Consider the real number that is represented by a zero and a decimal point, followed by a never-ending string of nines:
It may come as a surprise when you first learn the fact
that this real number is actually EQUAL to the integer 1.
A common argument that is often given to show this is
as follows. If S = 0.999..., then 10*S = 9.999...
so by subtracting the first equation from the second, we get
9*S = 9.000...
and therefore S=1. Here's another argument.
0.1111... = 1/9,
so if we multiply both sides by 9, we obtain 0.9999...=1.
You might also mention that by similar arguments,
every rational number with a terminating decimal expansion
has another expansion that ends in a never-ending string of 9's.
So, for instance, the rational 7/20 can be represented
as 0.35 (the same as 0.35000...) or 0.34999...
The Math Behind the Fact:
When seeing these arguments,
many people feel that there is something shady going on
here. Since they do not have a clear idea what
a decimal expansion represents,
they cannot believe that a number can have two different representations.
We can try to clear that up by explaining what a decimal
Recall that the digit in each place of a decimal expansion
is associated with a (positive or negative) power of 10.
The k-th place to the left of the decimal
corresponds to the power 10^k.
The k-th place to the right of the decimal
corresponds to the power 10^(-k) or 1/10^k.
If the digits in each place are multiplied by
their corresponding power of 10 and then added together,
one obtains the real number that is represented by
this decimal expansion.
So the decimal expansion 0.9999... actually represents the
9/10 + 9/100 + 9/1000 + 9/10000 + ...
which can be summed as a geometric series
to get 1. Note that 1 has decimal representation 1.000...
which is just 1 + 0/10 + 0/100 + 0/1000 + ...
so if one realizes that decimal expansions are just
a code for an infinite sum, it may be less surprising
that two infinite sums can have the same sum.
Hence 0.999... equals 1.
How to Cite this Page:
Su, Francis E., et al. "Why Does 0.999... = 1?."
Math Fun Facts.