existence and uniqueness of decimal expansion

existence and uniqueness of decimal expansion

The

existence and uniqueness of decimal expansions

(or more generally, base-

b

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

n

,

m

∈

ℕ

0

,
which we will use many times implicitly:

n

<

m

⇔

n

≤

m

-

1

This fact can be proven by mathematical induction on

m

.

Contents:

1

Proof of Existence

1.1

Expansions for non-negative integers

1.2

Reduction to numbers in

[

0

,

1

)

1.3

Expansion of numbers in

[

0

,

1

)

2

Proof of uniqueness

2.1

Uniqueness for non-negative integers

2.2

Near-uniqueness for non-negative numbers

3

Every base-

b

expansion represents a real number

1

Proof of Existence

Let

x

be a number for which we want to write a base-

b

expansion
for any natural number

b

greater

than one.

We shall assume

x

≥

0

, since the base-

b

expansion of a

negative number

is by

definition

the

negative

of the expansion for its

absolute value

.

1.1

Expansions for non-negative integers

First we prove the

existence

of expansions of the form

x

=

∑

i

=

0

k

a

i

⁢

b

i

,

0

≤

a

i

<

b

,

a

i

∈

ℕ

0

for non-negative

integers

x

,
using mathematical induction. (This proof is essentially the formal
statement of how to do

addition

by base-

b

digits.)

The number

x

=

0

obviously has the expansion

0

.

Suppose that we know the existence of expansions for a number

x

-

1

.
We prove the existence of an expansion for

x

.

Let

x

-

1

be

expanded

as

x

-

1

=

∑

i

=

0

k

a

i

⁢

b

i

,

0

≤

a

i

<

b

,

a

i

∈

ℕ

0

,

a

k

+

1

=

0

.

From the above equation, add

1

to both sides:

x

=

(

a

0

+

1

)

+

∑

i

=

1

k

a

i

⁢

b

i

.

If

a

0

<

b

-

1

, then we are done. Otherwise,

(

a

0

+

1

)

=

b

, and therefore we may write instead

x

=

0

+

(

a

1

+

1

)

⁢

b

+

∑

i

=

2

k

a

i

⁢

b

i

.

If

a

1

<

b

-

1

, then we can stop. Otherwise,
repeat the process and continue carrying digits
until we reach some

i

for which

a

i

<

b

-

1

.
Since

a

k

+

1

=

0

, this process is guaranteed to stop.
At the end we will have expressed

x

in base

b

.

1.2

Reduction to numbers in

[

0

,

1

)

Let

⌊

x

⌋

be the

greatest integer less than or equal to

x

,
otherwise known as the floor of

x

.
We prove that the floor of

x

exists.

The set

A

=

{

n

∈

ℕ

0

:

n

≤

x

}

is bounded above by

x

.
However, by the

Archimedean
property

, the set of natural numbers is

not

bounded
above, so

ℕ

0

∖

A

must be non-empty,
and have a smallest

element

u

(formally, by the

well-ordering principle

).
For every

n

∈

A

, we have

n

≤

x

<

u

.
The latter condition is

equivalent

to

n

≤

u

-

1

<

x

,
so

u

-

1

is the

maximum

element of

A

.
In other words,

⌊

x

⌋

=

u

-

1

.

Since

u

-

1

≤

x

<

u

, we have

0

≤

x

-

⌊

x

⌋

<

1

.
We shall obtain the base

b

expansion of

x

as the sum of the expansion of

⌊

x

⌋

and

x

-

⌊

x

⌋

.

1.3

Expansion of numbers in

[

0

,

1

)

Given

x

∈

[

0

,

1

)

,
let

a

1

=

⌊

b

⁢

x

⌋

.
Then

0

≤

a

1

≤

b

⁢

x

<

b

, so we can take

a

1

as the first
digit of the base-

b

expansion of

x

.
Next, write

x

=

a

1

⁢

b

-

1

+

y

⁢

b

-

1

,

and observe that

0

≤

x

-

a

1

⁢

b

-

1

=

b

-

1

⁢

(

b

⁢

x

-

⌊

b

⁢

x

⌋

)

<

b

-

1

,
so it is possible to get the next digit of the expansion by
expanding

y

.
We do this recursively, leading to these recursive relations:

y

i

=

a

i

⁢

b

-

1

+

y

i

+

1

⁢

b

-

1

,

0

≤

a

i

=

⌊

b

⁢

y

i

⌋

<

b

,

0

≤

y

i

<

1

,

y

1

=

x

.

More explicitly, we have

x

-

(

a

1

⁢

b

-

1

+

a

2

⁢

b

-

2

+

⋯

+

a

k

⁢

b

-

k

)

=

b

-

1

⁢

(

y

2

-

(

a

2

⁢

b

-

1

+

⋯

+

a

k

⁢

b

-

k

+

1

)

)

=

b

-

2

⁢

(

y

3

-

(

a

3

⁢

b

-

1

+

⋯

+

a

k

⁢

b

-

k

+

2

)

)

=

⋯

=

b

-

k

+

1

⁢

(

y

k

-

a

k

⁢

b

-

1

)

=

b

-

k

⁢

y

k

+

1

.

It is easy to prove that the expansion

a

1

⁢

b

-

1

+

a

2

⁢

b

-

2

+

⋯

+

a

k

⁢

b

-

k

+

⋯

converges

to

x

:

0

≤

x

-

∑

i

=

1

k

a

i

⁢

b

-

i

=

b

-

k

⁢

y

k

+

1

<

1

b

k

→

0

,

as

k

→

∞

.

(Formally, the “

→

0

” part appeals to the Archimedean property.)

2

Proof of uniqueness

2.1

Uniqueness for non-negative integers

Suppose that

x

=

∑

i

=

0

k

a

i

⁢

b

i

,

0

≤

a

i

<

b

.

Now

a

k

⁢

b

k

≤

∑

i

=

0

k

a

i

⁢

b

i

≤

a

k

⁢

b

k

+

∑

i

=

0

k

-

1

(

b

-

1

)

⁢

b

i

=

a

k

⁢

b

k

+

(

b

k

-

1

)

<

(

a

k

+

1

)

⁢

b

k

,

and the

intervals

[

a

k

⁢

b

k

,

(

a

k

+

1

)

⁢

b

k

)

are

disjoint

for each
value of

a

k

, so

a

k

is uniquely determined by where

x

lies
in amongst these intervals.

Then we can consider

x

-

a

k

⁢

b

k

=

∑

i

=

0

k

-

1

a

i

⁢

b

i

.

Repeating the previous

argument

with

k

replaced by

k

-

1

,
we see that

a

k

-

1

is uniquely determined.
Then we can consider

x

-

a

k

⁢

b

k

-

a

k

-

1

⁢

b

k

-

1

and so on.
Continuing this way, we see that all the digits

a

i

are uniquely
determined.

2.2

Near-uniqueness for non-negative numbers

If

x

=

a

k

⁢

b

k

+

⋯

+

a

1

⁢

b

+

a

0

+

a

-

1

⁢

b

-

1

+

a

-

2

⁢

b

-

2

+

⋯

then

a

0

,

…

,

a

k

are uniquely determined,
since

a

k

⁢

b

k

+

⋯

+

a

1

+

a

0

is the expansion for the non-negative integer

⌊

x

⌋

.

The argument to prove that

a

-

i

are uniquely determined proceeds
similarly as before.
We have

a

-

1

⁢

b

-

1

≤

a

-

1

⁢

b

-

1

+

a

-

2

⁢

b

-

2

+

⋯

≤

a

-

1

⁢

b

-

1

+

∑

i

=

2

∞

(

b

-

1

)

⁢

b

-

i

(geometric series)

=

a

-

1

⁢

b

-

1

+

(

b

-

1

)

⁢

b

2

1

-

b

-

1

=

(

a

-

1

+

1

)

⁢

b

-

1

,

where equality on the second line occurs if and only if

a

-

i

=

b

-

1

for every

i

≥

2

.
If we insist that

a

-

i

is never

eventually

the same
digit

b

-

1

,
then this shows that the digit

a

-

1

is uniquely determined by
where the original number

x

in the disjoint intervals

[

a

-

1

⁢

b

-

1

,

(

a

-

1

+

1

)

⁢

b

-

1

)

.

This argument may be repeated, to show that

a

-

i

are uniquely
determined, under the

assumption

that the expansion
does not end in all digits being

b

-

1

.

If the assumption is not made, then
numbers which have an expansion ending in all digits

0

have an alternate expansion ending in all digits

b

-

1

,
but other numbers still have unique base-

b

expansions.

3

Every base-

b

expansion represents a real number

We also want to prove that for every

sequence

of digits

a

k

,

a

k

-

1

,

…

,

a

1

,

a

0

,

a

-

1

,

a

-

2

,

…

there exists a real number

x

with the base-

b

expansion

x

=

∑

i

=

0

k

a

i

⁢

b

i

+

∑

i

=

0

∞

a

-

i

⁢

b

-

i

.

This is the where we use the least

upper bounds

property

of the
real numbers. (So far we have only used the Archimedean property,
so what we have done so far is also valid for

ℚ

.)

Consider the sequence

{

s

n

}

with the

s

n

=

∑

i

=

0

k

a

i

⁢

b

i

+

∑

i

=

0

n

a

-

i

⁢

b

-

i

.

This sequence, considered as a set,
is bounded above, for

s

n

≤

a

k

⁢

b

k

+

⋯

+

a

0

+

1

.
So it has a least upper bound

x

.
Since the sequence

{

s

n

}

is also

increasing

,
its least upper bound is the same as its
limit.

Title

existence and uniqueness of decimal expansion

Canonical

name

ExistenceAndUniquenessOfDecimalExpansion

Date of creation

2013-03-22 15:42:12

Last modified on

2013-03-22 15:42:12

Owner

stevecheng (10074)

Last modified by

stevecheng (10074)

Numerical id

8

Author

stevecheng (10074)

Entry type

Theorem

Classification

msc 11A99

Synonym

every

decimal expansion

represents a real number

Related topic

CantorsDiagonalArgument

Related topic

DecimalExpansion

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