### 0.999… is equal to 1

This is a hard one to wrap your head around. 0.999… with infinite decimals, often written as 0.(9) is actually equal to 1. How does that work? Well, there’s a couple of ways you can look at it.

Firstly, one divided by three is 0.333…, with infinite decimals. Then, 0.(3) multiplied by 3 is 0.(9)… but it’s also 1 because a number multiplied and divided by the same number remains unchanged. To put it mathematically:

1:3 = 0.333, or 0.(3)

0.(3) x 3 = 0.(9),

but 1:3*3 = 1, so 1 = 0.(9).

You can also think about it this way:

x = 0.999…

10x = 9.999…

10x – x = 9.999… – 0.999…

9x = 9

x = 1

If that’s not convincing enough, then think about it another way. If 0.(9) is smaller than 1, then what would you have to add to it to get to 1? No matter what number you add, it’s still not small enough because such a number doesn’t exist – 0.999… = 1.

### 111111111 * 111111111 = 12345678987654321

Now this one almost seems made up. By multiplying numbers made of “ones” you end up with a number made of all the figures from 1 to 9 and back to 1. More properly, it would be written like this:

111,111,111 * 111,111,111 = 12,345,678,987,654,321

It still doesn’t seem to make much sense, but it does if you take it step by step. So:

1*1 = 1. Of course.

11 * 11 = 121. Hmm…

111 * 111 = 12,321. A pattern is starting to emerge.

1,111 * 1,111 = 1,234,321. Ok, you probably get the idea by now. Just to finish the whole thing:

11,111 * 11,111 = 123,454,321,

111,111 * 111,111 = 12,345,654,321,

1,111,111 * 1,111,111 = 1,234,567,654,321,

11,111,111 * 11,111,111 = 123,456,787,654,321, and of course:

111,111,111 * 111,111,111 = 12,345,678,987,654,321. There you have it!

### The Fibonacci sequence is encoded in the number 1/89

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