existence and uniqueness of decimal expansion
The existence and uniqueness of decimal expansions (or more generally, base- expansions) is taken for granted by most grade school students, but they are facts that need to be rigorously proven if one wants to understand the real numbers thoroughly.
We mention the following fact about natural numbers , which we will use many times implicitly:
This fact can be proven by mathematical induction on .
1 Proof of Existence
Let be a number for which we want to write a base- expansion for any natural number greater than one.
1.1 Expansions for non-negative integers
First we prove the existence of expansions of the form
The number obviously has the expansion .
Suppose that we know the existence of expansions for a number . We prove the existence of an expansion for .
Let be expanded as
From the above equation, add to both sides:
If , then we are done. Otherwise, , and therefore we may write instead
If , then we can stop. Otherwise, repeat the process and continue carrying digits until we reach some for which . Since , this process is guaranteed to stop. At the end we will have expressed in base .
1.2 Reduction to numbers in
Let be the greatest integer less than or equal to , otherwise known as the floor of . We prove that the floor of exists.
is bounded above by . However, by the Archimedean property, the set of natural numbers is not bounded above, so must be non-empty, and have a smallest element (formally, by the well-ordering principle). For every , we have . The latter condition is equivalent to , so is the maximum element of . In other words, .
Since , we have . We shall obtain the base expansion of as the sum of the expansion of and .
1.3 Expansion of numbers in
Given , let . Then , so we can take as the first digit of the base- expansion of . Next, write
and observe that , so it is possible to get the next digit of the expansion by expanding . We do this recursively, leading to these recursive relations:
More explicitly, we have
It is easy to prove that the expansion
converges to :
(Formally, the “” part appeals to the Archimedean property.)
2 Proof of uniqueness
2.1 Uniqueness for non-negative integers
Then we can consider
Repeating the previous argument with replaced by , we see that is uniquely determined. Then we can consider and so on. Continuing this way, we see that all the digits are uniquely determined.
2.2 Near-uniqueness for non-negative numbers
then are uniquely determined, since is the expansion for the non-negative integer .
The argument to prove that are uniquely determined proceeds similarly as before. We have
where equality on the second line occurs if and only if for every . If we insist that is never eventually the same digit , then this shows that the digit is uniquely determined by where the original number in the disjoint intervals .
This argument may be repeated, to show that are uniquely determined, under the assumption that the expansion does not end in all digits being .
If the assumption is not made, then numbers which have an expansion ending in all digits have an alternate expansion ending in all digits , but other numbers still have unique base- expansions.
3 Every base- expansion represents a real number
We also want to prove that for every sequence of digits there exists a real number with the base- expansion
Consider the sequence with the
This sequence, considered as a set, is bounded above, for . So it has a least upper bound . Since the sequence is also increasing, its least upper bound is the same as its limit.
|Title||existence and uniqueness of decimal expansion|
|Date of creation||2013-03-22 15:42:12|
|Last modified on||2013-03-22 15:42:12|
|Last modified by||stevecheng (10074)|
|Synonym||every decimal expansion represents a real number|