Why does 0.999… = 1? (Part 2)

In this series, I discuss some ways of convincing students that 0.999\dots = 1 and that, more generally, a real number may have more than one decimal representation even though a decimal representation corresponds to only one real number. This can be a major conceptual barrier for even bright students to overcome. I have met a few math majors within a semester of graduating — that is, they weren’t dummies — who could recite all of these ways and were perhaps logically convinced but remained psychologically unconvinced.

Methods #2 and #3 are indirect methods. We start with a decimal representation that we know and end with 0.999\dots.

Method #2. This technique should be accessible to any student who can do long division. With long division, we know full well that

\displaystyle \frac{1}{3} = 0.333\dots

Multiply both sides by 3:

\displaystyle 3 \times \frac{1}{3} = 3 \times 0.333\dots

\displaystyle 1 = 0.999\dots

Though not logically necessary, this method could be reinforced for students by also considering

\displaystyle 1 = 9 \times \frac{1}{9} = 9 \times 0.111\dots = 0.999\dots

green line

Method #3. With long division, we know full well that

\displaystyle \frac{1}{3} = 0.333\dots \quad and ~ \quad \displaystyle \frac{2}{3} = 0.666\dots

Add them together:

\displaystyle \frac{1}{3} + \frac{2}{3} = 0.333\dots + 0.666\dots

\displaystyle 1 = 0.999\dots

Though not logically necessary, this method could be reinforced for students by also considering any (or all) of the following:

1 = \displaystyle \frac{1}{9} + \frac{8}{9} = 0.111\dots + 0.888\dots = 0.999\dots

1 = \displaystyle \frac{2}{9} + \frac{7}{9} = 0.222\dots + 0.777\dots = 0.999\dots

1 = \displaystyle \frac{4}{9} + \frac{5}{9} = 0.444\dots + 0.555\dots = 0.999\dots

Leave a comment

1 Comment

  1. Why Does 0.999… = 1? (Index) | Mean Green Math

Leave a Reply

Fill in your details below or click an icon to log in:

WordPress.com Logo

You are commenting using your WordPress.com account. Log Out /  Change )

Google photo

You are commenting using your Google account. Log Out /  Change )

Twitter picture

You are commenting using your Twitter account. Log Out /  Change )

Facebook photo

You are commenting using your Facebook account. Log Out /  Change )

Connecting to %s

This site uses Akismet to reduce spam. Learn how your comment data is processed.

%d bloggers like this: