One number can have two decimal names

Write 0.9, then 0.99, then 0.999. Each is a little short of 1. What if the 9s never stop?

The rule

Rule: The decimal 0.999… with endless 9s is exactly 1. A positive decimal that stops has a second name in endless 9s.

The gap to 1 is how far a decimal is below 1. After one 9, the gap is 0.1. After two 9s, the gap is 0.01. Each new 9 makes the gap ten times smaller. Endless 9s make the gap smaller than any positive number. Only 0 is that small.

Any decimal that stops

Take 0.5. Lower its last non-zero digit by 1. That gives 0.4. Now add endless 9s: 0.4999…

Check it. 0.4999… is 0.4 plus 0.0999…. Now 0.0999… is one tenth of 0.999…, which is 1. So 0.0999… is 0.1, and the total is 0.5.

Compare two decimals. Are they the same number?

Try the lab to finish this lesson.