Another weblog

20 February 2010

When many maths students encounter the expression 0·999… = 1·000…, they are a bit uneasy. It challenges their preconceived notion that each number can be represented in one and only one decimal way. One stumbling-block might be that they think of 0·999… as a large but finite string of digits—even if they accept an infinite string of nines, they may still think in terms of there being a last digit ‘at infinity’. Also, some students imagine 0·999… as more of a ‘process’ than a representation of a number.

There are however, many ways of proving the identity. One of the most satisfying for students is this one:

⅓ = 0·333…
⅓ × 3 = 0·333… × 3
1 = 0·999…

And here’s another one:

Let x = 0·999…
Then 10x = 9·999…
10x − x = 9·999… − 0·999…
9x = 9·000…
x = 1·000…
And since x = 0·999… (as we defined at the beginning),
0·999… = 1·000…

Of course there’s got to be the light-bulb joke:

How many mathematicians does it take to change a light bulb?
0·999…

All of which means I’ll be 16·999… years old tomorrow at exactly the same instant I’ll be 17·000… years old.

By the way, 17 years = 536 467 742 seconds, not 536 112 000 seconds, because of leap years and leap seconds. And unfortunately I don’t know the exact time of day I was born at, so I can’t celebrate the right second. But in case you want to figure out how many days old you are, you should check out Wolfram|Alpha. Try putting in something like ‘22 February 1732 to 4 July 1776’ or ‘1000 weeks from 21 February 1993’—or ‘Apple and Microsoft’, for that matter—and see what you get.

Posted in Science |