Catégorie

Arithmétique

L'arithmétique est la branche des mathématiques qui étudie les nombres entiers naturels , relatifs et rationnels , voire réels , ainsi que leurs relations et propriétés, en lien avec quelques opérations élémentaires : addition (+), soustraction (−), multiplication (×), division (÷, /, ou :), puissance et racine (). Le terme inclut parfois d'autres concepts de la théorie des nombres. Le mot arithmétique vient du grec ancien , « nombre ». L’origine de l'arithmétique semble être une invention phénicienne. Dans l'école pythagoricienne, à la deuxième moitié du , l'arithmétique était, avec la géométrie, l'astronomie et la musique, une des quatre sciences quantitatives ou mathématiques (Mathemata). Celles-ci furent regroupées au sein des sept arts libéraux par Martianus Capella () et plus précisément désignées sous le nom de quadrivium par Boèce. Les trois autres disciplines étaient littéraires (grammaire, rhétorique, dialectique) et firent l'objet des travaux de Cassiodore et, plus tard, Alcuin qui leur donna le nom de trivium. L'arithmétique désigne l'étude des nombres, de leurs relations et de leurs propriétés individuelles et collectives. Dans sa définition commune, elle se limite aux entiers naturels ou relatifs, et aux nombres rationnels, voire aux nombres réels, et à quelques opérations associées (addition, soustraction, multiplication, division, puissance et racine). Dans une acception plus rare, elle est utilisée dans un sens plus large par des mathématiciens, dont Carl Friedrich Gauss, pour inclure des concepts plus pointus de la théorie des nombres. Au-delà de l'arithmétique élémentaire, le mot peut alors faire référence à l'arithmétique modulaire, à la théorie algébrique des nombres ou encore à l'arithmétique des polynômes. Ensemble Les nombres obtenus par dénombrement (comptage) d'objets d'ensembles divers donnent la liste des entiers naturels non nuls . Cet ensemble est représenté par le symbole , ou si on lui ajoute le nombre « ».

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Exercices associés (32)

Le liévre et la tortue

A hare and a turtle race over a distance L. The turtle starts running at constant speed

v_{\rm t}

and the hare at constant speed

v_{\rm h}

. The hare is not taking this seriously, and so

v_{\rm h} < v_{\rm t}

. When the turtle reaches a bridge that stands at a distance L′u2004

Unités et analyse dimensionnelle

Dimensional analysis is a physical concept that helps understanding physical situations in which various physical quantities are used. It is also used to hypothetically link different physical quantities that can then be verified experimentally. For every physical quantity, there is a corresponding dimension, associated in mechanics to length (L), mass (M), time (T) and their combinations. For instance, the physical dimension of “velocity” v is length/time (L/T). The unit and dimension of a physical quant

L échelle

A ladder with mass m and length L is resting on a vertical wall. The angle between the wall and the ladder is α and the friction coefficient between the ladder’s foot and the floor isxa0μ. There is no friction between the wall and the ladder. What’s the maximal value

\alpha_{\rm max}

angle α can take such that the ladder’s bottom doesn’t slip on the floor and fall? We consider the ladder as the system, seen from the referential of the floor and the wall. We choose a Cartesian reference frame with a horiz

Le corbeau et le renard

A few days after the events narrated in the famous fable[1], the crow and the fox meet again. Having learned from its mistakes, the crow is determined not to be fooled again by the fox’s flattery. The fox, however, is set on getting a second piece of cheese. This time , it chooses a more direct method and throws a stone at the crow to scare it. It works beyond expectations: as soon as the stone is thrown, the crow drops the cheese. The crow is perched on a tree of height H; the fox is a distance L from the

Cylindres sur une feuille de papier

Let consider two cylinders of mass M, of radius R and of moment of inertia I_(G)u2004=u2004λu2006Mu2006R². One is solid (i.e. λu2004=u20041/2) and the other one is hollow (i.e. λu2004=u20041). They are laying on a sheet of paper initially at rest. The sheet is pulled with a known speed V_(P). The two cylinders roll without slipping on the paper.

Express the angular speed ψ̇ of the cylinders in terms of the speed V_(P) of the paper and of the speed V_(G) of the centre of mass of the cylinder.

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Notebooks associés (31)

Digital Simulation of Guitar Feedback in I Feel Fine by The Beatles

I Feel Fine, Digitally[1]¶ This notebook is a little musing inspired by the "internal mechanics" of a very famous audio snippet, namely the guitar sound at the beginning of the song "I Feel Fine", written and recorded by the Beatles in 1964. The historical significance of the record lies in the following claim by John Lennon:

 "I defy anybody to find a record... unless it is some old blues
 record from 1922... that uses feedback that way. So I claim it for
 the Beatles. Before H

Understanding the Haar Basis for Grayscale Image Compression

A Basis for Grayscale Images[1]¶

                                      [2]Paolo Prandoni, [3]LCAV, EPFL

The concept of "basis" for a vector space is one of the most profound and fruitful ideas in linear algebra. Basis vectors are the "building blocks" that all other vectors are built from; by "disassembling" a vector into a linear combination of basis elements we are able to "look inside" the vector and highlight many of its relevant features. A vector space has an infinite number o

Voice Transformation Techniques Using Digital Signal Processing

A Gallery of Voice Transformers[1]¶

                                      [2]Paolo Prandoni, [3]LCAV, EPFL

In this notebook we will explore various ways to implement a system that changes the voice of a speaker. Typical applications for these techniques, strictly sorted by plausibility, include international spy missions and prank telephone calls, and devices such as the one displayed here can be easily bought on the web... or in a novelty shop. No description has been provided for t

Numerical Precision in DFT Computation

Numerical Precision in DFT Computation[1]¶

                                      [2]Paolo Prandoni, [3]LCAV, EPFL

In [1]:

first our usual bookkeeping

%matplotlib inline import matplotlib import matplotlib.pyplot as plt import numpy as np In this notebook we will look at some numerical issues associated to the DFT; in particular we we will look at the differences in precision between the "naive" way of computing the DFT and the FFT algorithm. As a quick reminder, the definitions of t

Implementing a Fractional Resampler using Local Lagrange Interpolation

FIR Resampling[1]¶

                                      [2]Paolo Prandoni, [3]LCAV, EPFL

In this notebook we will implement a simple fractional resampler that uses local Lagrange interpolation. The resampler can be used to perform any rational sampling rate change (but beware of aliasing when downsampling!) In [1]: import numpy as np import IPython from scipy.io import wavfile Local Lagrange interpolation[4]¶ To perform resampling we will need to compute subsample values and, in

Exercices associés (32)

Le liévre et la tortue

A hare and a turtle race over a distance L. The turtle starts running at constant speed

v_{\rm t}

and the hare at constant speed

v_{\rm h}

. The hare is not taking this seriously, and so

v_{\rm h} < v_{\rm t}

. When the turtle reaches a bridge that stands at a distance L′u2004

Unités et analyse dimensionnelle

L échelle

\alpha_{\rm max}

Le corbeau et le renard

Cylindres sur une feuille de papier

Express the angular speed ψ̇ of the cylinders in terms of the speed V_(P) of the paper and of the speed V_(G) of the centre of mass of the cylinder.

Afficher plus

Arithmétique

Chattez avec Graph Search

first our usual bookkeeping

first our usual bookkeeping

Vertical profiles of broadband albedo in snow-covered complex terrain

GHT-SELEX demonstrates unexpectedly high intrinsic sequence specificity and complex DNA binding of many human transcription factors

PALUTE: Processing-In-Memory Acceleration via Lookup Table for Edge LLM Inference