Mathematical Notation

References, credits and the end of the book

Further reading Martzloff, Jean-Claude. A History of Chinese Mathematics. Berlin: Springer, 1997. Shen, Kangshen; Crossley, John N.; Lun, Anthony W.-C. The Nine Chapters on the Mathematical Art: Companion and Commentary. Oxford: Oxford University Press, 1999. Ifrah, Georges. Histoire universelle des chiffres. Paris: Robert Laffont, 1994. Ascher, Marcia; Ascher, Robert. Code of the Quipu. Ann Arbor: University of Michigan Press, 1981. Alder, Ken. The Measure of All Things: The Seven-Year Odyssey and Hidden Error That Transformed the World. New York: Free Press, 2002. Einstein, Albert. Die Grundlage der allgemeinen Relativitätstheorie. Annalen der Physik, 49, 1916, pp. 769–822. Weil, André. Souvenirs d’apprentissage. Basel: Birkhäuser, 1991. Iverson, Kenneth E. A Programming Language. New York: Wiley, 1962. Iverson, Kenneth E. Notation as a tool of thought. Communications of the ACM, 23(8), 1980, pp. 444–465. Knuth, Donald E. Two notes on notation. American Mathematical Monthly, 99(5), 1992, pp. 403–422. Graham, Ronald L.; Knuth, Donald E.; Patashnik, Oren. Concrete Mathematics. Reading: Addison-Wesley, 1989. Backus, John et al. The FORTRAN automatic coding system. Proceedings of the Western Joint Computer Conference, 1957, pp. 188–198. Stigler, Stephen M. Stigler’s law of eponymy. Transactions of the New York Academy of Sciences, 39, 1980, pp. 147–157. Carraher, Terezinha N.; Carraher, David W.; Schliemann, Analúcia D. Mathematics in the streets and in schools. British Journal of Developmental Psychology, 3, 1985, pp. 21–29. Carraher, Terezinha N.; Carraher, David W.; Schliemann, Analúcia D. Na vida dez, na escola zero. São Paulo: Cortez, 1988. D’Ambrosio, Ubiratan. Etnomatemática: elo entre as tradições e a modernidade. Belo Horizonte: Autêntica, 2001. Gerdes, Paulus. Geometry from Africa: Mathematical and Educational Explorations. Washington: Mathematical Association of America, 1999. Appel, Kenneth; Haken, Wolfgang. Every planar map is four colorable. Illinois Journal of Mathematics, 21, 1977, pp. 429–567. Hales, Thomas et al. A formal proof of the Kepler conjecture. Forum of Mathematics, Pi, 5, 2017, e2. Moura, Leonardo de; Ullrich, Sebastian. The Lean 4 theorem prover and programming language. CADE 28, Springer, 2021. Image sources Image Source Licence Rod numerals in Qin Jiushao’s Shushu jiuzhang (1247) Wikimedia Commons public domain Page 62 of the Dresden Codex, reproduction by Sylvanus Morley Wikimedia Commons public domain Inca quipu in the Larco Museum, Lima Wikimedia Commons CC BY 2.0, bobistraveling Decimal time dial by Hanin (c. 1793) Wikimedia Commons public domain Albert Einstein in 1921, by Ferdinand Schmutzer Wikimedia Commons public domain Punched cards with Fortran Wikimedia Commons public domain The triangle in Zhu Shijie’s Siyuan yujian (1303) Wikimedia Commons public domain Ubiratan D’Ambrosio receiving the Kenneth O. May Medal (2001) Wikimedia Commons CC BY-SA 4.0, ICHM Map coloured with four colours Wikimedia Commons CC BY-SA 3.0, Inductiveload The interactive figures (the three scripts of a number and decimal time) work in all four languages of the site. The record of the downloaded images, with the hash of each file, is in data/semiotica_imagens.json.

October 26, 2026 Read

Notation and culture: a guide and a glossary

This chapter read notation as the product of people, places and tools. The guide below gathers the questions that help to make that reading of any sign. Five questions for a notation What tool was it born of? The rod board, the punched card, the APL keyboard, the blackboard. The shape of the signs often keeps the shape of the tool. What community uses it? Physicists, engineers, programmers, market sellers. The same letter can mean different things in each tribe, like $\log$ or $j$. What did it replace, and why? The metre replaced local measures because trade gained from it; decimal time replaced nothing because nobody gained. Whom does it leave out? Every abbreviation charges a toll to those who do not know the convention. Whoever writes for many readers undoes the shortcut at least once. What does it make thinkable? Red and black rods made negative numbers thinkable; the Iverson bracket turns counts into sums; formal writing makes it possible for a machine to check a proof. A complete reading of a sign: the equals sign in a program Consider the line total = total + price, in a supermarket checkout program.

October 26, 2026 Read

Notation for machines: proofs the computer checks

For two thousand years, a mathematical proof was a text written to be read by people. It could skip “obvious” steps, lean on a drawing, use the conventions of the community. The reader filled in the rest. In the last fifty years a reader has appeared that fills in nothing: the computer. The four colours How many colours are enough to colour any map so that neighbouring countries have different colours? The answer, four, was conjectured in 1852 and resisted for over a century. In 1976 Kenneth Appel and Wolfgang Haken, of the University of Illinois, proved it by reducing the problem to almost two thousand cases that a computer checked one by one, in more than a thousand hours of machine time.

October 26, 2026 Read

Street mathematics and school mathematics: mathematics off the page

In the early 1980s three researchers at the Federal University of Pernambuco, in Brazil, Terezinha Nunes Carraher, David Carraher and Analúcia Schliemann, went to the markets of Recife to talk to children and teenagers aged 9 to 15 who sold coconuts, corn and popcorn. They posed as customers and set sums at the moment of purchase. Later they came back with the same sums written on paper, school-style. The result, published in 1985 under the title Mathematics in the streets and in schools, became famous. In the market, the children got about 98% of the sums right. With the same sums presented as word problems, about 74%. As written computation exercises with no story, about 37%. The book the authors published later, in Brazil, summed up the situation in its title: Na vida dez, na escola zero (1988), “ten out of ten in life, zero at school”.

October 26, 2026 Read

The names on theorems: Stigler's law

The names of theorems, formulas and numbers are a part of notation that almost nobody questions. We write “Pascal’s triangle”, “Pell’s equation”, “L’Hôpital’s rule” as if the name were a neutral label. It is not: the name records who was read, by whom and in what language. A law that confirms itself In 1980 the American statistician Stephen Stigler published a paper with a provocative law: no scientific discovery is named after its original discoverer. And, to round it off, he attributed the law to the sociologist Robert K. Merton, who had already studied the phenomenon. Stigler’s law of eponymy is therefore an example of itself. It exaggerates, but it is right with uncomfortable frequency.

October 26, 2026 Read

The equals sign that gives orders: x = x + 1 and programming languages

For a mathematician, the line $$x = x + 1$$is an equation with no solution: subtracting $x$ from both sides leaves $0 = 1$. For a programmer, it is one of the most common lines there is, and it means “take the value of $x$, add one and store the result in $x$”. The same sign, in the two communities, does opposite jobs: there it asserts an equality; here it orders an assignment.

October 26, 2026 Read

Notation as a tool of thought: Iverson

In 1962 the Canadian Kenneth Iverson, who had taught at Harvard and was then working at IBM, published a book called A Programming Language. Despite the title, it was a book of mathematical notation: a way of writing algorithms concisely and precisely, with new symbols for operations on whole lists and tables of numbers. A few years later IBM itself turned this notation into a language a computer could run, and it took the name of the book: APL.

October 26, 2026 Read

Tribal conventions: Einstein, Bourbaki and the initiated reader

A notation does not belong to all mathematicians in the same way. Each field, sometimes each school, has its shortcuts, and outsiders trip over them. This is not a flaw: it is what happens with any living language, which splits into accents and jargons. But it has consequences for readers and for learners. Einstein erases the summation sign In 1916, in the paper that presented general relativity, Albert Einstein had to write hundreds of expressions like

October 26, 2026 Read

The metre won, decimal time did not

In 1793, at the height of the French Revolution, the National Convention decided that time too would be decimal. The day would have 10 hours, each hour 100 minutes and each minute 100 seconds. Clockmakers made new dials, some with both scales side by side so that people would get used to them. A paper dial presented by the clockmaker Hanin to the National Convention, with the new hours in Arabic numerals and the old ones in Roman numerals, for comparison: “relation of the new hours to the old, adopted by the National Convention”.Hanin · Wikimedia Commons · public domain Decimal time became compulsory at the end of 1794 and was suspended in April 1795, seventeen months after it was decreed. Nobody wanted to change their household clock, church clocks went on striking twelve, and the new time had no practical advantage to make up for the bother. The Republican calendar, with months of 30 days divided into three “decades” of 10, lasted a little longer: Napoleon abolished it at the start of 1806.

October 26, 2026 Read

Other scripts, other mathematics: rods, dots and knots

The way we write numbers today seems natural because it is the only one we learned. But other cultures solved the same problem, writing quantities and calculating with them, in different ways, and each solution left its mark on what those cultures were able to think. Chinese counting rods For almost two thousand years, Chinese mathematicians calculated with rods: little sticks of bamboo, bone or metal laid out on a board divided into columns. Each column was a decimal place, and a digit from 1 to 5 was made with the same number of rods; from 6 to 9, with one rod worth five plus a few more. So that neighbouring columns would not be confused, the rods alternated position: upright in the units, lying down in the tens, upright again in the hundreds.

October 26, 2026 Read

References and credits

Further reading Seron, Xavier; Fayol, Michel. Number transcoding in children: a functional analysis. British Journal of Developmental Psychology, 12, 1994, pp. 281–300. Miura, Irene T.; Kim, Chungsoon C.; Chang, Chih-Mei; Okamoto, Yukari. Effects of language characteristics on children’s cognitive representation of number: cross-national comparisons. Child Development, 59, 1988, pp. 1445–1450. Skemp, Richard R. Relational understanding and instrumental understanding. Mathematics Teaching, 77, 1976, pp. 20–26. Gray, Eddie M.; Tall, David O. Duality, ambiguity, and flexibility: a “proceptual” view of simple arithmetic. Journal for Research in Mathematics Education, 25(2), 1994, pp. 116–140. Halliday, Michael A. K. Language as Social Semiotic. London: Edward Arnold, 1978. The chapter on the mathematics register. Pimm, David. Speaking Mathematically: Communication in Mathematics Classrooms. London: Routledge & Kegan Paul, 1987. Baruk, Stella. L’âge du capitaine: de l’erreur en mathématiques. Paris: Seuil, 1985. Brousseau, Guy. Théorie des situations didactiques. Grenoble: La Pensée Sauvage, 1998. Pólya, George. How to Solve It. Princeton: Princeton University Press, 1945. Matz, Marilyn. Towards a process model for high school algebra errors. In: Sleeman, D.; Brown, J. S. (eds.). Intelligent Tutoring Systems. London: Academic Press, 1982, pp. 25–50. Gray, Jeremy J. A commentary on Gauss’s mathematical diary, 1796–1814, with an English translation. Expositiones Mathematicae, 2, 1984, pp. 97–130. Berndt, Bruce C. Ramanujan’s Notebooks, parts I to V. New York: Springer, 1985–1998. Kanigel, Robert. The Man Who Knew Infinity: A Life of the Genius Ramanujan. New York: Scribner, 1991. Halmos, Paul R. How to write mathematics. L’Enseignement Mathématique, 16, 1970, pp. 123–152. Knuth, Donald E.; Larrabee, Tracy; Roberts, Paul M. Mathematical Writing. Washington: Mathematical Association of America, 1989. Nemeth, Abraham. The Nemeth Braille Code for Mathematics and Science Notation. Louisville: American Printing House for the Blind, 1972. Image sources Image Source Licence The Ishango bone Wikimedia Commons CC BY-SA 4.0, Joeykentin Mental Arithmetic, by Nikolai Bogdanov-Belsky (1895) Wikimedia Commons public domain First entry in Gauss’s diary (1796) Wikimedia Commons public domain The “ΕΥΡΗΚΑ” in Gauss’s diary (1796) Wikimedia Commons public domain A page from Ramanujan’s notebooks Wikimedia Commons public domain The cubarithm in Recherches et inventions (1926) Wikimedia Commons public domain The interactive figures (numbers written as spoken, the false rules and Gauss’s eureka) work in all four languages of the site. The record of the downloaded images, with the hash of each file, is in data/semiotica_imagens.json.

October 26, 2026 Read

How to learn to read and write mathematics: a guide and a glossary

This chapter followed the learner: the child transcribing numbers, the pupil reading a word problem, the student writing a solution. The guide below gathers the habits that help at all these moments. Six habits Read with a pencil in hand. A mathematics text is dense: one line may be worth a paragraph. Paul Halmos advised: “Don’t just read it; fight it!” Redo the calculations, invent examples, ask why each step holds. Read aloud. Saying $\sum_{k=1}^{n} k^2$ as “the sum of $k$ squared, for $k$ from one to $n$” forces you to recognise each part. Where speech stalls, reading stalled too. Look at the shape before calculating. What is the last operation? Are there repetitions, symmetries, a number that appears twice? Bogdanov-Belsky’s blackboard is solved by looking. Translate both ways. From text to formula, giving a name and a unit to each unknown; and from formula back to text, to check that it says what the problem says. Test with numbers. Before trusting a rule, substitute small values. A counterexample refutes a false rule; many examples do not prove a true one, but they give the confidence to look for a proof. Write for someone. Complete sentences, words between formulas, the goal announced before the calculations. If a classmate cannot follow the solution, it is not finished yet. A complete reading of a badly written solution A student hands in the following solution to “solve $2x + 3 = 11$”:

October 25, 2026 Read