Decimal representation

A decimal representation of a non-negative real number

Real number

In mathematics, the real numbers include both rational numbers, such as 42 and −23/129, and irrational numbers, such as pi and the square root of two; or, a real number can be given by an infinite decimal representation, such as 2.4871773339..., where the digits continue in some way; or, the real...

 r is an expression of the form
where a₀ is a nonnegative integer, and a₁,
a₂, … are integers satisfying 0 ≤ a_i ≤ 9 this is often written more briefly as
That is to say, a₀ is the integer part of r, not necessarily between 0 and 9, and a₁, a₂, a₃, … are the digits forming the fractional part of r.

Both notations above are, by definition, the following limit of a sequence

Limit

A limit can be:* In mathematics:** Limit of a function** Limit of a sequence** One-sided limit** Limit superior and limit inferior** Limit of a net** Limit point** Limit ** Direct limit...

:.

Any real number can be approximated to any desired degree of accuracy by rational number

Rational number

In mathematics, a rational number is any number that can be expressed as the quotient a/b of two integers, with the denominator b not equal to zero. Since b may be equal to 1, every integer corresponds to a rational number. The set of all rational numbers is usually denoted .Formally each rational...

s with finite decimal representations.

Assume .

A decimal representation of a non-negative real number

Real number

In mathematics, the real numbers include both rational numbers, such as 42 and −23/129, and irrational numbers, such as pi and the square root of two; or, a real number can be given by an infinite decimal representation, such as 2.4871773339..., where the digits continue in some way; or, the real...

 r is an expression of the form
where a₀ is a nonnegative integer, and a₁,
a₂, … are integers satisfying 0 ≤ a_i ≤ 9 this is often written more briefly as
That is to say, a₀ is the integer part of r, not necessarily between 0 and 9, and a₁, a₂, a₃, … are the digits forming the fractional part of r.

Both notations above are, by definition, the following limit of a sequence

Limit

A limit can be:* In mathematics:** Limit of a function** Limit of a sequence** One-sided limit** Limit superior and limit inferior** Limit of a net** Limit point** Limit ** Direct limit...

:.

Finite decimal approximations

Any real number can be approximated to any desired degree of accuracy by rational number

Rational number

In mathematics, a rational number is any number that can be expressed as the quotient a/b of two integers, with the denominator b not equal to zero. Since b may be equal to 1, every integer corresponds to a rational number. The set of all rational numbers is usually denoted .Formally each rational...

s with finite decimal representations.

Assume . Then for every integer there is a finite decimal such that
Proof:

Let , where .
Then , and the result follows from dividing all sides by .
(The fact that has a finite decimal representation is easily established.)

Non-uniqueness of decimal representation

Some real numbers have two infinite decimal representations. For example, the number 1 may be equally represented by 1.000... as by 0.999...

0.999...

In mathematics, the repeating decimal 0.999… which may also be written as $0.\backslash bar\{9\}\; ,\; 0.\backslash dot\{9\}$ or $0.\backslash ,\backslash !$ denotes a real number equal to one. In other words, the notations 0.999… and 1 represent...

 (where the infinite sequences of digits 0 and 9, respectively, are represented by "..."). Conventionally, the version with zero digits is preferred; by omitting the infinite sequence of zero digits, removing any final zero digits and a possible final decimal point, a normalized finite decimal representation is obtained.

Finite decimal representations

The decimal expansion of non-negative real number 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 non-negative 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 ,
for some n.
By ,
x will end in zeros.

Recurring decimal representations

Some real numbers have decimal expansions that eventually get into loops, endlessly repeating a sequence of one or more digits:

¹/₃ = 0.33333...

¹/₇ = 0.142857142857...

¹³¹⁸/₁₈₅ = 7.1243243243...

Every time this happens the number is still a rational number

Rational number

In mathematics, a rational number is any number that can be expressed as the quotient a/b of two integers, with the denominator b not equal to zero. Since b may be equal to 1, every integer corresponds to a rational number. The set of all rational numbers is usually denoted .Formally each rational...

 (i.e. can alternatively be represented as a ratio of a non-negative and a positive integer).

External links

• Plouffe's inverter tries to identify a number given the start of its decimal representation. For instance, given 3.14159265 it will say that your input probably came from one of the following and list π
  Pi
  Pi or π is a mathematical constant whose value is the ratio of any circle's circumference to its diameter in Euclidean space; this is the same value as the ratio of a circle's area to the square of its radius. The symbol π was first proposed by the Welsh mathematician William Jones in 1706...
  
  as the simplest.