# What is decimal expansion: Definition and 17 Discussions

A decimal representation of a non-negative real number r is its expression as a sequence of symbols consisting of decimal digits traditionally written with a single separator:

Here . is the decimal separator, k is a nonnegative integer, and

b

0

,

,

b

k

,

a

1

,

a

2

,

{\displaystyle b_{0},\ldots ,b_{k},a_{1},a_{2},\ldots }
are digits, which are symbols representing integers in the range 0, ..., 9.
Commonly,

b

k

0

{\displaystyle b_{k}\neq 0}
if

k
>
1.

{\displaystyle k>1.}
The sequence of the

a

i

{\displaystyle a_{i}}
—the digits after the dot—is generally infinite. If it is finite, the lacking digits are assumed to be 0. If all

a

i

{\displaystyle a_{i}}
are 0, the separator is also omitted, resulting in a finite sequence of digits, which represents a natural number.
The decimal representation represents the infinite sum:

Every nonnegative real number has at least one such representation; it has two such representations (with

b

k

0

{\displaystyle b_{k}\neq 0}
if

k
>
0

{\displaystyle k>0}
) if and only if one has a trailing infinite sequence of 0, and the other has a trailing infinite sequence of 9. For having a one-to-one correspondence between nonnegative real numbers and decimal representations, decimal representations with a trailing infinite sequence of 9 are sometimes excluded.

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2. ### I Non unicity of decimal expansion and extremes of intervals

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4. ### Decimal Expansion Homework: Terminating & Non-Terminating 9's

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5. ### Prove that every real number x in [0,1] has a decimal expansion.

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6. ### How to Compute Base 3 Floating Point Expansion of 1/4?

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7. ### MHB Decimal expansion to rational number

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8. ### Multiple Decimal Expansions: Explained

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9. ### What is meaning of pai decimal expansion

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11. ### Periodicity of Decimal Expansion

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12. ### Is Every Real Number's Decimal Expansion Key to Mapping S into (0,1)?

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14. ### Number sequence is present in the decimal expansion of pi?

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15. ### Why Does the Supremum Definition Make Sense for Infinite Decimal Expansions?

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16. ### Rational numbers - periodic decimal expansion

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17. ### Number if rational iff it has periodic decimal expansion

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