**Balanced ternary** is a ternary numeral system (i.e. base 3 with three digits) that uses a balanced signed-digit representation of the integers in which the digits have the values −1, 0, and 1. This stands in contrast to the standard (unbalanced) ternary system, in which digits have values 0, 1 and 2.
The balanced ternary system can represent all integers without using a separate minus sign; the value of the leading non-zero digit of a number has the sign of the number itself. While binary numerals with digits 0 and 1 provide the simplest positional numeral system for natural numbers (or for positive integers if using 1 and 2 as the digits), balanced ternary provides the simplest self-contained positional numeral system for integers. The balanced ternary system is an example of a non-standard positional numeral system. It was used in some early computers and also in some solutions of balance puzzles.

Different sources use different glyphs used to represent the three digits in balanced ternary. In this article, T (which resembles a ligature of the minus sign and 1) represents −1, while 0 and 1 represent themselves. Other conventions include using '−' and '+' to represent −1 and 1 respectively, or using Greek letter theta (Θ), which resembles a minus sign in a circle, to represent −1. In publications about the Setun computer, −1 is represented as overturned 1: "1".

Balanced ternary makes an early appearance in Michael Stifel's book *Arithmetica Integra* (1544). It also occurs in the works of Johannes Kepler and Léon Lalanne. Related signed-digit schemes in other bases have been discussed by John Colson, John Leslie, Augustin-Louis Cauchy, and possibly even the ancient Indian Vedas.

A mathematician thinks of the cards in a pack, not in terms of their face value and suit, but in terms of their position in the original pack. So if you have a pack of 52 cards, label each card 0 to 51 by its position (the top card is labelled 0, rather than 1, as this makes the maths simpler). “I might want to know where each of these cards might lie after different combinations of the shuffles,” says Praeger. “So each outcome is like a reordering, a

permutation, of the cards in the pack. I want to know how many different permutations there are. And perhaps what sort of structure this whole set of reorderings, thispermutation group, has.”The first things we can do is count all the possible permutations you can create by shuffling the cards. If we start with a pack of

kcards, then we know there arekdifferent positions in the shuffled pack to place the top card. Then there’s onlyk-1slots left for the second card,k-2slots for the third card, and so on. The total number of possible reorderings isk x (k-1) x (k-2) x … x 2 x 1, otherwise known as thefactorialofk, writtenk!. The collection of all possible permutations of a pack ofkcards is called the symmetric group – and the size of the symmetric group for a pack ofkcards isk!. For example, for a pack consisting of 6 cards, there are 6! = 6x5x4x3x2x1 = 720 possible permutations in the symmetric group….

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The Curry-Howard correspondence can be pithily described as saying “proofs = programs”. The statement is short, alliterative, and rather beautiful. As a professional mathematician I know exactly what a proof is; as an amateur programmer I have a pretty clear idea about what a program is, and the concept that they might be in some sense the same thing is both enigmatic and appealing. A proof is a logical sequence of statements deducing a theorem from the the rules of maths; a program is a logical sequence of commands producing valid code from the rules of the programming language being used. When learning Lean I saw the correspondence in action many times, and so it actually took quite some time for the penny to drop: the Curry-Howard correspondence is not actually true.

Of course

somethingis true, but what is not true is any statement of this form where “proof” is interpreted as “what the pure mathematicians in my department do”. Whatistrue is that if you stick to constructive maths or intuitionistic logic or one of the flavours of maths where you’re not allowed to use the axiom of choice or the law of the excluded middle — in other words a version of maths which is unrecognisable to over 95% of pure mathematicians working in mathematics departments — then you might be in good shape. But that is not what I see happening in my mathematics department or the other mathematics departments which I’ve had experience of. Some proofs are not programs.

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The Mathemagix project aims at the development of a “computer analysis” system, in which numerical computations can be done in a mathematically sound manner. A major challenge for such systems is to conceive algorithms which are both efficient, reliable and available at any working precision. In this paper, we survey several older and newer such algorithms. We mainly concentrate on the automatic and efficient computation of high quality error bounds, based on a variant of interval arithmetic which we like to call “ball arithmetic”.

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In the foundations of mathematics, **Russell's paradox** (also known as **Russell's antinomy**), discovered by Bertrand Russell in 1901, showed that some attempted formalizations of the naive set theory created by Georg Cantor led to a contradiction. The same paradox had been discovered in 1899 by Ernst Zermelo but he did not publish the idea, which remained known only to David Hilbert, Edmund Husserl, and other members of the University of Göttingen. At the end of the 1890s Cantor himself had already realized that his definition would lead to a contradiction, which he told Hilbert and Richard Dedekind by letter.

According to naive set theory, any definable collection is a set. Let *R* be the set of all sets that are not members of themselves. If *R* is not a member of itself, then its definition dictates that it must contain itself, and if it contains itself, then it contradicts its own definition as the set of all sets that are not members of themselves. This contradiction is Russell's paradox. Symbolically:

- ${\text{Let }}R=\{x\mid x\not \in x\}{\text{, then }}R\in R\iff R\not \in R$

In 1908, two ways of avoiding the paradox were proposed: Russell's type theory and the Zermelo set theory. Zermelo's axioms went well beyond Gottlob Frege's axioms of extensionality and unlimited set abstraction; as the first constructed axiomatic set theory, it evolved into the now-standard Zermelo–Fraenkel set theory (ZFC). The essential difference between Russell's and Zermelo's solution to the paradox is that Zermelo altered the axioms of set theory while preserving the logical language in which they are expressed, while Russell altered the logical language itself. The language of ZFC, with the help of Thoralf Skolem, turned out to be first-order logic.