# Another proof that .999…=1

Jim Ward sends this nice proof:

1/3=.3333333…

`.33333333333333333333...`

+ .33333333333333333333...

+ .33333333333333333333...

------------------------------------------

= .99999999999999999999....but 1/3 + 1/3 + 1/3 = 1

Of course this proof requires that we understand that 1/3=.333… but that at least can be shown by long-division. Steve Landsburg, who has a PhD in mathematics in addition to being a brilliant economist, writes to warn, however, that we haven’t defined what we mean by an infinite series like .999… nor have we proven that multipying by 10 (in the earlier proof) is equivalent to moving the decimal point. These are all valid points. As a lay consumer of mathematics, rather than a producer, I find these “proofs” helpful but do take them with a grain of salt.

Addendum: Here is more on Kakutani’s theorem for the mathematically strong of heart.