Prove that the repeating decimal 0.999999… is equal to 1.
Solution
There are a number of proofs that show this to be true, but many people still struggle with the concept. Here’s a good one:
x = 0.999…
10x = 9.999…
10x – x = 9.999… – 0.999…
9x = 9
x = 1
1/3 = 0.333…
3 x 1/3 = 3 x 0.333…
1 = 0.999…
10x = 9.999…
10x – x = 9.999… – 0.999…
9x = 9
x = 1
The reason that many people struggle with this idea is that the concept of infinity is difficult for our primate minds to grasp. On some level, most people just imagine one final 9 somewhere down the line.
- Numbers can often look very different when expressed another way, and this is no exception.
- The reason is closely tied to the concepts of infinity and limits, which already tug at the brain pretty strongly.
Here’s another proof:
3 x 1/3 = 3 x 0.333…
1 = 0.999…
Source: thisinsider.com