Does 0.999... Really Equal 1? A Mathematician Explains

When you have an infinitely decreasing number does it ever hit a point where it is indistinguishable from the number it's approaching? Does 0.999999999..., for instance, ever finally just become 1? A mathematician has an answer for us.

Mathematician Jordan Ellenberg joined us today to take our questions about math in everyday life, including one about what becomes of infinite sums as they converge on finite numbers. The answer, says Ellenberg, is not just about mathematical realities, but about our mathematical definitions:

previouslyprommie

How can an infinite sum, even of infinitely decreasing numbers, converge to a finite number? Like, as in, the infinite sum of the series 1 over N to the X?

JSEllenberg

There's a lot of stuff in the book about the craziness of infinite series. Like, do you think 0.9999.... extended for ever is equal to 1? I have seen people get SUPER FURIOUS about this question.

But the main point is, it's not the right question. We shouldn't ask what the value of an infinite sum IS, but what it should be DEFINED to be. The number theorist G.H. Hardy has the money quote on this:

"…it does not occur to a modern mathematician that a collection of mathematical symbols should have a 'meaning' until one has been assigned to it by definition. It was not a triviality even to the greatest mathematicians of the eighteenth century. They had not the habit of definition: it was not natural to them to say, in so many words, `by X we mean Y.' … it is broadly true to say that mathematicians before Cauchy asked not 'How shall we define

1 – 1 + 1 – 1 + …

but

'What is 1 – 1 + 1 – 1 + …

and that this habit of mind led them into unnecessary perplexities and controversies which were often really verbal."

Image: Abacus / ansik