Equality

References and credits

Further reading Euclid. The Elements. Portuguese translation by Irineu Bicudo. São Paulo: Editora Unesp, 2009. The common notions and the structure of the propositions. Proclus. A Commentary on the First Book of Euclid’s Elements. Translated by Glenn R. Morrow. Princeton: Princeton University Press, 1970. The six parts of a proposition. Frege, Gottlob. Über Sinn und Bedeutung. Zeitschrift für Philosophie und philosophische Kritik, 100, 1892, pp. 25–50. Brazilian translation in: Lógica e filosofia da linguagem. São Paulo: Cultrix/Edusp, 1978. Frege, Gottlob. Die Grundlagen der Arithmetik. Breslau: Koebner, 1884. The quotation from Leibniz is in §65. Wittgenstein, Ludwig. Tractatus Logico-Philosophicus. London: Kegan Paul, 1922. On identity, propositions 5.53 to 5.5303. Gauss, Carl Friedrich. Disquisitiones Arithmeticae. Leipzig: Gerhard Fleischer, 1801. The definition of congruence and the sign ≡ are in articles 1 and 2. Pascal, Blaise. Traité du triangle arithmétique. Paris: Guillaume Desprez, 1665 (written in 1654). Induction appears in consequence 12. Dijkstra, Edsger W.; Scholten, Carel S. Predicate Calculus and Program Semantics. New York: Springer, 1990. The proof format with hints. de Bruijn, Nicolaas G. Asymptotic Methods in Analysis. Amsterdam: North-Holland, 1958. The one-way equalities of big $O$. Halmos, Paul R. I Want to Be a Mathematician: An Automathography. New York: Springer, 1985. The origin of $\blacksquare$. Kieran, Carolyn. Concepts associated with the equality symbol. Educational Studies in Mathematics, 12, 1981, pp. 317–326. Healy, Lulu; Hoyles, Celia. A study of proof conceptions in algebra. Journal for Research in Mathematics Education, 31(4), 2000, pp. 396–428. Fischbein, Efraim. Intuition and proof. For the Learning of Mathematics, 3(2), 1982, pp. 9–18 and 24. Cajori, Florian. A History of Mathematical Notations. 2 vols. Chicago: Open Court, 1928–1929. On the ≡ of Gauss and of Riemann and on Rahn’s $\therefore$. Earliest Uses of Symbols of Relation, Jeff Miller. Image sources Image Source Licence Portrait of Gottlob Frege, by Emil Tesch (c. 1905) Wikimedia Commons public domain Title page of Gauss’s Disquisitiones Arithmeticae (1801) Wikimedia Commons public domain First page of Pascal’s Triangulus Arithmeticus, Bibliothèque du patrimoine de Clermont Auvergne Métropole Wikimedia Commons CC BY-SA 4.0 The diagram of the sum of odd numbers and the interactive figures (the congruence clock and the steps of a solution) were produced for this book and 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 16, 2026 Read

How to read a proof: a guide and a glossary

This chapter followed the equals sign from a line of calculation to a whole proof. The guides below gather the questions that help read both. Five questions for an equals sign Does it ask, state or name? If it asks, it is an equation, and the task is to find the solutions. If it states something for all values, it is an identity, and the task is to prove it. If it names, it is a definition, and the task is to accept it and remember it. Do the two sides have the same sense or only the same reference? If the sense is the same, the equality is empty. If it is different, the equality teaches something, and it is worth asking what. Is it a true equality or an equivalence? $\frac12=\frac24$ is an equality between numbers, but an equivalence between fractions. $17\equiv2\pmod5$ says that two different numbers fall into the same drawer. Can I replace one side by the other here? Inside a calculation, always. Inside a sentence about what someone knows or wrote, not always. Do the three laws hold? If the relation is not symmetric ($:=$, the $=$ of big $O$) or not transitive ($\approx$), it is not a true equals sign, and chains $A=B=C$ stop working. Five questions for a proof What exactly is being claimed? Find the statement and its letters: which are arbitrary, which are given. Where is the “let”? That is where the general becomes particular. Check that nothing was assumed about the particular case beyond what the statement gives. What kind of arrow links the lines? A chain of $=$ and $\Leftrightarrow$ can be read both ways. A chain of $\Rightarrow$ only works from beginning to end. Where does each hypothesis open and close? Every time “suppose” appears, look for where it ends. Letters introduced inside do not exist outside. What was left out? Every “it is easy to see”, every ellipsis, every “similarly” is a step the reader should be able to fill in. A complete reading of a short proof Claim. For every integer $n$, the number $n^3-n$ is divisible by $6$. Proof. Let $n$ be an integer. Then $n^3-n=(n-1)\,n\,(n+1)$, the product of three consecutive integers. Among three consecutive integers, one is a multiple of $3$ and at least one is even. Hence the product is a multiple of $2$ and of $3$, and therefore of $6$. $\blacksquare$

October 16, 2026 Read

Examples do not prove: Euler, Pascal and induction

Calculate $n^2+n+41$ for $n=0,1,2,3,\ldots$ $$41,\ 43,\ 47,\ 53,\ 61,\ 71,\ 83,\ 97,\ 113,\ 131,\ \ldots$$They are all prime. Keep going up to $n=39$, and the result, $1601$, is still prime. Forty cases in a row, not one exception. In 1772, Leonhard Euler drew attention to this remarkable formula. And yet it does not produce only primes: with $n=40$, $$40^2+40+41=1600+40+41=1681=41^2.$$Forty examples proved nothing. One counterexample brought everything down. The temptation of cases Euler’s polynomial is not an isolated curiosity. In 1640, Pierre de Fermat observed that the numbers $2^{2^n}+1$ gave

October 16, 2026 Read

The words of a proof: let, suppose, hence, ∎

Read this short proof: Claim. The sum of two even numbers is even. Proof. Let $m$ and $n$ be even numbers. Then there are integers $a$ and $b$ such that $m=2a$ and $n=2b$. Hence $m+n=2a+2b=2(a+b)$. Since $a+b$ is an integer, $m+n$ is even. $\blacksquare$ Take away the calculations and a skeleton of words is left: let, then there are, hence, since, and a little black square at the end. None of them is mathematics in the sense of calculating, and all of them are indispensable. They are what tell the reader what each line is doing: introducing an object, drawing a consequence, invoking a fact, closing.

October 16, 2026 Read

The chain of equalities: how an identity is proved

How do you prove that $(a+b)^2=a^2+2ab+b^2$ for all numbers $a$ and $b$? Not by testing values: no amount of examples covers all numbers. You prove it by writing a chain: $$\begin{aligned} (a+b)^2&=(a+b)(a+b)\\ &=a(a+b)+b(a+b)\\ &=a^2+ab+ba+b^2\\ &=a^2+2ab+b^2. \end{aligned}$$Each equals sign links two neighbours, and each link is guaranteed by a known rule: the definition of the square, the distributive law, the distributive law again, the commutativity of multiplication. By transitivity, Euclid’s law that text 1 presented, the first link equals the last. The proof is the chain.

October 16, 2026 Read

⇒ and ⇔: what is gained and what is lost when an equation is transformed

Solve the equation $$\sqrt{x+2}=x.$$The usual route is to get rid of the root by squaring both sides: $$\begin{aligned} \sqrt{x+2}=x\quad&\Longrightarrow\quad x+2=x^2\\ &\Longrightarrow\quad (x-2)(x+1)=0\\ &\Longrightarrow\quad x=2\ \text{or}\ x=-1. \end{aligned}$$The calculations are all correct. The answer is not. With $x=2$: $\sqrt{4}=2$, and the equation holds. With $x=-1$: $\sqrt{1}=1$, which is not $-1$. The original equation has one solution, and the method found two. Where did the intruder come from? One equation per line Someone solving an equation is not calculating with numbers: they are writing a sequence of equations, each obtained from the previous one. What links one line to the next is not an equals sign, it is a statement about the sentences: “if this one is true, that one is too”. Logic has two signs for this.

October 16, 2026 Read

Equivalence: fractions, clocks and Gauss's ≡

The pieces of writing $\frac12$ and $\frac24$ are different: one has the digits $1$ and $2$, the other $2$ and $4$. A pizza cut into two slices, of which you eat one, is not the same thing as a pizza cut into four, of which you eat two: there is one more cut. And yet we write without hesitation $$\frac12=\frac24=\frac36=\frac{50}{100}.$$What this equals sign states is not that the pieces of writing are equal, nor that the pizzas are equal. It states that, for what matters, the amount eaten, they cannot be told apart.

October 16, 2026 Read

Equation, identity, definition: three jobs for the same sign

One letter, $f$, and three sentences: $$f(x)=x^2,\qquad f(x)=x\cdot x,\qquad f(x)=4.$$The first gives a name: from here on, $f$ is the function that squares. The second states a law: for every $x$, the value of $f$ is $x$ times $x$. The third asks a question: for which $x$ is the function equal to $4$? The answer is $x=2$ or $x=-2$. The sign is identical in all three. The reader’s task is not:

October 16, 2026 Read

Frege and the morning star: what a = b says

Compare two sentences: $$1024=1024\qquad\text{and}\qquad 2^{10}=1024.$$Both are true, and both say that a number is equal to itself: $2^{10}$ is $1024$. Even so, nobody learns anything from the first, and the second is a small discovery that costs ten multiplications. If the equals sign only said “these two things are the same”, the two sentences would say exactly the same thing. So what does the second one have that the first lacks?

October 16, 2026 Read

One sign, many equals

Robert Recorde invented $=$ in 1557 with a simple justification: no two things can be more equal than two parallel lines of the same length (chapter 1 tells this story, in the text on the signs of the operations). The sign caught on so well that today it does jobs Recorde never imagined. Look at these lines: $$3+4=7,\qquad 2x+1=7,\qquad (a+b)^2=a^2+2ab+b^2,$$$$f(x)=x^2,\qquad \tfrac12=\tfrac24.$$The sign is the same in all of them. What it says is not.

October 16, 2026 Read