Hacker News Board Hacker News, browsed like a classic message board
Live data from Hacker News

Basic Math Textbook: The Napkin Project

web.evanchen.cc

Post reply on HN
Print view View original

21–30 of 76 posts

Re: Basic Math Textbook: The Napkin Project

#21

Post by ColinWright »

qsort wrote: post #10
auggierose wrote: post #7

> The set ℕ is the set of positive integers, not including 0. Hell yeah! I've agonised over this quite a lot over the decades. Not including 0 is more intuitive, but including 0 is more convenient. Of course, both approaches are correct. My main reason for not including 0 is that I hate seeing sequences numbered starting with 0.

I used to write and review problems for math competitions. This is why we avoided saying "natural numbers". We used "nonnegative integers" or "positive integers" instead.

You need to be careful about this ... I believe that in France (for example) zero is regarded as both positive and negative. So in France:

Non-negative integers: 1, 2, 3, 4, 5, ...

Positive integers: 0, 1, 2, 3, 4, 5, ...

Similarly, for some countries "Whole Numbers" is equivalent to all the integers, while in other countries it's the set { 0, 1, 2, 3, 4, ... } while in still other countries it's { 1, 2, 3, 4, ... }

There is no approach that uses "natural language" and is universal, and being aware of this is both frustrating and useful. Whether it is important is up to the individual.

Re: Basic Math Textbook: The Napkin Project

#22

Post by gjm11 »

auggierose wrote: post #7

> The set ℕ is the set of positive integers, not including 0. Hell yeah! I've agonised over this quite a lot over the decades. Not including 0 is more intuitive, but including 0 is more convenient. Of course, both approaches are correct. My main reason for not including 0 is that I hate seeing sequences numbered starting with 0.

I never write ℕ, for exactly this reason. I write ℤ with a subscript ">0" or ">=0". Doesn't take up much more space, and completely unambiguous.

Re: Basic Math Textbook: The Napkin Project

#23

Post by auggierose »

thaumasiotes wrote: post #16
auggierose wrote: post #7

> The set ℕ is the set of positive integers, not including 0. Hell yeah! I've agonised over this quite a lot over the decades. Not including 0 is more intuitive, but including 0 is more convenient. Of course, both approaches are correct. My main reason for not including 0 is that I hate seeing sequences numbered starting with 0.

From a technical perspective you frequently need 0 in there. From a pure convenience perspective, it doesn't make sense to assign ℕ to the positive integers when they're already called ℤ⁺. Now you have two convenient names for the smaller set and none for the larger set.

By convenience I mean "convenient from a technical perspective", and yes, you often need 0 in there.

Your other argument doesn't make much sense. I learnt both in school and at university ℕ, ℕ₀, and ℤ as THE symbols for the natural numbers, the natural numbers including 0, and the whole numbers.

Fuck convenience. ℕ, ℕ₀, and ℤ it is :-) It is just so much prettier (ℤ⁺ is a really ugly symbol for such a nice set). It is actually also not inconvenient if you don't use static types.

Re: Basic Math Textbook: The Napkin Project

#24

Post by aap_ »

Really cool! This is the sorta thing that, just yesterday, I wished existed. And it's already on the HN frontpage. It's hard to see the forest for the trees in many math books, a bird's eye view is a really valuable perspective.

I highly appreciate this approach: "As i have ranted about before, linear algebra is done wrong by the extensive use of matrices to obscure the structure of a linear map. Similar problems occcur with multivariable calculus, so here I would like to set the record straight"

Math education and textbooks are doing an awesome job obscuring simple ideas by focusing on weird details and bad notation. Always good to see people trying to counter this :)

Re: Basic Math Textbook: The Napkin Project

#25

Post by rossant »

auggierose wrote: post #7

> The set ℕ is the set of positive integers, not including 0. Hell yeah! I've agonised over this quite a lot over the decades. Not including 0 is more intuitive, but including 0 is more convenient. Of course, both approaches are correct. My main reason for not including 0 is that I hate seeing sequences numbered starting with 0.

I didn't know that. In French textbooks, I believe ℕ always includes 0. I didn't even know that not including it was another possible convention.

Re: Basic Math Textbook: The Napkin Project

#26

Post by yorwba »

commandersaki wrote: post #17
ptsneves wrote: post #12

Earlier quoted context omitted.

> Centred around AI Where do you see the centered around AI? I have used it a lot and have not touched a single subject around AI. > - Seems geared around edutech (which is what I gather from the site) What is edutech and why is it unsuitable? Finally, have you _used_ MathAcademy at all?

Where do you see the centered around AI? From https://www.mathacademy.com/how-it-works : > Math Academy is an AI-powered, fully-automated online math-learning platform. Math Academy meets each student where they are via an adaptive diagnostic assessment and introduces and reinforces concepts based on each student’s individual strengths and weaknesses. What is edutech and why is it unsuitable? I don't want a computer…

As far as I can tell, most of its value comes from having a reasonably thorough dependency tree of math topics and corresponding exercises (which can be solved with pen and paper) and describing it as "AI" is how you get investors to fund a math textbook.

See also How Math Academy Creates its Knowledge Graph https://www.justinmath.com/how-math-academy-creates-its-know... "We do it manually, by hand."

Re: Basic Math Textbook: The Napkin Project

#28

Post by thaumasiotes »

ColinWright wrote: post #21
qsort wrote: post #10

Earlier quoted context omitted.

I used to write and review problems for math competitions. This is why we avoided saying "natural numbers". We used "nonnegative integers" or "positive integers" instead.

You need to be careful about this ... I believe that in France (for example) zero is regarded as both positive and negative. So in France: Non-negative integers: 1, 2, 3, 4, 5, ... Positive integers: 0, 1, 2, 3, 4, 5, ... Similarly, for some countries "Whole Numbers" is equivalent to all the integers, while in other countries it's the set { 0, 1, 2, 3, 4, ... } while in still other countries it's { 1, 2, 3, 4, ... }…

> I believe that in France (for example) zero is regarded as both positive and negative.

That would cause all kinds of problems, so I'd be pretty surprised if it turned out to be true.

I note that this is the heading of the relevant wikipedia page:

> Un nombre négatif est un nombre réel qui est inférieur à zéro, comme −3 ou −π.

( https://fr.wikipedia.org/wiki/Nombre_n%C3%A9gatif )

It'd be hard to be more explicit that zéro is not a negative number.

Re: Basic Math Textbook: The Napkin Project

#29

Post by diegof79 »

I love projects like these. Even when I took algebra and calculus in university, it’s good to refresh and go deeper into the concepts many years later.

However, a small critique to the author: the audience of this book is not clear. It says “basic” math, but then in chapter 1, the group's explanation starts with this sentence: “The additive group of integers (Z,+) and the cyclic group Z/Zm.” Maybe it was a draft note. To be fair the paragraphs that follow attempt a more basic explanation of groups, but even my “Algebra I” book at the university was friendlier than that.

Re: Basic Math Textbook: The Napkin Project

#30

Post by auggierose »

auggierose wrote: post #23
thaumasiotes wrote: post #16

Earlier quoted context omitted.

From a technical perspective you frequently need 0 in there. From a pure convenience perspective, it doesn't make sense to assign ℕ to the positive integers when they're already called ℤ⁺. Now you have two convenient names for the smaller set and none for the larger set.

By convenience I mean "convenient from a technical perspective", and yes, you often need 0 in there. Your other argument doesn't make much sense. I learnt both in school and at university ℕ, ℕ₀, and ℤ as THE symbols for the natural numbers, the natural numbers including 0, and the whole numbers. Fuck convenience. ℕ, ℕ₀, and ℤ it is :-) It is just so much prettier (ℤ⁺ is a really ugly symbol for such a nice set). It i…

On the other hand, even for writing a perfectly fine natural number like "10", you need the zero... Maybe it is just ℕ and ℤ after all.

And round we go.

Post reply on HN