Why does algebra have letters in sums?

My answer to a question on 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 know 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.


No intrinsic gender differences in children’s earliest numerical abilities

Alyssa J. Kersey, Emily J. Braham, Kelsey D. Csumitta, Melissa E. Libertus & Jessica F. Cantlon. No intrinsic gender differences in children’s earliest numerical abilities. npj Science of Learning  3, Article number: 12 (2018).


Recent public discussions have suggested that the underepresentation of women in science and mathematics careers can be traced back to intrinsic differences in aptitude. However, true gender differences are difficult to assess because sociocultural influences enter at an early point in childhood. If these claims of intrinsic differences are true, then gender differences in quantitative and mathematical abilities should emerge early in human development. We examined cross-sectional gender differences in mathematical cognition from over 500 children aged 6 months to 8 years by compiling data from five published studies with unpublished data from longitudinal records. We targeted three key milestones of numerical development: numerosity perception, culturally trained counting, and formal and informal elementary mathematics concepts. In addition to testing for statistical differences between boys’ and girls’ mean performance and variability, we also tested for statistical equivalence between boys’ and girls’ performance. Across all stages of numerical development, analyses consistently revealed that boys and girls do not differ in early quantitative and mathematical ability. These findings indicate that boys and girls are equally equipped to reason about mathematics during early childhood.


What are some real-world uses of the determinant of a matrix?

My answer to a question on Quora:

What are some real-world uses of the determinant of a matrix?

At the time of writing, I am engaged in a small debate with a colleague on one of the LinkedIn discussion groups: he teaches students to solve systems of 2 linear equations with 2 variables using Cramer’s rule (that is, via determinants), without giving any justification or proof for it, but I personally prefer self-justified solutions: for systems of 2 linear equations with 2 variables, the honest Gaussian elimination is quick, and it is easy to explain to students why it gives the right solution. Moreover, every intermediate step of Gaussian elimination can be naturally interpreted in terms of the original system of equations.

And this goes to the heart of the matter: in life, determinants are almost never used in computation. Someone said that

mathematics is the art of avoiding calculations;

in that sense,

linear algebra is the art of avoiding calculations with matrices,

and the rule of thumb is

avoid calculations with determinants!

For example, you can invert a matrix in essentially the same time as compute its determinant; after that the use of the cofactor formula for the inverse of a matrix and Cramer’s rule for solving systems of linear equations becomes waste of time.

However, determinants provide extremely efficient tools for thinking about problems of linear algebra, including those in practical applications. Linear algebra in its development or exposition goes through more and more compressed expression of relevant mathematical meaning, and the value of the determinant: zero or not zero is perhaps the most compressed form of expression of linear dependence / independence of \(n\) vectors in \(\mathbb{R}^n\).

Determinants have wonderful algebraic properties and occupied their proud place in linear algebra because of their role in higher level algebraic thinking.

In this thread on Quora some uses of determinants were mentioned, for example, computation of eigenvalues of a matrix; I am not an expert in numerical linear algebra, but I have a feeling that most methods for computation of eigenvalues do not even mention the word “determinant”. Even at a theoretical level, determinants can be excluded from the standard treatment of linear algebra, see Sheldon Axler’s paper Down with Determinants!

So, let me summarise:

  • If you need more that just application of existing computer programs for solving practical problems of linear algebra and have to think about the process of solution, you may find determinants very useful indeed.
  • Determinants can be meaningfully used for compact formulation of mathematical models of physical phenomena (perhaps this applies not only to physics). This thread in Quora contains some nice examples.
  • But is is best avoid calculation of determinants.

Why is mathematical illiteracy socially acceptable?

My answer to a question on Quora, with some amendments:
Why is mathematical illiteracy socially acceptable?

Because what is known as “mathematical literacy” is economically redundant: having it, or not having it does not affect earnings of 95% of people. An ever decreasing pool of jobs which require “mathematical literacy” is filled (at least in Britain) by recruiting university graduates from mathematics-intensive disciplines, like mathematics, or physics, or electronic engineering. These jobs require no more than basic school level mathematics skills which are supposed to be given to every school leaver. However, big employers do not trust school marks, and rightly so.

However, a small number of professional occupations require knowledge of mathematics far beyond “mathematical literacy”. Critically, for many nations, this includes certain jobs in the defense and security sector.

The summary: everyone is taught mathematics at school not for his/her personal advancement and enjoyment, but for alleged future employment – which is a fiction for majority of learners. This is misselling of “educational product” on a grandiose scale. Wide acceptance of “mathematical illiteracy” is a natural, and healthy, reaction of the society to this scam.

I support the idea that mathematics has to be taught as music: for learner’s enjoyment and personal development, without any promise of future employment in “music-intensive industries”. This will make mathematics more popular — and much more expensive to teach, which, of course, kills this idea at its roots.

There are also crucially important skills  which  beg to be taught: mathematics for citizenship, mathematics for protection ones’ rights as a responsible human and a member of the society. Should I explain why this is not part of the curriculum?

A colleague wrote to me recently that “anti-math” campaign had reached his university and was pushed by the university administration. It is easy to explain: they see their task as training, on the cheap, future workforce for businesses; they are not interested in educating citizens or helping young people to boost their spiritual and intellectual potential.

I apologise for plugging my papers, but they contain more on that:

Mathematics for makers and mathematics for users, bit.ly/2qYHtst

Calling a spade a spade: Mathematics in the new pattern of division of labour, goo.gl/TT6ncO

As a taster, a quote from one of these papers:

If banks and insurance companies were interested in having numerate customers – as they occasionally claim – we would witness the golden age of school mathematics: fully funded, enjoying cross-party political support, promoted and popularised by the best advertising companies in all forms of mass and social media. But they are not; banks and insurance companies need numerate workforce – but even more so they need innumerate customers. 25 years ago in the West, the benchmark of arithmetic competence at a consumer level was the ability to balance a chequebook. Nowadays, bank customers can instantly get full information about the state of their accounts from an app on a mobile phone together with a timely and tailored to individual circumstances advice on the range of recommended  financial products. This kind service can be described in a logically equivalent form: a bank can instantly exploit the customer’s vulnerability.

Perhaps I have to add a disclaimer:

Views expressed are my own and do not necessarily represent position of my employer, or any other person, corporation, organisation, or institution.


What is your favorite visual mathematical proof? Perhaps it is Tennenbaum’s proof of the irrationality of the square root of 2

My answer to a question in Quora, slightly edited:

What is your favorite visual mathematical proof?

Perhaps it is Tennenbaum’s proof of the irrationality of the square root of 2. What follows is reposted from a blogpost by David Richeson at his wonderful blog Division by Zero.

Suppose \(\sqrt{2} = a/b\) for for some positive integers a and b. Then \(a^2=2b^2=b^2+b^2\).

Geometrically this means that there is an integer-by-integer square (the pink  \(a\times a\) square below) whose area is twice the area of another integer-by-integer square (the blue \(b \times b\) squares).

Assume that our \(a\times\) square is the smallest such integer-by-integer square. Now put the two blue squares inside the pink square as shown below. They overlap in a dark blue square.

By assumption, the sum of the areas of the two blue squares is the area of the large pink square. That means that in the picture above, the dark blue square in the center must have the same area as the two uncovered pink squares. But the dark blue square and the small pink squares have integer sides. This contradicts our assumption that our original pink square was the smallest such square. It must be the case that \(\sqrt{2}\) is irrational.

Added on 06 July 2018: Nicolas Miari wrote in a comment on Quora:

First time I’ve seen a “Visual proof by infinite descent”!

In that case, you will have fun in turning this diagram into an infinite descent proof of irrationality of \(\sqrt{2}\):

The quickest proof by infinite descent is perhaps proof of irrationality of the Golden Ratio (I found it in a beautiful little book by Tim Gowers Mathematics: A Very Short Introduction).

By definition, the rectangle \(ABCD\) is a golden rectangle, if after cutting off the square \(B’BCC’\), the remaining rectangle \(ADC’B’\) is similar to the original one, \(ABCD\), see the picture:

The ratio of lengths of sides of a golden rectangle is called the golden ratio.

Theorem. The golden ratio is irrational.

Proof.  If the golden ratio were rational, a “golden rectangle” could be drawn on square grid paper, as on picture above.  After cutting a square from it we get a smaller “golden rectangle” drawn on square grid paper. By principle of infinite descent, this is impossible – hence the golden ratio is irrational.

Hmm, we did not even care about the numeric value of the golden ratio  \(\dots\)


Why do irrational numbers exist? Why does math exist that is only conceptual and not applicable in nature?

My answer to a question in Quora, slightly edited:

Why do irrational numbers exist? Why does math exist that is only conceptual and not applicable in nature?

There are many good answers in this thread; I am trying to give a an answer that is simpler than most of them, but has deep historic roots.

Irrational numbers had been introduced into mathematics because it was discovered, quite a long time ago, that measurement of magnitudes cannot be reduced to counting of units of measurements. As it was explained in this thread many times, it was a great discovery of Pythagoreans that there is no unit of length which can be used for simultaneous absolutely precise measurement of the side and diagonal of the square.

Magnitudes and numbers are two very different species and need to be treated with equal respect. A simplest example: temperature is a magnitude, but, strictly speaking, is not a number. Why? because we expect from numbers that they can be compared by size, added, and multiplied. We can can compare two temperatures (“water in jar A is warmer than water in jar B”), but addition is already a problem: if we pour water from both jars into jar C, resulting warmth of water depends on the amount of water in each of jars. And what about multiplication of two temperatures? Can you suggest a real life situation when it is needed and used?

Finally, there is one particular magnitude for which even units of measurement do not exist: strength of smell (or odour).

It is one of the misconceptions about mathematics: it is frequently claimed that all mathematical magnitudes can be measured by real numbers. There is a simple and striking counterexample: there are rigorous, in the most modern sense, axiomatisations of Euclidean geometry, where angles cannot be measured by distances. Remarkably, Euclid never claimed that angles and lengths were the same magnitude.

So, acceptance of irrational numbers is a form of mental hygiene: it is equivalent to acceptance of the sad fact of reality that there is no one universal unit (with subunits centi- milli-, micro- , nano-) of measurement for everything.

The most prolific and popular responder to mathematics questions on Quora,  Alon Amit

There’s no physical significance to the product of two temperatures, but there’s a lot of significance to the product of temperature and mass, for example. This provides ample motivation for attaching numerical value to temperature.

What follows is my reply to him:

Yes, of course. But measurement of time is remarkably imprecise in comparison with measurement of time and distance. And specific heat depends on temperature. And actually no phenomenon of thermodynamic nature could be measured with absolute precision.

Unit of time, second, is now defined via counting:

Since 1967, the second has been defined as exactly 9,192,631,770 times the period of the radiation corresponding to the transition between the two hyperfine levels of the ground state of the caesium-133 atom.

Wikipedia further says:

A set of atomic clocks throughout the world keeps time by consensus: the clocks “vote” on the correct time, and all voting clocks are steered to agree with the consensus, which is called International Atomic Time (TAI).

As we can see, there is no escape from the number/magnitude and counting/measurement dichotomies, Historically, mathematicians of Ancient Greece had to handle geometric proportions and discovered that they cannot be expressed by rational numbers. In modern terminology (and in a very crude description) Eudoxes dealt with real numbers as equivalent classes of proportions. His approach was brilliantly revived very recently, by Schanuel who suggested a new construction of the field of real numbers as the factor ring of the ring of “almost additive” maps on the additive group of integers (that is, maps from integers to integers such that the difference

\(f(m + n) – f(m) -f(n)\)

takes only finitely many values), by the ideal of maps which take only finitely many values. The reverse interpretation is obvious: if  \(t\) is a real number, we can associate with it a map from integers to integers defined as

\(f(n) = [tn]\),

where square brackets denote, as usual, the integer part of real number, and give a counting approximation to measurement.


How do mathematicians avoid circular reasoning when proposing a new proof for an already proved theorem?

My answer (slightly edited) to a question on Quora:

How do mathematicians avoid circular reasoning when proposing a new proof for an already proved theorem? In particular, this seems to happen among students when solving a homework or trying to reprove theorems they’ve studied.

Avoiding circular reasoning is a result of education, training, upbringing; mathematicians avoid circular reasoning because if the did not, they would never become mathematicians. To compare, there are certain things which every driver never does: for example, pressing accelerator and brakes simultaneously — for otherwise he/she would have difficulty in passing the driving test. Conveniently, the accelerator pedal and the brakes pedal are usually positioned in a car in such a way that it makes it difficult to press the accelerator with the left foot, and even harder to press the accelerator and brakes (with the right foot) simultaneously.

Perhaps it needs to be explained that mathematicians not just remember  a particular theorem (actually, very frequently only an idea of the theorem, without technical details – they know that they can recover details when they need them), but, crucially, they care about, and keep in order, relations between theorems; they keep in their minds not just facts, but relations and analogies between facts, and, moreover, analogies between analogies. They sustain in their minds a multidimensional image of a theory, a complex, strongly interlinked, hierarchically built and dynamically developing system. This is like a living organism; to keep it alive, one has to follow certain rules of mental hygiene; avoidance of circular reasoning is perhaps one of the most important.

Here lies one of the principal contradiction of mathematics teaching: too frequently, teachers find that it easier to make students to memorise and mechanically use individual mathematical facts rather than opening to them the rich, vibrant, full of colours world of connections between these facts (and, to say the truth, the teachers themselves are frequently unaware about the existence of this world). Not surprisingly, students frequently have difficulties with circular arguments because it simply does not matter not only for them, but to their teachers, too, which comes first: statement A or statement B.

Students could be even more confused when they encounter completely legal circular proofs of equivalence of several different definitions (or characterisations) of the same concept or object. When writing this answer, I have checked lecture notes of the my undergraduate course of linear algebra: it contains 19 equivalent characterisations of invertible matrices, and I was able to produce, on the spot, two more. It is a kind of algebraic merry-go-round, and a deeper structural understanding is needed to avoid a dizzy spell.


“Universities are essentially massaging the figures”

This assessment  by an unnamed expert is quoted in the short on-line version of the report A degree of uncertainty: an investigation into grade inflation in universities from Reform, a UK think-tank. A fuller quote:

There is considerable evidence to suggest that ‘degree algorithms’ (which translate the marks achieved by students during their degree into a final classification) are contributing to grade inflation. Approximately half of universities have changed their degree algorithms in the last five years “to ensure that they do not disadvantage students in comparison with those in similar institutions”. Research has also identified serious concerns about how these algorithms treat ‘borderline’ cases where a student’s overall mark is close to the boundary of a better degree classification. One expert concluded that “universities are essentially massaging the figures, they are changing the algorithms and putting borderline candidates north of the border”.


Abstraction is multiple realisability

A brilliant blogpost from   Double-entry bookkeeping and Galileo: abstraction vs idealization.  He gives a definition of abstraction, as used in computer science:

abstraction is multiple realisability.

I am currently designing a first year, first semester course, something like Introduction to Algebra. One part of it will be about integers and polynomials:

  • start with division with remainder of integers and polynomials
  • develop, in parallel, divisibility theory and uniqueness of  prime factorisation – at every step  stating two theorems, one for integers, another for polynomials, but proving only one of them and leaving writing up a proof of the other as an exercise for students;
  • conclude this part of the course by proving the Chinese Reminder Theorem for integers and Lagrange’s Interpolation Formula for polynomials and explaining why this is one and the same theorem;
  • perhaps only then introduce rings, fields, homomorphisms, and Euclidean rings;
  • and give one more exercise to students: uniqueness of prime factorisation of Gaussian integers.

I think that, at start of undergraduate mathematics, an abstract concept should be introduced only after students have well familiarised themselves with its several realisations.