Why does 0.999… = 1? (Part 5)

Here’s one more way of convincing students that 0.\overline{9} = 1. Here’s the idea: how far apart are the two numbers?

First off, since 1 \ge 0.\overline{9}, we know that 1 - \overline{9} \ge 0.

Of course, we know that 1-0.9 = 0.1. Since 0.\overline{9} must lie between 0.9 and 1, we know that 1 - 0.\overline{9} must be less than 0.1.

Second, we know that 1-0.99 = 0.01. Since 0.\overline{9} must lie between 0.99 and 1, we know that 1 - 0.\overline{9} must be less than 0.01.

Third, we know that 1-0.999 = 0.001. Since 0.\overline{9} must lie between 0.999 and 1, we know that 1 - 0.\overline{9} must be less than 0.001.

By the same reasoning, we conclude that

0 \le 1 - 0.\overline{9} < \displaystyle \frac{1}{10^n}

for every integer n. What’s the only number that’s greater than or equal to 0 and less than every decimal of the form 0.00\dots001? Clearly, the only such number is 0. Therefore,

1 - 0.\overline{9} = 0, or 0.\overline{9} = 1.

green lineI like this approach because it really gets at the heart of the difference between integers \mathbb{Z} and real numbers \mathbb{R}. 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, 15), I can tell you the largest integer that’s less than your number (in our example, 14) and the smallest integer that’s bigger than your number (16).

Real numbers, however, do not have this property. There is no real number to the immediate right of 0. This is easy to prove by contradiction. Suppose x > 0 is the real number to the immediate left of 0. That means that there are no real numbers between 0 and x. However, x/2 is bigger than 0 and less than x, providing the contradiction.

(For what it’s worth, the above proof doesn’t apply to the set of integers \mathbb{Z} since x/2 doesn’t have to be an integer.)

By the same logic — visually, you can imagine reflecting the number line across the point x = 0.5 — there is no number to the immediate left of 1. So while 0.\overline{9} would appear to be to the immediate left of 1, they are in reality the same point.

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

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