writing , but I think it works in office settings. Check out this talk from Simon Sinek: Then you get into the what and how it's accomplished. With math, it can be similar: Humans in general prefer a narrative to a list of facts Hacker News readers excepted: Thank you sir! That is revealing and informative. Keep up the good work: Thanks, I appreciate it! ThrustVectoring on Dec 22, Euler's Identity is often explained in terms of mysterious language as well.
I hate the Taylor Series explanation of Euler's formula. Euler's Formula becomes "obvious" dare I say "obvious after the greatest mathematician figured it out for us".
I've wasted so much of my life. Me too. I was even a math geek but didn't understand most of it. I knew all the rules for equations plus heuristics for how to apply most of them. I can only imagine how much more effective I'd have been if there was a constant, parallel learning process focusing on intuitive understanding of all foundational concepts in various math branches.
I think I left a palm-print on my forehead when I saw that. I love this approach to learning. It's something I've been really working to take on in my studies and my own projects. I think it's a general principle blurry-to-sharp that works for many fields.
Paid you a great compliment in my main comment. Also can visualize it as a rectangle or stack of boxes that I take one chunk out of. I took a chunk out of all of that stack. All of this chunk equals the chunk I took. Too obvious. Stay on the harder shit like e, trig, etc. Excellent example. I understood it almost entirely in equations back when I did it. Outside of some examples with trees and stuff we rarely got to sit on what the terms mean.
So, let's see if I follow that.
So, a sine of 0. If I did it visually, that is. And where do the other two start? Or is my intuition screwing with me? Just saw your earlier comment, thank you! Mostly, I like it because we've overlooked something that's been under our noses for years or decades.
What else have we been missing? Let me know if that link above clears thins up. Yeah, to be honest, I'm writing for someone who saw the textbook definition of trig but didn't have it click. I'm not sure how it would work as a sole treatment. However, I suspect the vast majority of people reading the article are learning trig in school alongside an existing lesson, vs. The people who google "trig" for fun having never heard of it This is fascinating!
I wanted to know more about the history of that function at id, which lead me to this beyond3d post which was the topic of the reddit post that BetterExplained listed as a source: A slight correction: The paper was written by Chris Lomont in , when the gamedev.
There is a different paper that published a similar idea in Floating Point Tricks , but its not the one linked. The code first appeared in the wild on comp. I remember this because I followed the discussion on Gamedev. Some years after, I found the Beyond3D articles where they traced the origin back to Ardent Computer. Here's the best explanation I've seen: Why did you use the archive. TuringTest on Dec 22, Judging by the reactions to the BetterExplained site, other people agree with that.
I need to be shown and eventually understand why something was developed when it was developed. I need chronology of thought and ideas. Learning about Leibniz's and Newton's calculus without learning about infinitesimals.
And so on. If you think about your maths classes, sometimes you're instructed to learn a method because it is useful and because it has real-world applications but it I don't think anybody is ever first taught algebraic geometry properly, if I may use that word. I don't think kids are taught the geometry is one thing and algebra is another and that different spaces can have different metrics. Am I making sense here? Do people see what I'm trying to get at?
Great point. I find myself looking at the history of the idea when writing up a post. Did you realize negative numbers were only accepted in the late s? That the Fourier Transform was originally rejected as untrue when first presented, by world-famous mathematicians even?
Yet we require students to internalize it without issue in a single lecture. Historical context is huge. I did not know that! This makes total sense. I'd like to know more about that. When you think about it, only whole positive numbers make sense from a quantitative perspective.
One thing, two things, three things, and so on. What's half-a-thing? And how can no thing nothing be a number? And how can negative numbers be "numbers".
It has always struck me that imaginary numbers are really badly named. Zero and the negative numbers are just as 'imaginary', equally unintuitive from a certain perspective. I applaud what you're doing. I think there is a metric-tonne of dogma and bad naming schemes in the standard maths curriculum. Remember in software engineering they say that naming things is one of the hardest parts of the task? I think the same applies to maths, perhaps more so.
There's a quote from a famous mathematician at the time that the negatives "Darken the very whole doctrines of the equations". If positive is good, negative must be evil right?
Math, Better Explained: Learn to Unlock Your Math Intuition
And how can "less than nothing" exist? I love the philosophical implications of it. Ugh, tell me about the naming. Nobody complains "Hey, when will I ever use the second dimension? But "imaginary numbers" are setup to be eye-rolled. I'd recommend "The Joy of x" by Steven Strogatz.
It's full of that sort of story. I was really hesitant to force an acronym it actually started as ADE but then I realized I could work my way up to the full technical definition. Really glad to hear it's resonating. Sometimes things happen out of order, i. But the idea is to have all 5 parts if you want to truly master a concept. IkmoIkmo on Dec 22, Agreed, but it depends. The analogy can make things harder if you're not intimately familiar with the analogy. Take the below, an excerpt from the explanation on prime numbers for example, and consider you know 0 chemistry not unlikely if you're reading an intro on prime numbers: Chemical elements have properties based on their location in the periodic table of the elements: Atoms in group 8A Neon, Argon are the noble gases.
They don't react and won't blow up in your face. Atoms in group 4A Carbon, Silicon bond well. They're great building blocks for other elements. Atoms in group 1 Sodium, Potassium, etc. Drop 'em in water and see them explode. And in organic chemistry there's an idea of a functional group: For example: Alcohols are a certain carbon-hydrogen chain with an OH group at the end.
Methanol, ethanol, and other alcohols share similar properties because of this OH functional group. Those are the basics, if I didn't mess it up. Now let's see what happens when we treat numbers like chemicals. First Example: Guessing Evenness In general, an organic chemical contains carbon not quite, but it's a good starting point. No matter what elements you mix together, if you never add any carbon then you can't create an organic compound.
I love analogies in learning, but one has to be careful to pick analogies from a level of understanding way below what you're trying to explain. I guess kids are introduced to primes and chemistry at roughly the same age, but I'd have picked a non-academic analogy to explain an academic concept.
But even then, it's tricky. For example I've been confused by my fair share of 'sports analogies' in secondary school books, for sports I happened not to have ever tried or knew the rules for. But really, the analogy should be completely supplemental, and if possible marked off in a side box that people can, but not should, read for better understanding if it helps them. I find many school books do this really well, but I haven't seen it translated to web content as much somehow.
For example, on Evenness he'll continue by explaining how if you have a factor of 2 in your number e. I don't think that analogy is very strong, it's confusing if you don't know chemistry, and it's pretty redundant if you do. In fact I'd personally be better of without it, and understood Kalid's normal explanation without issue. Yet I had to read through something about Atoms in group 4A and their properties, unsure whether I could just skip it or whether it was important to grasp some larger point.
Anyway I was already familiar with primes but my 12 year old self probably would've been confused with the chemistry analogy. Thanks for the feedback! Agree analogies are context and time sensitive. As soon as you make a reference the clock starts ticking about how long it would remain relevant. For this specific example, I was writing to a high-school version of myself who wanted to really get an intuition for primes.
What can we deduce from a prime factorization, are there other ways to think about it? Number theory is studied later, even though numbers are introduced early. For a younger child, I'd probably use Lego or Minecraft to show how numbers can have "building blocks".
Looking for other ways to read this?
And if you didn't use any Redstone as a building block, there won't be any Redstone in the result. Thanks for the comment!Is this viewpoint necessary for survival? For every foot we ask for the input. As I said, assuming the teacher is capable and passionate, it still depends on the students. A rich body of content knowledge about a subject area is a necessary component of the ability to think and 1 The research on which these principles are based has been summarized in How People Learn: Mind, Brain, Experience and School Expanded Edition NRC, b.
For example, in most linear algebra textbooks, you are given matrix and are asked you to process it.
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