Three Ways of Proving 0.999... = 1

More often than not, people don't believe me when I tell them that 0.99999... (followed by an infinite number of 9's) exactly equals 1. Over the years, I've accumulated 3 different ways to convince people this is the case.

Two of these are pretty much exactly the same, only the first is much more intuitive than the second. I can hear the skeptics yelling "Bring it on!", so here it goes.

The First Way Of Proving 0.999... = 1

Everybody knows that 1/3 = 0.333.... Just multiply both sides by 3 to get 3*1/3 = 3*0.333... which results in 1 = 0.999....

The Second Way Of Proving 0.999... = 1

Almost exactly the same as the first, only I'll be using a letter in this proof and people are afraid of letters when it comes to math. Intimidation is by far my favorite proof technique...

(1):  x = 0.999...
(2):  => 10x = 9.999...

Now (2) minus (1) gives 10x-x = 9.999... - 0.999..., so 9x = 9.000 and thus x = 1.

The Third Way of Proving 0.999... = 1

For the third proof, we'll need the formula for the sum of an infinite, converging, geometric series. Like Wikipedia says, the formula for this is (when -1 < r < 1)

S = \sum_{k=0}^\infty ar^k = \frac{a}{1-r}.

Now, let's rewrite 0.999...

0.999... = 0.9 + 0.09 + 0.009 + \cdots = {9 \over 10} + {9 \over 100} + {9 \over 1000} + \cdots = \sum_{n=1}^\infty \frac{9}{10^n}= \sum_{n=1}^\infty 9 \times \frac{1}{10^n}

Notice how the sum index in the last equation starts from 1 and the one in the Wikipedia formula starts from 0? No problem, we'll merely have to manufacture the first term out of thin air...

0.999... = \sum_{n=1}^\infty 9 \times \frac{1}{10^n} = 9 - 9 + \sum_{n=1}^\infty 9 \times \frac{1}{10^n} = 9 \times \frac{1}{10^0} - 9 + \sum_{n=1}^\infty 9 \times \frac{1}{10^n}= -9 + \sum_{n=0}^\infty 9 \times \frac{1}{10^n}

So a = 9 and r = 1/10 (which is between -1 and 1, so no problem there).

0.999 = -9 + \sum_{n=0}^\infty 9 \times \frac{1}{10^n} = -9 + \frac{a}{1-r} = -9 + \frac{9}{1-{1 \over 10}} = -9 + \frac{9}{9 \over 10} = -9 + 10 = 1

There are undoubtedly other ways to prove 1 = 0,999... but these three are my favorites and should be enough to convert even the strongest of unbelievers. Isn't math lovely?