I heard a conjecture once that the best textbook you'll find on any given topic is your third. The point, of course, is that it simply takes about three serious attempts to make it click - but it is a fallacy to give all the credit to the third book.

Probably additional skills and experience you picked up all these years by solving/understanding hard problems

It is very difficult to take that step back and divorce myself from years of experience and tough lessons, and to present the subject matter in a way that can be grasped without an innate understanding that took me years to reach.

Ah, but there are books, just making you want to go tp sleep by just looking at them and some are able to spark passion (in me).

My point being, it is definitely about motivation and discipline, but a good didactic book, helps with that.

And since we are all different (types of lerners), there definitely isn't one book to rule them all. And its been a while since I studied from a book, but I could usually tell from skimmimg over a few pages, of whether this book can help me, or not.

So it gets you over a hump which might make the more advanced / detailed books more accessible.

It is free online. It starts with recursion and dynamic programming. I felt like dynamic programming really clicked for me after reading the first few chapters but this was probably the 3rd or 4th book I read on the subject.

[1] https://youtu.be/yFVXsjVdvmY?t=431

Although I also dont want to discourage anyone from trying this either. Anything that'll get people to learn is better than apathy. :)

I've experienced the '3rd text' phenomenon, but I could also point to specific features affecting the wide variance in math text effectiveness.

And this book is a pretty good example of exactly that: it has specific unique features allowing it to fulfill its promise of being an effective 'translation' guide for a programmers to a bunch of otherwise typically implicit ideas relating to methods or foundational concepts in mathematics that can be extremely difficult stumbling blocks for the self-taught.

IMO a good strategy: take people's glowing praise about particular texts with a grain of salt—but, if specific beneficial features can be pointed out, which would be advantageous to you as a learner, know that can mean striking gold sometimes (in terms of not wasting time).

I'm learning Rust now and feel like I need to do the same thing. Read "the rust book" plus Rust by Example plus one or two Rust books on the O'Reilly website. It takes a lot longer to read multiple books in parallel but I find I remember better this way.

https://www.google.com/amp/s/byorgey.wordpress.com/2009/01/1...

Walter Rudin’s Principles of Mathematical Analysis (chapters 1 through 7). A mental torture on the first exposure, but like a fine wine when the palate is mature.

For any integer n >= 0 and any list of n + 1 points (x[1], y[1]), ... , (x[n+1], y[n+1]) in R^2 with x[1] < x[2] < ... < x[n+1], there exists a unique polynomial p(x) of degree at most n such tat p(x[i]) = y[i] for all i.

So, it seems the author assumes that a reader will have math maturity of a good senior high-school student, as most of students wouldn't need to worry about property of existence. The book also covers the proof of such theorem with formal notations and the proof is built up with previous theorems -- a pretty standard way in math books which nonetheless requires math maturity of a good high school senior. The table of contents also shows that the book will cover linear algebra, calculus, and group theory in a whirlwind. Again, such content demands close-to-college-level math maturity. I'm also generous here, as public schools of the US do not really teach that much formal math.

So, here is the dilemma: people with this level of maturity should already be good at math or have access to other materials to help them with math. People who do not possess such maturity will not go through the book anyway, or have more beginner-friendly materials to read. Note I'm sure there are exceptions, but I question the percentage of such exception.

I did not connect with math when I was in High School. I liked geometry, but never took anything after that. I didn’t really learn anything new in college algebra.

I program at an accomplished level- I’m doing senior dev work in a complex ___domain, and have led teams of developers.

But I feel bad when trying to go through this book. Like I had huge gaps in knowledge that the author assumed I had, which led me to wonder why that is.

I will probably try to struggle through this again, in hopes that eventually it clicks. If anyone knows of a book I could use as a prerequisite or intermediate step, I would appreciate that!

It certainly hasn't made me a mathematician by any stretch, but it's helped me fill in a lot of gaps left behind by awful maths teachers in school, and it's helped rekindle an interest that I'd long since forgotten.

You are right that this book requires high school level math knowledge, as it's the equivalent of a first (and maybe second) year math course. Most programmers I have met do display that amount of knowledge however. What other starting point (or topic) would you suggest for teaching someone mathematics that can be related to programming?

> He also discusses the fact that the language of mathematics is looser than programming in a lot of ways. In code, things have to be expressed a very exact way or they just don’t compile. Variables and functions have to be fully and explicitly defined if you expect the computer to run them. But in math, there’s a lot of tacit agreement and assumptions that go on. Lots of shortcuts and conventions.

This kind of context around how math is done as a human activity in practice, especially in contrast to programming, is extremely helpful orientation for programmers trying to self-teach mathematics.

It would've saved me tons of time and trouble if I'd known the above while trying to work through math texts after graduating with a CS degree: instead I wasted a tone of time writing over-detailed proofs, always feeling as if I were doing something wrong if every tiny step weren't explicit (more closely matching my experience with programming).

Sign up for the mailing list here if you're interested in getting updates: https://jeremykun.us11.list-manage.com/subscribe?u=99aa071e9...

And some more notes on the process and ideas behind this book: https://buttondown.email/j2kun/archive/a-week-of-book-writin...

A Good Year for “A Programmer’s Introduction to Mathematics” - https://news.ycombinator.com/item?id=21676384 - Dec 2019 (51 comments)

On Self-Publishing “A Programmer’s Introduction to Mathematics” - https://news.ycombinator.com/item?id=18642481 - Dec 2018 (24 comments)

A Programmer's Introduction to Mathematics - https://news.ycombinator.com/item?id=18579076 - Dec 2018 (214 comments)

Might be of interest to a similar group of people as the OP

https://jeremykun.com/2013/04/03/homology-theory-a-primer/

Homotopy has is origin in topology (but eventually transcended it), so learning about it first in a more “tangible” setting of where it came from might indeed be helpful.

[1] https://jeremykun.com/primers/

Learning math in some prescribed way (book or sequence of books) is the mathematics equivalent of "what programming language should I learn first". The most important thing is simply doing anything at all! Don't get planning paralysis!

If you think you'll actually do or read anything, then give it a try. Definitely push yourself some, but if it becomes a slog don't be afraid to move on to something that looks more interesting. Youll find your way back to anything that was actually important anyway. :)

What's the equivalent for learning mathematics? A lot of mathematics seems only useful for learning more advanced mathematics.

I don't know exactly what you like to work on, but perhaps theres some related mathematical area youre curious to know more about?

But that may be because my education always put the pure math and applications in close proximity.

Eg Galois theory kind of looks like this: Applying an abstract model of symmetry to a model of polynomials to answer a question about solvability.

https://www.youtube.com/playlist?list=PLMcpDl1Pr-viA25VUkHNm...

Just found it. So no clue as to the quality

I learned in a class and we didnt use a book so I cant recommend one. I dont think it should matter too much though.

Aside from that, what kind of things are you curious about?

Another intro book I thought of was Peter Eccle's introduction to mathematical reasoning. Might be worth looking at.

If you want a nice leisurely introduction to groups Nathan Carter's Visual group theory is nice.

I got a lot of use out of the princeton encyclopedias of math. Dont expect for them to really teach you anything, but the articles are nice for seeing whats out there.

Definitely try to learn basic analysis and algebra. I dont think any book I know is amazing, but basically any will do the job.

Importantly, dont be afraid to try to learn something out of your depth. In fact I think its important to try! If something really grabs you, try to read more and backfill what you don't know.

However, it's an intense read. I strongly recommend it, but if you don't have some college level math under your belt, it can be harder to understand than its title makes it seem.

This issue has been bothering me for years. In a typical math forumla you find on wikipedia, there are many unnecessary symbols included + really critical things are left vague. I feel like mathematicians are at fault here. They should clean up their shit, maybe write these equations as code. When I translate one of these equations into code its always much much shorter, and it has the benefit of being 100% deterministic. Eg. one example f(X) (often drawn big and elaborate) means Y!!

Often specific symbols have implicit meanings, like theta θ is pretty commonly used for some angle, r is often used to mean a radius. So you'll often see something like "r sin θ" with no explanation. At first it's meaningless, but once you know the conventions, it's crystal clear. It's considered so basic, that nobody would waste the space explaining it. Same is if you're reading something about code and something says "const float x = 0.1" or something. The author is probably not going to go into an explanation of what a const or a float is or what x means. You're expected to know.

So what I like about the book is that helps someone without knowledge of all these conventions to begin to understand them.

Sometimes the right kind of intuition is all it needs to make it click. Sometimes it's that tiny bit of knowledge one is missing to get the whole picture and suddenly everything makes sense.

(Btw I think we might have met ages ago at a conference or two in Cologne)

Whereas for math it does not seem to be the same. There is no documentation, and no-one ever seems to explain those basics online. Eg. this book is a pretty obscure pdf.

Can you elaborate on what you mean here?

In order to draw the curve I iterate through values of X for each pixel, plug those values into the right hand side, and what the right hand side of equation is equal to is my Y value for that point on the curve. So if they had just written that big fancy f(X) as Y it would have been much clearer and easier to understand for me initially.

Eg One might have f(x,y,z) = (...). If f is in some class of functions with some properties (homogeneous, linear, smooth, etc) we can operate on it abstractly. We could even derive properties of surfaces f(x,y,z) = C.

You mean the function? It could just as well be something like:

    f(x, y) = 5xy + 3x^2 + 4y^2

With respect to the use of Greek letters and such, I do lament that many writers of mathematics fail to define their terms. Instead assuming that the reader is fully conversant in the ___domain, when often a single paragraph at the start would add a great deal to the clarity of their work. However, that doesn't mean that the use of such variables is bad, they just need to be defined.

The benefit of the mathematical notation is that it permits conciseness and lends itself well to symbolic manipulation (that is, a large portion of what we do when we do algebra and calculus). The former is a tricky subject, conciseness at the cost of clarity can be a net negative. But the latter is crucial to a lot of work, the way that we write programs does not lend itself well to symbolic manipulation and would be counterproductive for mathematics.

In fact, I've often had to translate programs into a symbolic notation in order to try and decipher them because the long descriptive names, as useful as they are in isolation, ended up rendering the total procedure nearly impenetrable. Or at least unanalyzable. And the conversion to a symbolic notation permitted me to simplify the program substantially because I was able to apply ideas from algebra to the program (often boolean algebra, in particular, this is a very useful practice for condition heavy code with lots of predicates).

As to your original criticism, I'll wager that more people in this world will understand f(x) = 5x + 3x^2 than will understand rudimentary code, as it is taught a lot more than programming is. I don't mean anything negative when I say this, but the only people I know who complain about math syntax are programmers. By changing these conventions, you are asking all mathematicians, physicists, chemists, most engineers, economists, etc to change. I will wager that over 95% of them will not prefer your style.

Finally, the trouble with replacing f(x) with Y is that the former tells me I'm dealing with a function. The latter does not. In fact, for many mathematicians, y = 5x + 3x^2 is a constraint, not a function.

I feel similarly about statistics. I of course, given my line of work, have a solid foundation there. But my area of expertise is... more complicated to explain or define. When it comes to statistics expertise though, apart from a solid foundation I simply know enough to know what tools to use, and how to research and evaluate such tools. For example, in a recent project I knew that LSTM was an appropriate tool, but I don't know more than a high level abstraction of how it works and the ___domain of problems it might help solve.

To give a very basic example sort of like knowing when to use the Pythagorean theorem but not knowing enough to prove it up from axioms.

https://mitpress.mit.edu/books/structure-and-interpretation-...

One wrong sign and instead of cosine u are calculating sine. Chas theory. Dynamic systems sensitive to initial values.

Windows is mostly functional despite lots of bugs.

I don't believe that an introduction to math has to be long, and here I will try to give an introduction, and overview, just as blog posts. That is this introduction is much shorter than a book.

So I introduce the main topics. Throughout, nearly always you can get a lot more from a simple Google search or Wikipedia article.

By far the most important topics are calculus and linear algebra. My view is that a 12 year old with good interest and some okay or better teaching and some good texts and access to Google and Wikipedia can do well with both calculus and linear algebra.

Both calculus and linear algebra have enough so that a person can spend their life in research, maybe in applications, in advanced parts. This is especially the case for calculus. E.g., calculus can include calculus on manifords, partial differential equations, differential geometry, and numercial and algorithmic topics for all of these.

What is described here could be covered in a good undergraduate math major. Such a student would have all or nearly all of a good math background for graduate work in math, physics, or other STEM fields.

None of this math is nearly new. So, for good texts, I recommend ones that have been regarded as among the best for at least 20 years. For some texts, 60 years is not too old. E.g., my favorite text in linear algebra was first published in 1942. So, look for old texts. Usually can buy used copies in good condition for less than $10.

How to use the texts: (1) Get a stack of blank paper, a sharp pencil, a big eraser, a comfortable chair, a good light, and a quiet room. (2) Read a section of the text, try to understand all or nearly all the section, when time is available try to get a first cut intuitive understanding good enough to explain to someone who knows no math at all, and then do the exercises. If your text proves theorems and is short on good exercises, then prove the theorems yourself as a good source of exercises. (3) For the more important topics, especially calculus and linear algebra, get 3-4 well recommended texts, pick one as your main text, and use the others for alternative explanations and more exercises. (4) Occasionally look back at what you have learned and find a good, intuitive view of what is going on.

Qualifications: I hold a Ph.D. in pure/applied math from a world class research university. While I've published peer-reviewed orginal research in pure/applied math, mathematical statistics, and artificial intelligence, my interests in math are mostly for applications; I've applied math for US national security, in business, and in computer science research. I've been programming for decades, and now I'm doing a startup, a novel Web site, that has some old pure and new applied math at its core (that users won't see) and have written and run the software for the Web site. The software is almost entirely in Microsoft's .NET and consists of 24,000 statements, in 100,000 lines of typing. Currently I'm collecting initial data before going live on the Internet.

Below are 17 numbered sections. Calculus is in section 11, and linear algebra, section 12. For sections 1-10, plane geometry in section 8 deserves careful study, e.g., as in a good high school course, but for most students for the other sections of 1-10 just the material here may be enough.

(1) Sets

A set is a collection, aggregation, box of, etc. of elements -- we're supposed to already know what the elements are.

In math, sets are useful in much the same way as little covered plastic containers are in cooking, say to store some left over peas, cookies, pizza sauce, etc.

So, a set is a way to identify, keep track of what we are talking about.

Given a set A and some element x, the x is either in the set A or it is not -- there are no other alternatives. If element x is in set A, then it is in set A exactly once. e.g., never 2 or 3 times.

The elements in a set are not in any particular order. E.g.,

{1, 3, -4} = {3, -4, 1}

Given two sets A and B, an element x might be in both A and B, might be in many sets.

We also permit a special set, the empty set that has no elements. The empty set is curiously useful, plays a role roughly like 0 does for numbers.

We can define sets with notation, e.g.,

A = {x| x > 7}

We read this as "A is the set of all elements x such that x > 7."

(2) Foundations

Near 1900 a lot of math was known. A nagging question was, what are the foundations that let us be confident that what we are talking about makes sense?

B. Russell noticed we could write

A = {x| x is not a element of x}

Then A is an element of A if and only if it is not an element of A. So, we have a contradiction, angst, tummy aches, heart burn, head aches, etc. This is the Russell Paradox.

The problem was solved by first deciding what the elements were and then making sets out of just those elements. Net, we rule out asking of set A is an element of set A. Of course this work was all just conceptual.

Also there was work to set up some axioms (properties) to be assumed for sets; this was axiomatic set theory. Then there was an effort, successful, to use axiomatic set theory to define everything else in math. So, roughly, if the axioms are solid, so is the rest of math.

Some deep, difficult, and profound questions were found and some answers were found for some of these.

In the end, except for people specializing in foundations, the approach taken for set theory is Zermelo–Fraenkel set theory usually assuming also the axiom of choice. I know that that work is solid way down in the basement, foundations, of math, but not many people go down there very often. I've been there for a good tour and have no intention of going back.

Roughly, axiomatic set theory is to add credibility to the math you already knew in the 8th grade.

It may be that this material gets taught to give a deeper understanding to people who intend to teach in K-12. E.g., Newton invented calculus, and I doubt that he knew about the Russell Paradox or the Axiom of Choice.

(3) Numbers

The most important concept in math is numbers, and the most important numbers are the real numbers. To understand these numbers it is nearly always enough to visualize them as just the points on a line. That is essentially the same as using a yard stick. No biggie.

We can talk about the set of real numbers; maybe we let R denote the set of real numbers.

We can also consider the whole or natural numbers -- 1, 2, 3, .... We might agree that N is the set of natural numbers.

We might define the set N of natural number by: 1 is in N and, for each n in N, n+1 is also in N. We will use this in proofs by mathematical induction.

By including 0 and the negatives of the whole numbers we get the integers

... -3, -2, -1, 0, 1, 2, ...

Maybe we say that Z is the set of integers.

The rational numbers are all the p/q where p and q are integers and q is not zero. We can say that Q is the set of rational numbers.

If we let i be or act something like the square root of -1 all the complex numbers are all the x + iy for reals x and y. We can let C denote the set of all the complex numbers.

Of course, there is no square root of -1. But we can regard the complex numbers as a clever bookkeeping trick that at times is curiously useful in defining the sine and cosine in trigonometry, electrical circuit theory, analysis of wave motion, quantum mechanics, etc.

This use of R, N, Z, Q, and C is common, but there is no rule that says we could not use other notation.

The most important properties of these numbers you learned by the 9th grade and learned most of those in grade school. No biggies.

There is one more property that is crucial in calculus: The set of real numbers R is complete. There are several equivalent definitions of complete. Here is maybe the simplest definition: Given a subset S of R and some x in R, if for all a in S we have that a <= x, then x is an upper bound of set S.

The completeness property is that each non-empty subset S of R with an upper bound has a least upper bound.

E.g., let

S = {x| x in R and x < 2}

Then 3 is an upper bound of S and 2 is the least upper bound.

Perhaps a better way to explain completeness is to say, intuitively, that whenever we move in steps to be as close as we please to some point on the line, there is a real number there we are getting close to.

A big point is that the rational numbers are not complete. E.g., let

S = {a | a rational and a < square root of 2}

Then the square root of 2 would be the least upper bound but is not rational. So, the rational numbers are not complete.

The concept of completeness, especially viewing it as converging to something, is a major concept in advanced work in math analysis.

There is something of a joke that "Calculus is the elementary consequences of the completeness property of the real number system."

(4) Infinity

During the days near 1900 struggling with axiomatic set theory, there was also struggling with the concept of infinity or infinite.

We say that two sets A and B have the same cardinality, essentially are the same size, if they can be put into 1-1 correspondence.

Okay, set B is a subset of set A provided each element of B is an element of A. B is a proper subset of A if A contains some elements not in B, that is, if B is not all of A.

Then a set is infinite if it can be put into 1-1 correspondence with a proper subset of itself. A common example is to let A be the set of natural numbers and B be the set of even (can be divided by 2 with no remainder -- 4th grade math) natural numbers. Then B is a proper subset of A, and, for each natural number n in A, let it correspond to the natural number 2n in B. Presto, bingo, we have put A and B into 1-1 correspondence. So, set A is infinite.

A set that is not infinite is finite.

Similarly sets of integers, rationals, reals, and complex numbers are all infinite.

Not very difficult to see. the set of natural numbers has the same cardinality as the set of integers, that is, the natural numbers and the integers can be put into 1-1 correspondence.

Any set that can be put into 1-1 correspondence with the natural numbers, has the same cardinality as the natural numbers, is said to be countably infinite.

Curiously, the rationals are countably infinite. To show this, use the clever Cantor diagonal process (lots of hits at Google).

Is the set of reals countably infinite? No, and there is a short, clever argument that shows this. So, the cardinality of the set of reals is larger (after we handle some details) than the set of natural numbers.

Is there a set D with cardinality greater than the natural numbers but smaller than the cardinality of the set of real numbers? Difficult question. That there cannot be such a set D is the continuum hypothesis (CH). As can see at Wikipedia, this question was settled by work of Kurt Gödel in 1940 and Paul Cohen in 1963: The result is that the CH is independent of Zermelo–Fraenkel set theory with the axiom of choice. By independent we mean that we can assume that the CH is true or false and cannot get a contradiction. This work has been a bit amazing, even philosophical.

Understanding infinity is important, and we have done a good job at that. Unless we are down in the basement of foundations, we at most rarely consider the continuum hypothesis.

There is a theorem that for any natural number n, we have

1^3 + 2^3 + ... + n^3 = n^2(n + 1)^2 / 4

We can prove this by mathematical induction. We want a proof that works for any natural number n, and for this we use the definition (above) of the natural numbers.

So, first we check if the equation is true for n = 1:

1^3 = 1

and when n = 1

n^2(n + 1)^2 / 4 = 1 (2)^2 / 4 = 1

So, the equation holds for n = 1.

Suppose the equation holds for some n and check the equation for n + 1:

1^3 + 2^3 + ... + n^3 + (n + 1)^3 =

n^2(n + 1)^2 / 4 + (n + 1)^3 =

n^2(n + 1)^2 / 4 + 4(n + 1)(n + 1)^2 / 4 =

(n^2 + 4(n + 1))(n + 1)^2 / 4 =

(n + 1)^2 (n^2 + 4(n + 1)) / 4 =

But

n^2 + 4(n + 1) =

n^2 + 2n + 1 + 2(n + 1) + 1 =

((n + 1) + 1)^2

So

(n + 1)^2 (n^2 + 4(n + 1)) / 4 =

(n + 1)^2((n + 1) + 1)^2 / 4

and the equation also holds for n + 1.

Then by the definition of the natural numbers, the equation holds for all natural numbers.

Done.

So, that's an example of proof by mathematical induction.

(5) Fundamental Theorem of Arithmetic

A prime number is a natural number like 1, 3, 5, 7, 11, 13, 17, 19, 23, ..., that is, evenly divisible by only itself and 1.

Prime numbers have been studied since the ancient Greeks, and there are many difficult questions easy to state; now we have answers to some of the questions.

Consider 45:

45 = 3 * 3 * 5

So, we have written 45 as a product of prime numbers. It is easy enough to see that any natural number can be written as a product of prime numbers.

The Fundamental Theorem of Arithmetic is that each natural number can be written as a product of prime numbers in just one way, e.g., where we neglect the order of the prime numbers in the product.

So,

45 = 3 * 3 * 5

is the only way to write 45 as a product of prime numbers.

(6) High School Algebra

A quadratic is an algebraic expression of the form

ax^2 + bx + c

A polynomial is the same except can have higher natural number powers of x, e.g.,

7x^5 - 4x^3 + 2x - 1

Polynomials play an important role in linear algebra -- e.g., via determinants and eigenvalues.

Polynomials are also used in error correcting codes and related work.

There have been attempts to use polynomials in curve fitting, but for large natural numbers n the x^n term means that as go on the real line a long way from 0, commonly the value of the polynomial goes a very long way from 0, so far from 0 that it is usually unrealistic for applications. That is, polynomials don't provide good, versatile means of curve fitting.

Also interesting and useful is the binomial, for natural number n and numbers x and y

(x + y)^n

Presumably the name binomial is from having the two numbers, x and y.

Curiously, surprisingly, this binomial is important in various approximations and in combinations, permutations, probability, and statistics.

(7) Second Year High School Algebra

A root of a polynomial is a value of x that makes the value of the polynomial 0.

The fundamental theorem of algebra is that each polynomial can be written as

(ax + p)(bx + q) ... (cx + r)

where the p, q, ... r are roots of the polynomial. The theorem holds for polynomials with either real or complex coefficients. The roots, even with only real coefficients, can be complex.

I've omitted some details due to the challenge of typing math.

Mathematicians worked for centuries to get good proofs of this result.

Also might be covered are the algorithms for least common multiple and greatest common divisor. These were known to the ancient Greeks and get used currently.

We can define a function f by, for any number x,

f(x) = ax^2 + bx + c

So, function f is a look-up table -- plug in a value for x and get back the value of

ax^2 + bx + c

Since the x is a number, we say that the ___domain of the function f is the set of numbers, usually the set of real numbers. In this case, the range of function f is also the set of real numbers.

We can have look-up tables for telephone numbers -- the ___domain a set of names of people and the range a set of telephone numbers. We can have a function for Internet addresses, ___domain URLs and range, say, the 64 bit internet addresses.

So, functions have enormous generality -- given any two non-empty sets A and B, we can have a function with ___domain A and range B and to indicate that f is a function with ___domain A and range B write

f: A --> B

(8) High School Plane Geometry

High school plane geometry contains some nice, useful math and an excellent first lesson in mathematical proofs.

(9) Abstract Algebra

In grade school, we learned that

2 + 3 = 5

Abstract algebra calls the +, addition, an operation. Multiplication is also an operation.

We learned that for real numbers a and b

a + b = b + a

So, addition is a commutative operation. It is also associative: For real numbers a, b, c we have

a + (b + c) = (a + b) + c

Multiplication is also commutative and associative.

For addition, 0 is the identity element since for any number a we have

a + 0 = 0 + a = a

And -a is the inverse of a since

a + -a = 0.

Etc. with the rest of the usual properties of the operations of addition and multiplication we knew in grade school.

Well, abstract algebra studies operations on sets more general than the numbers we have considered so far. So, there are groups, rings, fields, etc. The rationals, reals, and complex numbers are all examples of fields.

E.g., a group is a nonempty set with just one operation. The operation is associative and there is an identity element and inverses.

Some groups are commutative, and some are not. Some groups are on a set that is infinite and some are on a set that is finite.

Given a common definition of vectors, with vector addition the set of all the vectors forms a commutative group.

Similarly for the rings, fields, etc. -- that is, we have sets with operations and properties.

There are some important applications, e.g., in error correcting coding.

(10) Calculus

Calculus was invented mostly by Newton and mostly for understanding the motions of the planets etc.

The first part of calculus has to do with derivatives. So, suppose we have an object that for each time t is at distance t^2. Then the speed of the object at time t is 2t and the acceleration is 2. The acceleration is directly proportional to the force on the object.

We got 2 and 2t from t^2 by taking calculus derivatives.

We write

d/dt t^2 = 2t

d/dt 2t = 2

So at time t, consider increment in time h. Then at time t + h, the object is at position (t + h)^2. So in time h, the object has moved from t^2 to (t + h)^2, that is, has moved distance

(t + h)^2 - t^2

= t^2 + 2th + h^2 - t^2

= 2th + h^2

Since speed is distance divided by time, the object has moved at speed

(2th + h^2) / h

= 2t + h

Then as h is made small and moves to 0, the speed moves to just

2t

So for very small h, the speed is approximately just 2t, and as h moves to 0 the speed is exactly 2t.

So, 2t is the instantaneous speed at time t.

So, 2t is the calculus derivative of t^2.

To reverse differentiation we can do the second half of calculus, do integration, start with the 2t and get back to t^2.

Calculus is the most important math in physics.

Later chapters in a calculus book show how to find various areas, arc lengths, and surface areas.

Calculus with vectors is crucial for heat flow, fluid flow, electricity and magnetism, and much more.

Einstein's E = mc^2 can be derived from just some thought experiments and the first parts of calculus. There is an amazing video at

https://www.youtube.com/watch?v=KZ8G4VKoSpQ

Calculus is a pillar of civilization.

(11) Linear Algebra

This topic starts with systems of linear equations in several variables, e.g., for two equations in three variables:

     2x + 5y - 2z = 21

     3x - 2y + 2z = 12

A solution can be regarded as a vector, e.g.,

(x, y, z)

A function f is linear provided for x and y in the ___domain of f and numbers a and b we have

f(ax + by) = af(x) + bf(y)

It can be claimed that the two pillars of the analysis part of math are continuity and linearity.

Linearity is enormously important, essentially because it is such a powerful simplification.

The linear equations of linear algebra are linear in this sense.

Differentiation and integration in calculus are linear.

Linearity is a key assumption in physics and engineering, especially signals and signal processing.

In linear algebra we take the two equations in two unknowns and define matrices so that we can write the equations as

     / 2  5 -2 \ / x \   / 21 \
     |         | |   | = |    |
     \ 3 -2  2 / | y |   \ 12 /
                 |   |
                 \ z /

We define matrix multiplication so that multiplying the first and second matrix gives the same results as in the linear equations:

     2x + 5y - 2z = 21

     3x - 2y + 2z = 12

The 3 x 1 and 2 x 1 matrices are examples of vectors.

Matrix multiplication is linear.

The known properties of matrices and their multiplication and addition and of associated vectors are amazing, profound, and enormously powerful.

There is a joke, surprisingly close to true, that all math, once applied, becomes linear algebra.

(12) Differential Equations

In the subject differential equations, we are given some equations satisfied by some function and its derivatives and asked to find the function.

The classic application was for the motion of a planet: Newton's laws of motion specified the derivatives of the position of the planet as a function of time, and the goal was to find the function, that is, predict where the planet would be.

Science and engineering have many applications of differential equations.

Some of the models for the spread of a disease are differential equations.

We have seen that

d/dx t^2 = 2t

Well, we may have function

f(s, t) = st^2

We can still write

d/dx f(s, t) = d/dx st^2 = 2st

but usually we say that we are taking the partial derivative of f(s, t) with respect to x.

(13) Mathematical Analysis

This topic is really part of calculus and has the fine details on the assumptions that guarantee that calculus works. The main important assumption is continuity. The next most important assumption is that the function has ___domain a closed interval or a generalization to what is called a compact set; a closed interval is, for real numbers a and b,

{x| a <= x <= b}

Also important are infinite sequences and series that make approximations as close as we please to something we want. So, in this way we define sine, cosine, etc.

(14) Advanced Calculus

This topic is also part of calculus but concentrates on surfaces, volumes, and maybe the math of flows of heat, water, etc.

Commonly this part of calculus makes heavy use of vectors and matrices.

Also covered can be Fourier series and integrals.

A pure tone in music, say, from an organ, has a fundamental frequency, say, 440 cycles per second and also overtones at natural number multiples of that fundamental frequency. Each of the overtone frequenciess works like an axis of orthogonal coordinates and is a terrific way to analyze or synthesize such a tone.

The Fourier transform is closely related and has an orthogonal coordinate axis for each real number.

Engineering is awash in linear systems that do not change their properties over time -- time invariant linear systems. Such systems merely adjust some Fourier coefficients. This is a powerful result.

(15) Measure Theory

This topic develops integral calculus with a lot more generality.

(16) Probability Theory

In probability we imagine that we do some experimental trials and observe the results. We have a sample space, our set of trials.

We regard the set of all trials like a region with area, and we say that the area is 1.

Maybe we flip a coin and get Heads. The set of all trials that yield Heads is one event Heads.

The event Heads has area, probability

P(Heads)

With a fair coin we have

P(Heads) = 1/2

A random variable X is some outcome of a trial. Usually X is a number. In this case we can write

P(X >= 2)

for the probability that random variable X is >= 2.

Events A and B are independent provided

P(A and B) = P(A)P(B)

In probability we can have algebraic expressions with random variables, have a distance between random variables, have a sequence of random variables converge to a random variable, etc.

Given a real valued random variable X, for real number x we can let

F_X(x) = P(X <= x)

Then F_X is the cumulative distribution of X, and

f_X(x) = d/dx F_X(x)

is the probability density of X.

One of the major results is the central limit theorem that shows a distribution converging to the Gaussian bell curve.

Two more results are the weak and strong laws of large numbers that show convergence to average values (means, expectations).

One more result, amazing, is the martingale convergence theorem.

The Radon-Nikodym theorem of measure theory provides an amazing general theory of conditional expectation which is one way to look at the value of information.

In statistics we are given the values of some random variables and try to infer the values of something related.

The more serious work in probability and statistics is done on the foundation of measure theory.

(17) Optimization

We have a furniture factory and a new shipment of wood. We have some freedom in what we do with this wood. An optimization problem is how to use the freedom to make the greatest revenue from the wood.

More generally we have some work to do with some freedom in how we do the work, and an optimization problem is how best to exploit the freedom to have the best outcome, profit, for the work.

If the profit is linear and if the work and freedom are described by linear equations, then we have the topic of linear programming.

Linear programming has played a role in the Nobel prizes in economics.

The main means of solving linear programming problems is the simplex algorithm which is a not very large modification of Gauss elimination.

In practice the simplex algorithm is shockingly fast; in theory its worst case performance is exponential.

In the furniture making, we can't sell a fraction, say, 1/3rd, of a dining room chair. So, we want our solution to consist of integers. The simplex algorithm does not promise us integers. So we have a problem in integer linear programming.

In practice from the simplex algorithm we may get to make 601 + 1/3rd dining room chairs and round that to 601 or just 600. No biggie.

If we insist on integers, then, surprisingly, even shockingly, we have one of the early examples of a problem in NP complete.

In practice there are still ways to proceed; often we can get the optimal integer solutions we want easily; sometimes such solutions are challenging; sometimes we have to settle for approximations; often the approximations can be quite good, like forgetting about 1/3rd of a chair; in theory the ways have worst case performance that is exponential and nearly useless.

One integer linear programming problem I attacked has 600,000 variables and 40,000 constraints. I wrote some software that in 600 seconds on a slow computer found an integer solution within 0.025% of optimality. The solution made heavy use of the simplex algorithm, non-linear duality theory, and something called Lagrangian relaxation.

In addition to optimization, the simplex algorithm and related math make nice contributions to geometry, e.g., prove theorems of the alternative used in proving the famous Kuhn-Tucker conditions necessary for optimality.

Since it is possible that the work we are trying to optimize is complicated, its mathematical description may also be complicated and the optimization effort, difficult.

Optimization problems are common in animal feeding, cracking crude oil, transportation problems, and much more.

Game theory can be regarded as a special case. One proof of the famous Nash result (featured in the movie about Nash, A Beautiful Mind) is via linear programming.

Some problems are to find curves, say, how to climb, cruise, and descend an airplane to minimize fuel burned or the best curve for a missile to attack an enemy fighter jet. Newton found the brachistochrone curve solution to the problem of the frictionless curve that would let a bead slide down in minimum time.

Relevant fields are calculus of variations and deterministic optimal control. One famous result is the Pontryagin maximum principle.

Some problems have uncertainty; so the optimization may be to find the solution with the best expected outcome. For such problems there is the field of stochastic optimal control.

I am truly wondering how many (professional) programmers don't. (Not to say, of course, that the book is not good or not useful.)

And then there are all the non-technical majors who become programmers, like the many philosophy graduates I've worked with. This isn't to say they can't learn the math, but they often have even less exposure than the typical business major in the US.

And globally there are many people who come to professional programming without any degree at all beyond a high school diploma. And given the variance in high school curricula globally there's no way to say what level of math this group possesses, but they almost certainly lack college level academic math exposure, the majority at least.

Heh heh. Electrical engineer here. My first and last use of Karnaugh maps for a job was 14-15 years after taking my digital logic course. The irony was that it was for a routine programming problem: The customer had given me with an ugly flowchart and I felt I must reduce that monstrosity to something much simpler. I did, but then my fellow coworkers would always wonder if my result was identical to the flowchart. I would tell them to make a truth table and confirm it. Finally one coworker did that and left a comment pointing out he had already verified it. In one sense, it made the code less "readable", but no one wanted the job of doing a direct translation of the flowchart.

I did have to Google to remind myself how to do them, but it took only a few minutes to figure out.

Of course, this doesn't apply to domains where math is necessary (graphics, etc).

I remember in one engineering job I had, two engineers were tasked with the problem: Given an ellipse, assume a horizontal/vertical line "cuts" off the ellipse. What is the area of the remaining piece?

This is a standard Calc II problem, and I'm sure everyone there had taken it - we required a MS in that team, and some people had PhDs. The solution takes a bit of work but in the end you'll have an analytical formula with the exact answer. In code you'd just put one line with the formula.

Is that what they did? No. They wrote a program to approximate the area. I really doubt they took care of any floating point subtleties, but I knew that I wouldn't be popular if I probed their solution.

There are also a lot of people who got a degree in some other field and later moved to a programming career (I see a lot of Philosophy majors get into programming, interestingly.)