Decimal expansion

Builds on Real numbers Completeness axiom

Definition

Let $x$ be a real number with $x \ge 0$ . A decimal expansion of $x$ is a natural number $a_0$ and a sequence $(d_k)_{k \ge 1}$ of digits $d_k \in \{0, 1, \dots, 9\}$ with $x = a_0 + \sum_{k=1}^{\infty} \frac{d_k}{10^k}.$ The notation is $x = a_0.d_1 d_2 d_3 \dots$ , with $a_0$ in base $10$ .

Remarks

The series always converges, because its partial sums are increasing and bounded above by $a_0 + 1$ . A number can have two expansions. This happens exactly when one expansion ends in infinitely many $0$ and the other in infinitely many $9$ , as for 0.999... and 1. For a negative number $x$ , one writes $-$ followed by an expansion of $-x$ .

Reading path

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Real numbers object
Set bounded above definition
Supremum definition
Completeness axiom statement
Decimal expansion definition

Where this leads

2 pages build on this one.

Tag	`06XY4X9430`
URI	`m2://06XY4X9430`
Kind	definition
Writing	W1 stated
MSC	11A63
Defines	Decimal expansion
Uses	Real numbers, Completeness axiom

Decimal expansion

Definition

Remarks

Reading path

Where this leads

Further reading

What this page needs for W2

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