Here’s one more way of convincing students that . Here’s the idea: how far apart are the two numbers?
First off, since , we know that
.
Of course, we know that . Since
must lie between
and
, we know that
must be less than
.
Second, we know that . Since
must lie between
and
, we know that
must be less than
.
Third, we know that . Since
must lie between
and
, we know that
must be less than
.
By the same reasoning, we conclude that
for every integer . What’s the only number that’s greater than or equal to
and less than every decimal of the form
? Clearly, the only such number is
. Therefore,
, or
.
I like this approach because it really gets at the heart of the difference between integers
and real numbers
. For integers, there is always an integer to the immediate left and to the immediate right. In other words, if you give me any integer (say,
), I can tell you the largest integer that’s less than your number (in our example,
) and the smallest integer that’s bigger than your number (
).
Real numbers, however, do not have this property. There is no real number to the immediate right of . This is easy to prove by contradiction. Suppose
is the real number to the immediate left of
. That means that there are no real numbers between
and
. However,
is bigger than
and less than
, providing the contradiction.
(For what it’s worth, the above proof doesn’t apply to the set of integers since
doesn’t have to be an integer.)
By the same logic — visually, you can imagine reflecting the number line across the point — there is no number to the immediate left of
. So while
would appear to be to the immediate left of
, they are in reality the same point.
One thought on “Why does 0.999… = 1? (Part 5)”