03/14/19

Is there any beginner friendly books about algebra?

My answer to a question on Quora: Is there any beginner friendly books about algebra?

The answer depends on the level at which you wish to study algebra, and on your background. So I making two assumptions. The first one is

  • You wish to study abstract algebra at undergraduate level.

My second assumption involves a dichotomy. If you learn with an established education system and follow an established curricular path, then stick to textbooks which are normally used by colleges /universities in your country at the next stage of education.

If you are a self-learner or an advanced learner who is working ahead of curriculum, then I suggest to consider a truly classical book,

G. Bikhoff and S. Mac Lane, A Survey of Modern Algebra.

Why? On the first page of Preface they say:

We have tried throughout to express the conceptual background of the various definitions used. We have done this by illustrating each new term by as many familiar examples as possible. This seems especially important in an elementary text because it serves to emphasize the fact that the abstract concepts all arise from the analysis of concrete situations.

They start the book with the most classical of all algebraic structures: the ring of integers and use it to introduce commutative integral domains. They do that long before they introduce groups. In my opinion, this is a right approach, to start with something very familiar. However, here is the catch: look at this paragraph from page 3:

They take for granted that the reader understands understands without further explanation this sentence:

In \(\mathbb{Z}[\sqrt{2}], \quad a+b\sqrt{2} = c+d\sqrt{2} \) if and only if \( a=c, \quad b=d\).

Is it familiar to you? If you can recognise in these words a classical and very old fact of mathematics (frequently mentioned in secondary school mathematics) than the book is perhaps for you.

So if you are a self-learner of mathematics (for example, if you are in a college or university where teaching is below your level) then you also have to take responsibility for gauging the level of your readiness for study of something new.

03/14/19

Why does algebra have letters in sums?

My answer to a question in Quora: Why does algebra have letters in sums?

One should not underestimate the influence of François Viète who was the first to use algebraic notation (letters) not only for unknowns but also for parameters (knowns) in a problem. He also used, as Wiki states, “simplification of equations by the substitution of new quantities having a certain connection with the primitive unknown quantities”. Importantly, Viète is the first cryptographer and cryptanalist known to us by name. His decryption of intercepted diplomatic correspondence had direct effect on European politics of his time. A really juicy bit from the Wiki:

In 1590, Viète discovered the key to a Spanish cipher, consisting of more than 500 characters, and this meant that all dispatches in that language which fell into the hands of the French could be easily read.

Henry IV published a letter from Commander Moreo to the king of Spain. The contents of this letter, read by Viète, revealed that the head of the League in France, the Duke of Mayenne, planned to become king in place of Henry IV. This publication led to the settlement of the Wars of Religion. The king of Spain accused Viète of having used magical powers

At that time, encryption of texts mostly used substitution ciphers, and the idea of substitution of letters for numbers should be very natural for Vieta.

I modestly suggest that teachers could perhaps use this idea: teaching primary school children some basic substitution ciphers: it is fun, it is a natural spelling exercise, and, I believe, a good propaedeutic for later study of algebra and computer coding.

03/14/19

Why do many physicists disparage mathematical rigour?

My answer to a question on Quora: Why do many physicists disparage mathematical rigour?

Perhaps “disparage” is a wrong word, as it has already been pointed out in this discussion. However, the question is well pointed. My answer will be short and excessively general:

In mathematics, “rigour” is a tool for, and source of, consistency of mathematical results: they remain the same no matter how they are proven, results of calculation do not depend on the way the calculation was carried out, etc.

In physics, the nature itself, the experiment is a source of consistency. The world around us is remarkably consistent.

Perhaps the same can be said with smaller words: mathematicians build the ideal world, and rigour is a way to ensure that it does not collapse. Physicists have a world ready for them, and so far it does not show any sign of collapse.

03/7/19

What number is between 1/2 and 8/9?

My answer to a question on Quora: What number is between 1/2 and 8/9?

John K Williamsson gave a good answer: for what he called the “dirty sum” of \(\frac{1}{2}\) and \( \frac{8}{9}\) (the mathematical term for that is mediant):

\(\displaystyle{\frac{1}{2} < \frac{1+8}{2+9} < \frac{8}{9}}\)

He suggests to use algebra to prove the mediant inequality: if \(a,b,c,d\) are positive numbers and

\(\displaystyle{\frac{a}{b} < \frac{c}{d}}\)

then

\(\displaystyle{\frac{a}{b} < \frac{a+c}{b+d} < \frac{c}{d}}.\)

I wish to add that, at the primary school level, the mediant inequality does not need an algebraic proof, it is sufficiently self-evident.

Indeed consider fractions \(\frac{1}{2}\) and \(\frac{8}{9}\) as descriptions of real-life situations:

\(\frac{1}{2}\) : \(2\) children have \(1\) bag of fruits.

\(\frac{8}{9}\) : \(9\) children have \(8\) bags of fruits.

They come together and share equally: \(1 + 8\) bags of fruit between \(2+9 = 11\) kids, that is, they form the mediant:

\(\displaystyle{\frac{1+8}{2+9}}\)

In this sharing, which group of kids looses and and which one gains? Of course \(2\) children with \(1\) bag gain: they have \(\frac{1}{2}\) bags per head, the other group comes with bigger share per head: \(\frac{8}{9}\). For the same reason, kids in the second group lose.

I use an example with kids and bags of sweets in my lectures; here I replaced sweets by more politically correct fruits — perhaps I have to go further and use green vegetables in place of fruits. The original idea belonged to the great Israel Gelfand, and was stated in a more colorful language:

You can explain mathematics to everyone, even to drunkards. If you ask some people drinking vodka on a park bench, what is is bigger, \(\frac{2}{3}\) or \(\frac{3}{4}\), they will respond with expletives. But if you ask them, what is better, \(2\) bottles of vodka for \(3 \) people or \(3\) bottles of vodka for \(4 \) people, they will instantly give you the right answer: of course, \(3\) bottles for \(4\) people.

And this instant conclusion comes from an argument which is the reversal of the informal proof of the mediant inequality: how to get from the situation “\(2\) bottles for \(3\) people” to the situation “\(3\) bottles for \(4\) people”? Of course, it means that a fourth man comes and brings with him a whole bottle — can you imagine, a whole bottle of vodka! In the mediant inequality,

\(\displaystyle{\frac{2}{3} < \frac{2+1}{3+1} < \frac{1}{1}},\)

or

\(\displaystyle{ \frac{2}{3} < \frac{3}{4} < 1.}\)

I have seen some papers which confirm that this is a typical pattern of arithmetical thinking, as done by “normal” people in real life situations (for example, I have seen a claim that it is used by hospital nurses for comparing doses of medication, which one is bigger and which one is smaller).

Later addition: there are several version’s of Gelfand’s story floating on the Internet, this is one, from “Love and Math: The Heart of Hidden Reality” by Edward Frenkel, is interesting because it triggers the recursive use of the mediant inequality:

“People think they don’t understand math, but it’s all about how you explain it to them. If you ask a drunkard what number is larger, 2/3 or 3/5, he won’t be able to tell you. But if you rephrase the question: what is better, 2 bottles of vodka for 3 people or 3 bottles of vodka for 5 people, he will tell you right away: 2 bottles for 3 people, of course.”

03/7/19

What is a guaranteed way to learn algebra?

My answer to a question on Quora: What is a guaranteed way to learn algebra?

The question appears to be not particularly well posed: to learn algebra at what level? School? Undergraduate? Graduate? Research level? Starting from what kind of mathematical background? And what does it mean “guaranteed”? This expression refers to some level of expected outcome, to some criterion. For passing an examination? If so, what kind of examination? For further study of mathematics? If so – where, what kind of mathematics, at what level? For writing and publishing, in serious peer refereed journals, research papers on mathematics? For proving the Jacobian conjecture?

In my humble opinion, with all this variety of possible scenarios, in many of them there are well-tested efficient ways to learn algebra, but, to the best on my knowledge, in none of the scenarios there is a guaranteed way to learn.

03/7/19

What is a good way to help a student to remember to carry numbers in multiplication?

My answer to a question on Quora: What is a good way to help a student to remember to carry numbers in multiplication?

The answer was already given in this thread by Mark McLenaghan: place value. As soon as children understand place value, carries will be easier to understand. And another point: carries should appear first in addition.

Next come comes multiplication of a number by a single digit number – first without carries ( such as \(23 \times 3\)) and then with carries, first occurring only once and going to the empty position, (\(43 \times 3\)), then once with addition in the next place (\(23 \times 4\)), and several times (such as \(23 \times 9\) and \(123\times 9\)).

Finally, multiplication of arbitrary numbers is bolted together from multiplication by single digit numbers — of course, starting with multiplication by two-digit numbers.

I think experienced teachers can subdivide the learning process in even larger number of steps, avoiding sudden jumps in difficulty and making the progress smooth.

For learning place value, an useful exercise is playing with casino-style tokens of value 1, 10, 100, maybe even 1000 and counting pods with holes for 10 and 100 tokens (of the kind that were used in banks in pre-machine age for quick counting of coins); it could be fun.

Please notice that I am talking about the pre-machine age. This is the general principle of development of an individual: ontogeny recapitulates phylogeny (as formulated by Ernst Haeckel in case of evolution: a human embryo at early stages of development looks like a fish). So the use of archaic tools in learning is natural and perhaps unavoidable.

From the mathematical point of view, the long multiplication (and, actually, long addition) are immensely deep; I apologise for plugging in my old blog posts, but this one is illumination: The secrets of long multiplication. And this one, about long division: A tale about long division. A good discussion of place value can be found here: R. Howe, Three pillars of first grade mathematics.

03/7/19

Does this equation have an infinite number of rational solutions: x^2+y^2=2?

My answer to a question on Quora: Does this equation have an infinite number of rational solutions: x^2+y^2=2?

I want to add, as a comment to the excellent post by Senia Sheydvasser in this thread, a simpler proof of the following statement:

If \(Ax^2 + Bxy + Cy^2 = D\) is a quadratic curve with rational coefficients \(A, B, C, D\), and if it has a point \((a,b)\) with rational coordinates, then it has infinitely many points with rational coefficients.

As Senia Sheydvasser explained, we can make change of variables \(x \leftarrow x – a\), \(y \leftarrow y\) and assume, without loss of generality, that the curve has a rational point \((0, b)\) (see the diagram below). Now pick on the \(x\)-axis a point with rational coordinates \((t,0)\) and draw the line through the points \((t, 0)\) and \((0,b)\). It is well-known that this line has equation

\(\displaystyle{\frac{x}{t} + \frac{y}{b} = 1.}\)

(the so-called “equation of a straight line in terms of its intercepts”).

The points of intersection of the line and the curve are solutions of the system of simultaneous equations

\begin{eqnarray*} \frac{x}{t} + \frac{y}{b} &=& 1\\ Ax^2 + Bxy + Cy^2 &=& D \end{eqnarray*}

Expressing \(y\) in terms of \(x\) in the first equation and substituting it into the second equation yields a quadratic equation for \(x\) with rational coefficients, something like \(Px^2 +Qx +R = 0\); one of its solutions is \(x = 0\); hence \(R = 0\) and the second root is found from a linear equation \(Px + Q = 0\) and is therefore also rational, and then the corresponding value y is also rational — it is linked to \(x\) by a linear equation with rational coefficients.

We found a way to assign to every rational point \((t,0)\) on the \(x\)-axis a point on the curve with rational coordinates.

What we did is that we used what is known as the birationality property of the Stereographic projection, one of its many wonderful properties. In a proper system of mathematics education, stereographic projections should be taught to schoolchildren who plan to seriously study, later at university, mathematics or mathematically intensive subjects in sciences or engineering. It explains a lot of stuff, say, the so-called universal trigonometric substitutions of calculus, and, in its 3-dimensional version, is critically important for understanding of complex analysis.

03/6/19

How does category theory relate to other branches of mathematics?

My answer to a question in Quora: How does category theory relate to other branches of mathematics?

An excellent question. Category theory is important on its own, and has important applications in a number of other mathematical theories; however, the crucial and the most fundamental impact of category theory is invisible and under-reported, it is of cultural nature. It can be compared with the influence of set theory: 99% of mathematicians use only a modicum of naive set theory, ignoring deeply penetrating and frequently very hard results of set theory as the live research discipline which continues to develop and flourish.

There is a telling example: the theory of games of chance was created in the 17th century and gave birth to probability theory; the latter was already quite developed by the time when, in the 20th century, the concept of a deterministic game had finally crystallized – in a paper, of all people, by Zermelo, who proved that chess was a deterministic game: for one of the players, there is a strategy, that is, a function from the set of permissible position to the set of moves, which achieves at least a draw. His paper was published in 1913 (see Zermelo’s theorem (game theory) – Wikipedia). Why did this happen so late? The word “set” in the definition of a strategy came to use only in the second half of the 19th century.

The same is happening with category theory: the vast majority of mathematicians use its ideas and terminology in a very rudimentary and naive form, frequently even without realisation that they are doing so. In the work that I am doing, it had happened to be very important to remember that an algebraic group was a functor from the category of unital commutative rings to the category of groups. Some my colleagues who work in the same theory continue to insist that a group is a fixed set with some operations on it. When my co-author and I recently solved a certain problem which was open since 1999, we were able to do that only because for us a group in question was a functor – not much deeper than that. Why it was not solved by someone else earlier? Because for them a group was just a set.

03/6/19

Do subjects in the realm of pure mathematics stay that way?

My answer to a question in Quora: Do subjects in the realm of pure mathematics stay that way?

Many mathematical disciplines, traditionally considered being part of pure mathematics, left it and found applications (and sources of funding) elsewhere, sometimes under a new name. There are routine and well-known examples; a branch of mathematical logic became computer science, some parts of number theory were swallowed by cryptography, some highly theoretical parts of analysis became the mathematical machinery of quantum mechanics, etc. Also, a lot of stuff in what is seen as pure mathematics was born from attempts to solve very practical questions. Who remembers now that this standard bit of undergraduate mathematics, integration of rational functions by decomposing the fraction into a sum of partial fractions and then integrating these partial fractions was invented in the 17th century for solving certain mathematical problems of maritime navigation? Or that George Boole invented his Boolean algebra as a practical tool of thinking and argumentation in real life situations? (He was a social activists, and wanted to arm working class people by tools for understanding, and challenging, legal arguments in courts, etc.)

Summary: in the historic perspective, boundaries between “pure” and “applied” mathematics are blurred.

02/28/19

Zeno moving in the direction of gradient

My answer to a question on Quora: What’s going on with the gradient? My understanding is that it points you in the direction of the greatest change, but you only go a tiny bit in that direction before computing a new gradient. That’s really vague though and I need specifics.

What appears to be a weak point of your question is its unnecessary personification of the process described:

it points you in the direction of the greatest change, but you only go a tiny bit in that direction before computing a new gradient”:

notice repetition of the word you:

it points you, you go, you are computing.

This triggered similar associations and comparisons in some answers:

Think of yourself out in the rolling countryside

When I write this answer, a drop of sweat is sliding down by bald head. The drop does not think. It does not feel. It has no imagination. It computes nothing. It follows the gradient of the surface of my skull because it is a consequence of some basic principles of mechanics of which the drop is unaware — as well as it is unaware that its existence will be terminated, in a second, by a handkerchief.

The Nature does not compute. The wind does not solve Navier–Stokes equations – it is us, humans, who use equations for a very crude and approximate description of behaviour of fluids and gases.

However, this personification of the process of following the direction of the gradient points us to one of the greatest paradoxes in mathematics: Zeno’s Arrow Paradox, see Zeno’s paradoxes – Wikipedia.

I am using here one of its forms (it is called Dichotomy Paradox in the Wiki, and I adopt it there:

Before reaching the target, the arrow must get halfway there. Before it can get halfway there, it must get a quarter of the way there. Before traveling a quarter, it must travel one-eighth; before an eighth, one-sixteenth; and so on.

Wiki continues:

The resulting sequence can be represented as:

\(\displaystyle \left\{\cdots ,{\frac {1}{16}},{\frac {1}{8}},{\frac {1}{4}},{\frac {1}{2}},1\right\}\)

This description requires one to complete an infinite number of tasks […]

Do you see the similarity with your question? It is in the words requires one. Who is this one?

Zeno’s paradox also assumes personification, it describes the world through eyes of some sentient and intelligent being who sits on the arrowhead and counts tasks .

In the description of this counting, the physical time of the flight of the arrow is substituted, in sleight of hand, by a fiction, by imaginary time in which counting is taking place; indeed, just to say “one part out of thirty two” — or whatever the English words for \(\frac{1}{32}\) are — requires the use of more and more, and longer and longer words. This creates a psychological illusion that time stops.

But no-one sits on the head of Zeno’s arrow, and no-one completes any tasks, and no-one counts them.