Decimal number

A decimal number is a Nombre having a limited decimal Développement and which can be written in the form $a\; \backslash \; times\; 10^p$ (where has and p is relative entireties). It is not the case, for example, of the number $\backslash \; pi\; \backslash ,$ which has as many decimal approximations however one wants: 3,14; 3,14159; 3,14159235; etc

Characterization

The following properties are equivalent and characterize the fact that a rational Nombre has is decimal:

• has admits a limited decimal development (i.e. with a finished number of figures different from 0).
• There exists $m\; \backslash \; in\; \backslash \; mathbb\; Z$ and $p\; \backslash \; in\; \backslash \; mathbb\; N$ such as: $a\; =\; \backslash \; frac\; \{m\}\; \{10^p\}$.
• the irreducible Fraction of has is form $\backslash \; frac\; \{B\}\; \{5^m\; \backslash \; times\; 2^p\}$, where B is a relative entirety and m and p of the natural entireties.
• has has two distinct decimal developments.

Remarks

• It first assertion proves that 1,6666 is a decimal number and that 1,6666… (which would be written with an infinity of 6) is not one.
• the second assertion says to us that $\backslash \; frac\; \{765\}\; \{10^8\}$ is a decimal number, but it cannot be used to prove that a number is not decimal.
• the third assertion gives us a method to recognize if a rational number is decimal: it is enough to determine its irreducible fraction (for example by calculating PGCD of its numerator and its denominator), then to test if the denominator is only divisible by 2 and 5.
• the fourth assertion often has the appearance of “a curiosity”. Number 2,5 can also be written 2,4999… (with an infinity of 9). For details, to see the article decimal Development of the unit.

Algebraic structure

The Ensemble of decimal is often noted $\backslash \; mathbb\; D$. $(\backslash \; mathbb\; D,\; +,\; \backslash \; times)$ is a just Anneau commutative. Its Body of the fractions being $\backslash \; mathbb\; Q$.

Topology

$\backslash \; mathbb\; D$ is dense in $\backslash \; mathbb\; R$. In other words, all Real number is the limit of a succession of decimal numbers. This theorem is frequently used at the time of the search for approximate values.

See too

• decimal Development

• rational Number
• Decimal system

Random links:

Denis (first name) | Party Quebec Vision | Armand Lavergne | Kai Bracht | 1967 in Canada

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