Best book for understanding arithmetic?

[Antisthenes]

Is there any very basic arithmetic book (for dummies) written by respected mathematicians like for example Serge Lang, Gelfand and Allendoerfer? Looking for a book that really explains the key concepts and rules of arithmetic, in a pedagogical way, from scratch, without skipping any steps when showing (proving) how and why conclusions follow from premises.

I want to understand the logic behind even the most elementary concepts in arithmetic (and math in general), and understand this deep enough so that it will be easier to know which mathematical tools should be applied in new contexts.

I'm good with logic, but really struggle with problem solving each time I encounter a new context. Have read the book "A Mind for Numbers" by Barbara Oakley, but it doesn't actually focus much on math, so just now got the book "How To Think Like A Mathematician" by Houston, though I assume it will be too advanced for a noob like me. Therefore, can anyone please recommend a comprehensive and deep arithmetic book (and workbook) for self-study? It's a plus if it includes the history of arithmetics and basic algebra: how and why elementary mathematical concepts were developed.

Btw, I'm not taking any exams, just having math as a hobby, so have time to read many long books about this topic, as long as the books start from scratch and don't skip any steps when explaining key concepts.

 

[jedishrfu]

The best I can think of are the Openstax books at openstax.org

THese books are often used in homeschooling settings:

https://openstax.org/details/books/prealgebra

PreAlgebra is the first in a sequence of books upto Calculus III

 

[Antisthenes]

Thanks for reply :) Just downloaded the book, but have to ask if the book is just one of those regular school books that just presents how to solve a particular problem without any deeper explanations of why it works and when to apply it in a particular context? Prefer a book written by a highly respected mathematician, because then one can feel confident that the explanations are as deep and comprehensive as possible. At least that is my assumption, as a noob :)

 

[fresh_42]

I've seen some schoolbooks from 1920 and other pages with freely available pdf. Google got me an Amazon bestseller hit on my search for "books on arithmetic" before I added "free" to the search. The difficulty here is a bit the lack of specification. Your question reads a bit as if you should look for schoolbooks, because the mathematical term arithmetic might not necessarily be what you're looking for. On the other hand the common English word of arithmetic at school is algebra or pre-algebra. Arithmetic concentrates more on the logical foundations of mathematics, which can be quite abstract with probably a lot of logic and set theory - a bit boring if you ask me.

 

[jedishrfu]

Its really hard to find such a book written by an esteemed mathematician and which talks about things in a simple, understandable but deep manner.

There's another book by Jan Gullberg that goes through the history of Mathematics in great detail and that may be what you are looking for. He is no mathematician. He was an MD with a deep interest in Mathematics and this was his opus.

https://www.amazon.com/dp/039304002X/?tag=pfamazon01-20

and there's the book Math 1001 by Prof Elwes which is a survey of Mathematics.

https://www.amazon.com/dp/1554077192/?tag=pfamazon01-20

Lastly, there's the Princeton Companion to Mathematics which is written by multiple authors and is somewhat difficult to read for a layperson:

https://www.amazon.com/dp/0691118809/?tag=pfamazon01-20

 

[jedishrfu]

On Youtube there are several channels on math most notably Numberphile by Brady Haran where mathematicians talk about specific problems and show some deeper insight into them.

And there's 3blue1brown channel with some really cool math videos. One such video is on the Pythagorean triples relating them to complex numbers.

 

[Antisthenes]

@jedishrfu

The book in your first post looks promising. It's from Rice University and starts by saying:

"The order of topics was carefully planned to emphasize the logical progression throughout the course and to facilitate a thorough understanding of each concept."

@fresh_42

Don't mind if it's boring. I'm used to reading Kant and Habermas for example. But what I "hate" is to read a book that suddenly skips a step, which actually means that I have a hole in my knowledge, and therefore I want to start from the very scratch of mathematical reasoning, including me doing all the exercises that are necessary to truly understand a math problem.

 

[fresh_42]

Don't mind if it's boring. I'm used to reading Kant and Habermas for example.

You're a tough one, indeed! I once found an original publication, I think it was a book, from Russel on the internet. This was so full of specifically invented notations that it was almost impossible to read. I like books about the history of mathematics and especially the first half of last century is exciting, not the least because of arithmetic, Hilbert's program and Gödel's answer. I have Popper's book about the logic of science here, but one sees quickly, that it had been written by a philosopher and not a mathematician.

I would start a search on terms like "foundations of arithmetic" then. Also Bourbaki might be a possibility, as they explicitly attempted to write down mathematics from scratch, but I do not know if there is a book about arithmetic. If so, it's probably worth a look. And by the way, Serge Lang was part of Bourbaki.

 

[Antisthenes]

@fresh_42

Have written a currently unassessed PhD thesis in moral philosophy, so I'm used to hard work, but have very low self-confidence in regards to math, spatial thinking and puzzle problems. My mind is flexible and creative in abstract philosophy, but it freezes and gets stuck in math... So need to rewire my brain and attitude. Two years ago my ambition was to learn quantum physics, and could easily follow math explanations all the way up to the first university level, but hit a wall when trying to solve problems myself. So now it's back to basics. Just want to understand the foundation of arithmetic properly, and then take it from there, if I have the IQ.

The level of explanation I'm interested in is illustrated by Micromass here: Gelfand's book Algebra "doesn’t only cover what mathematics is, but also why mathematics is true. For example, did you ever wonder why a negative number times a negative number is a positive number? This book explains all this and more in detail."

Reference https://www.physicsforums.com/insights/self-study-basic-high-school-mathematics/

 

[The Bill]

I think you might find The Foundations of Arithmetic by Gottlob Frege useful for your purposes. The treatment therein is very philosophical and detailed.

Other than that, I'd recommend practice. The Openstax source above should be a good inspiration for that. I'm my experience, the best way to get a feel for why the axioms and methods are what they are is to apply them repeatedly in varied circumstances.

I like the video series mentioned above for helping to build mathematical intuition, also. Mathologer is another good YouTube mathematics series. Oh, and PBS Infinite Series, too. Not all of those channels' videos will apply to what you're working through right now, but you may find something useful in their back catalog.

 

[fresh_42]

For example, did you ever wonder why a negative number times a negative number is a positive number?

Yes, I gave a lot of tutorials to school kids and I like abstract algebra, which is basically the theory of structures defined by multiplication and addition. If we have an ordered field like the rationals or the reals are, then all squares have to be positive (e.g. van der Waerden). Thus $(-1)^2 > 0$ and therefore also $(-a)\cdot (-b) = (-1) \cdot a \cdot (-1) \cdot b = (-1)^2\cdot ab > 0$. But this explanation isn't what I told the kids - mostly. It's easier to use an indirect proof: if not, we run into contradictions, at least with integers.

 

[jedishrfu]

I always liked the necessity is the mother of invention approach where we went from natural numbers to zero and to integers because we needed those numbers to solve a problem...

 

[Antisthenes]

@The Bill

Thank you very much, will read Frege. And fully agree that practice is necessary. Passive "understanding" of math is nothing but an illusion of knowledge, as stated by Oakley, and Micromass. My first go at math was a good example of that. Will try a more humble and realistic approach now.

@fresh_42

Can't honestly comment on your explanation. Must first refresh the math I learned two years ago. Eventually gave up back then because I initially felt confident and surprised by being able to passively understand the most advanced high school math books in Norway (my country), but then lost hope when totally sucking at solving problems. Therefore returned to my comfort zone of philosophy, but need more challenge, so will try a second time now.

By the way, in Norway one can actually get a PhD degree in the humanities without any high school math education at all, so that's why I could write a PhD thesis in philosophy. The education system in Norway is not very demanding compared to many other countries. Which is a problem now that big data, synthetic biology, neuroscience and physics are having a huge impact on the humanities and the social sciences.

 

[smodak]

Is there any very basic arithmetic book (for dummies) written by respected mathematicians like for example Serge Lang, Gelfand and Allendoerfer? Looking for a book that really explains the key concepts and rules of arithmetic, in a pedagogical way, from scratch, without skipping any steps when showing (proving) how and why conclusions follow from premises.

I want to understand the logic behind even the most elementary concepts in arithmetic (and math in general), and understand this deep enough so that it will be easier to know which mathematical tools should be applied in new contexts.

I'm good with logic, but really struggle with problem solving each time I encounter a new context. Have read the book "A Mind for Numbers" by Barbara Oakley, but it doesn't actually focus much on math, so just now got the book "How To Think Like A Mathematician" by Houston, though I assume it will be too advanced for a noob like me. Therefore, can anyone please recommend a comprehensive and deep arithmetic book (and workbook) for self-study? It's a plus if it includes the history of arithmetics and basic algebra: how and why elementary mathematical concepts were developed.

Btw, I'm not taking any exams, just having math as a hobby, so have time to read many long books about this topic, as long as the books start from scratch and don't skip any steps when explaining key concepts.

What Is Mathematics? An Elementary Approach to Ideas and Methods by Courant is the book you are looking for.

 

[Antisthenes]

@smodak

Tnx, have that book now and will see if it's on my newbie level :)

 

[Mark44]

Is there any very basic arithmetic book (for dummies) written by respected mathematicians like for example Serge Lang, Gelfand and Allendoerfer?

Not that I'm aware of. @fresh_42 said that "the common English word of arithmetic at school is algebra or pre-algebra."
I disagree with this. To my mind arithmetic is simply the four operations of addition, subtraction, multiplication, and division, operating on whole numbers, fractions, and decimal fractions. One might also include square roots, but only the four operations I listed are included among the arithmetic operations.

Looking for a book that really explains the key concepts and rules of arithmetic, in a pedagogical way, from scratch, without skipping any steps when showing (proving) how and why conclusions follow from premises.

I would say you are overthinking this. The rules for arithmetic are pretty short. If you allow negative numbers (additive inverses), you can completely eliminate subtraction. Likewise, once you recognize reciprocals (multiplicative inverses), you can do away with division.

Students typically study arithmetic in the early grades (here in the US), starting with arithmetic of whole numbers, and progressing through arithmetic of fractions and decimal numbers in about grades 5 and 6. It's been a long time since I was in elementary school, so I'm not certain how they do things these days. I can say, though, that a lot of students have trouble with adding fractions.

Here are the important concepts:
Addition
a + b = b + a Commutative property of addition
a + (b + c) = (a + b) + c Associative property of addition
a + 0 = a Zero is the additive identity
a + (-a) = 0 Additive inverse

Multiplication
a * b = b * a Commutative property of multiplication
a * (b * c) = (a * b) * c Associative property of multiplication
a * 1 = a 1 is the multiplicative identity
a * (1/a) = 1, if $a \ne 0$ Multiplicative inverse

Both
a * (b + c) = a * b + a * c Distributive property (multiplication distributes over addition)
What I've listed here are essentially the properties of a field, a term that has a specific meaning in mathematics, and is a topic that would usually be studied long after a student studies "plain old" arithmetic.

 

[Mark44]

Have written a currently unassessed PhD thesis in moral philosophy

How is it that you are working on a PhD, but are uncertain about what is probably the most basic level of mathematics? That seems very odd to me.

 

[Antisthenes]

@Mark44

There are many topics in the humanities which don't rely on math at all. Unless you want to go terribly deep that is. Anyway, don't think I'm overthinking it. Just new to math and therefore want to get an initial and rough map of the arithmetic territory, to see how wide and deep it is. And then learn it thoroughly, without gaps, before moving on to algebra. If it's as simple as you said in your first post, then excellent, but want to be sure that I have covered all the bases.

 

[Mark44]

There are many topics in the humanities which don't rely on math at all.

Yes, I understand that, but anyone with a university degree should have a good grasp on basic arithmetic, IMO. It's a shame that this isn't always the case. I remember meeting a young woman who was a senior at the college where I earned my degree in Mathematics. She told me that she didn't know how to do long division. That was shocking to me, how someone could get through 12 years of elementary and high school, and 3+ years of college, with such a gap in her knowledge.

Just new to math and therefore want to get an initial and rough map of the arithmetic territory, to see how wide and deep it is.

Not wide and not deep...

About a hundred years ago, people were expected to be very knowledgeable in arithmetic by around the fifth or sixth grades (at about the ages of 11 and 12 here).

 

[Antisthenes]

@Mark44

See your point, and in many ways agree with what you are saying. On the other hand, have never needed math, and never missed it. Now I'm only learning it because it actually interests me, but not out of any necessity, despite clearly recognizing that more and more fields within the humanities and social sciences will rely on math and big data. Without math, one will gradually be left behind in those fields, but it's not a must if one studies the history of ideas, literature or religions for example. If one has talents, one can still contribute to scientific research, without math.

 

[Mark44]

On the other hand, have never needed math, and never missed it.

We're not really talking "math" here -- just plain ol' arithmetic, one of the three pillars of the "three R's" -- readin', writin', and 'rithmethic.

If one has talents, one can still contribute to scientific research, without math.

I'm at a loss to see how. Your statement seems to me to be akin to "If one has talents, one can still contribute to scientific literary research, without math being able to read."

 

[mathwonk]

it is hard to know what you will like but the classic book making basic arithmetic look hard, or making it rigorous, is the one by Landau, Foundations of analysis: this is not a book about how to carry out arithmetic operations for elementary schooklers, but is rather a book about the logic underlying the theory of arithmetic aimed at senior or graduate students in math.

http://bookstore.ams.org/chel-79/

another book i like a lot, that teaches arithmetic and algebra from scratch, and is written by one of the greatest mathematicians of all time, is Elements of Algebra by Euler. here is a free copy: and there are bound ones available from amazon. i treasure my own copy.

https://archive.org/details/elementsofalgebr00eule

 

[Antisthenes]

@Mark44

Study the humanities (more), and perhaps you will see how. But this is becoming too off topic. Right now I'm more interested in learning math than debating philosophy of science, if that's okay :)

 

[Antisthenes]

@Mark44

Just realized that my last post might appear insufficiently humble, and this is a friendly and helpful forum, so it's only proper to explain what I mean: using qualitative methods for example, such as participant observations, one can contribute to fields like psychology and anthropology without remembering the formal rules and details of arithmetic. But one should at least have studied logic, philosophy of science and critical thinking (Tetlock / Stanovich for instance).

 

[fresh_42]

This is a difficult to answer question. Any application, such as psychology, anthropology or any other, often simply uses parts of mathematics chosen by the special needs of its models. E.g macroeconomics is full of higher analysis, in biology or social sciences you can find algebra(s), and every field which uses statistics, also implicitly uses calculus, measure theory and stochastic. And especially the latter is full of misunderstandings on the users' part, simply because measure theory, analysis and stochastic are usually treated quite superficial in other sciences if at all.

I still don't get, whether you want to refresh or learn, what is normally part of a school education, or the principles and foundations of arithmetic, which will lead you to set theory and logic, rather than calculations, or the mathematics of nowadays applications, which would rise the follow-up question about further specification, as these would be too many and too different ones. Closest to philosophy are set theory and logic, and logicians often have been considered to be both, mathematicians and philosophers, e.g. B. Russell.

 

[Antisthenes]

@fresh_42

The social sciences need good quality data, gathered with the help of fieldwork usually, or else you get "garbage in, garbage out". A street smart and logical researcher can provide such data and thereby contribute to scientific progress. Stephen Grey writes in "The New Spymasters" that conflicts have been lost because Western governments have relied too much on sigint and not humint.

Anyway, I don't know enough about arithmetic/number theory to have a clear picture of what I actually need to genuienly understand math, and work efficiently on higher levels. Basically I just want to have enough knowledge and practice on one level to not have gaps and holes when entering the next level. But since I lack confidence in my math skills, I want a really good foundation of understanding math, so that I can apply it in new contexts. I suck at creative puzzle solving, and my IQ in regards to pattern recognition is perhaps not higher than 110, so guess I must compensate by having a very deep and wide understanding, through practice and reading.

 

[Mark44]

Just realized that my last post might appear insufficiently humble, and this is a friendly and helpful forum, so it's only proper to explain what I mean: using qualitative methods for example, such as participant observations, one can contribute to fields like psychology and anthropology without remembering the formal rules and details of arithmetic.

I'm not saying that it's an either/or situation. Certainly someone can use the qualitative techniques that you mention, but to limit oneself to just that type of investigation reduces the effectiveness and impact of the work. Relative to the fields of study you mentioned, a psychological study will typically involve some analysis of the data that has been gathered, which will involve statistics. As for anthropology, my stepdaughter received a PhD in Anthropology. Her thesis involved gathering and analyzing seeds and bone remnants from a site in Utah more than 8000 years old. She used statistics to analyze the types of seeds and pollen found to made an inference about what the climate was like during the times when the site was populated. She used carbon dating techniques to get estimates of the dates of the various layers.

But one should at least have studied logic, philosophy of science and critical thinking (Tetlock / Stanovich for instance).

A basic understanding of logic is useful, but I don't see the utility of philosophy of science. As for critical thinking, one gets that by osmosis in any standard chemistry of physics class. Back about 20 years, when "critical thinking" was alll the rage, it seemed to me that some of the loudest proponents of this were completely incapable of actually thinking critically.

IMO, because you have little background in mathematics, you are making too much of the "formal rules and details of arithmetic." I take it as an article of faith, perhaps unjustified, that anyone who has obtained a high school diploma will be competent at arithmetic, provided that he or she wasn't asleep during the whole process.

Anyway, I don't know enough about arithmetic/number theory to have a clear picture of what I actually need to genuienly understand math, and work efficiently on higher levels. Basically I just want to have enough knowledge and practice on one level to not have gaps and holes when entering the next level.

If you are able to add, subtract, multiply, and divide fractions, then you are probably already capable of moving on. Start with algebra, and then trigonometry. If you want to go further, start in with number theory or calculus.

 

[Antisthenes]

@Mark44

By "critical thinking" I don't mean the excesses of French postmodernism, cf Sokal. Though one can actually learn a lot from moderate postmodernism, if the goal is wisdom, open-mindedness and viewing human beings from different perspectives. But that can also get very onesided if not balanced by thinkers such as Karl Popper and the hard sciences.

My attitude is to learn the best from all traditions, while trying to notice the limits of each one of them. In other words, like many philosophers I'm a generalist who tries to be humble when learning things from specialists. I view math and physics as the king and queen of science. So by "critical thinking" I mean the rational attitude of researchers like Shermer, Kahneman, Gazzaniga, Loftus, and Asch, in addition to philosophers such as Berlin and Kant for example.

As already mentioned, in Norway one can study the humanities without high school math, if one has shown talent within a field that doesn't rely on math. My only excuse for lacking math skills are school teachers that acted intimidating when I asked questions as a kid, so in math class I read zoology books instead, and then started to study human primates.

In any case, what I'm looking for is a book which teaches when to apply different math tools in different contexts. I can understand the logic of each tool, but there are so many of them that I easily lose track of when to apply the correct one in order to solve a particular problem. Have tried to google "problem solving strategies" and even "learn to think like a math genius", but guess the only solution is practice, practice, practice.

 

[Buffu]

Can you add, subtract, multiply and divide ? There is nothing more to arithmetic than that.

 

[fresh_42]

In any case, what I'm looking for is a book which teaches when to apply different math tools in different contexts.

This reminds me a bit of a discussion I once had here. The question had been: Is there a book, which lists all differential equation systems according to their applications? I started to gather some across several very different fields of science and recognized soon, that such a book would need several volumes:
https://www.physicsforums.com/threa...n-if-i-know-its-solution.881146/#post-5539656

 

[fresh_42]

Can you add, subtract, multiply and divide ? There is nothing more to arithmetic than that.

Not from the logical point of view:
https://en.wikipedia.org/wiki/Axiomatic_system
https://en.wikipedia.org/wiki/Deduction_theorem
https://en.wikipedia.org/wiki/Gödel's_incompleteness_theorems

@Antisthenes
In case you've read Kant or Habermas in the original version, it might be worth to switch the language of those Wikipedia pages, although I haven't checked to which extend they differ. Usually they do quite a lot.

 

[Antisthenes]

@Buffu

Yes, can do that of course. But thought perhaps that arithmetic is much more than that, and that a math imbecile like me needs as much knowledge in elementary math as possible, before trying to master the art and creativity of math on higher levels.

@fresh_42

Good advice, tnx :) However, is there perhaps more books, for newbies, about the general mindset necessary for being a good mathematician? Read "A Mind for Numbers" by Oakley, but it contained almost no examples of how to solve math problems in creative and different ways. Really need to learn flexibility when visualising and thinking mathematically, it seems.

 

[Buffu]

Not from the logical point of view:
https://en.wikipedia.org/wiki/Axiomatic_system
https://en.wikipedia.org/wiki/Deduction_theorem
https://en.wikipedia.org/wiki/Gödel's_incompleteness_theorems

I won't call that arithmetic. That looks like Mathematical logic.

Yes, can do that of course. But thought perhaps that arithmetic is much more than that, and that a math imbecile like me needs as much knowledge in elementary math as possible

If you already know arithmetic(+,-,/,*) then what else are going to learn in it, you already know everything.

before trying to master the art and creativity of math on higher levels.

Arithmetic (+,-,/,*) is of very little use in maths. If you want to learn maths then you should learn algebra/calculus and leave all your arithmetic (+,-,/,*) to Stephen Wolfram.

 

[fresh_42]

However, is there perhaps more books, for newbies, about the general mindset necessary for being a good mathematician? Read "A Mind for Numbers" by Oakley, but it contained almost no examples of how to solve math problems in creative and different ways. Really need to learn flexibility when visualising and thinking mathematically, it seems.

Oversimplified we may say, that mathematics is a bit like playing LEGO, including the player's character. The textbooks can be compared with the accurate and detailed models of all kinds, from castles to space ships. Doing mathematics is then the big box which contains all kinds of bricks and you start building something without a detailed plan but a general idea of the result. Until then the structure is rebuilt many times, extended, changed and often a compromise due to the lack of optimal bricks. So the bigger your box is, i.e. the more deconstructed models are in it - the more textbooks you've read, the better you can build whatever you want to.

I would say: Grab an algebra book (van der Waerden as I quoted in post #11, if you like more text, or Serge Lang if you prefer more formulas, or a freely available version like in openstacks, cp. posts #2, #9 and the link there), then a book on calculus (usually people here recommend Spivak, but I don't know, since my elementary ones aren't in English, or which I like very much, Hewitt, Stromberg, (in engl.) especially because it contains a large section about the axiom of choice at the beginning, but gets very elaborated quickly, or again openstacks for free) and either read both of them or evaluate which kind of mathematics comes easier to you, because these are the main streams with different techniques.

 

[fresh_42]

I won't call that arithmetic.

Hilbert (and the mathematicians and logicians of his time) called it arithmetic. Who am I to correct those Grands? But it may well be a matter of language here. E.g. we do not use the word algebra for simple calculations. Or calculus for analysis. Arithmetic involves everything which mathematics is build upon, so yes it involves logic and set theory.