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A decimal representation of a positive real number r is an expression of the form Wikipedia does not have an article with this exact name. ... Decimal, or less commonly, denary, usually refers to the base 10 numeral system. ... In mathematics, the real numbers are intuitively defined as numbers that are in one-to-one correspondence with the points on an infinite line—the number line. ...

where a₀ is a nonnegative integer and are integers satisfying , usually written more briefly as follows:

Finite decimal approximations

Any real number can be approximated to any desired degree of accuracy by rational numbers with finite decimal representations. In mathematics, a rational number (or informally fraction) is a ratio or quotient of two integers, usually written as the vulgar fraction a/b, where b is not zero. ...

Assume . Then for every integer there is a finte decimal such that

Proof:

Let S be the set of all nonnegative integers . Then S is nonempty, since , and S is bounded above by x. Therefore S has a supremum, say . It is easily verified that , so a₀ is a nonnegative integer. We call a₀ the greatest integer in x, and we write . Clearly, we have.

Now let , the greatest integer in 10x − 10a₀. Since , we have and . In other words, a₁ is the largest integer satisfying the inequalities

.

More generally, having chosen with , let a_n be the largest integer satisfying the inequalities

Then and we have

where .

It is easy to verify that x is actually the supremum of the set of rational numbers .

Verification that a₀ ∈ S

By the approximation property of the supremum of a set of real numbers, for every z>0, there exists x in S such that a₀ − z < x. Therefore, and then for . In mathematics, the supremum of an ordered set S is the least element that is greater than or equal to each element of S. Consequently, it is also referred to as the least upper bound (also lub and LUB). ...

Verify that x is the supremum of the set of rational numbers r₁, r₂, ...

For every natual number i, , or x is the upper bound of the set of rational numbers r₁,r₂,.... Suppose that there is a real number y such that for every natural number i and y < x. Thus, 0 < x − y < 1 / 10ⁿ and then . This is a contradiction. Therefore, x is the least upper bound, or the supremum. In mathematics, the supremum of an ordered set S is the least element (not necessarily in S) which is greater than or equal to each element of S. Consequently, it is also referred to as the least upper bound. ...

Finite decimal representions

The decimal expansion of x will end in zeros(or in nines) if, and only if, x is a rational number whose denominator is of the form 2ⁿ5^m, where m and n are nonnegative integers.

Proof:

If the decimal expansion of x will end in zeros, or for some n, then the denominator of x is of the form 10ⁿ = 2ⁿ5ⁿ.

Conversely, if the denominator of x is of the form 2ⁿ5^m, for some p. While x is of the form p/10^k, for some n. By , x will end in zeros.