# An infinite question: Why doesn't .999~ = 1?

Based on Zeno’s paradox, could the light barrier be broken? In other words, just as the tortoise can be outran in real life even if the math looks odd, can light be outran even if the math says no?

http://www.straightdope.com/columns/030711.html

I don’t understand. Are you trying to argue that .9~ *doesn’t * equal 1?

I had an argument freshman year in college about this. I challenged my buddy (who believed it .9~ didn’t equal 1) to subtract .9~ from 1. His answer was .0~1. I said, “What’s that”? And he said, “Um, an infinitely small number.” I said, “Hm. Interesting. Moron.”

But back on topic: explain again how you’re going to use Zeno’s Paradox to get us to other galaxies? The math only looks “odd” in the tortoise v. Achilles if you frame it as Zeno’s Paradox. The impossibility of surpassing the speed of light isn’t framed as Zeno’s Paradox. It doesn’t look odd. …

I’m not even sure that there is any odd math in Zeno’s paradox (paradoxes). I seem to recall a mathematical “explanation” of the Tortoise v- Achilles race (similar to the one on this page) that demonstrates that, mathematically, there is no paradox; it is purely verbal.

I wouldn’t quite call him a moron (wrong, but not a moron). He probably heard about infinitesimals somewhere and thought it applied.

The OP is simply referencing the title of the article by Uncle Cece about .999 = 1. His post is not, however, discussing this part of the column, but, rather, discussing Zeno’s Paradox. He wonders if, since the Paradox turns out in the real world not to be a real paradox, the apparent speed limit of the universe might not be something untrue in actual fact? That is, have we bounded ourselves through creation of a self-imposed, mathematically-derived limit on speed?

I’ll let the wiser and brighter heads establish for the OP why it is that the speed limit of the universe is an actual limit and not a product of improper math.

Just in case it rears its ugly head, though…

Guests coming in to argue that 0.9999~ does not equal 1 have produced some of the longest and most aggravating threads in the history of the Dope.

Zut listed a full dozen old threads in this post and that was back in 2004.

Before anyone wants to go through this yet again, read those old threads and hopefully be enlightened.

Here’s one way to think of it: When the arrow gets halfway to the tree, or when Achilles gets to where the tortoise was before, it’s closer to its destination than it was before. With each successively smaller time interval, it keeps getting closer and closer, until, when the time interval size reaches zero, the arrow hits the tree, or Achilles catches up to the tortoise.

But you can’t get any closer to the speed of light, no matter how long a time interval you take to accelerate. If there’s a beam of light passing me, and I measure its speed, I’ll find that it’s exactly c, or 299792458 meters per second. Now, suppose I hop into a spaceship going 299792457 meters per second, in an effort to catch up. If I again measure the speed of that light beam, relative to my spaceship, I won’t measure 1 m/s. In fact, I’ll measure 299792458 m/s, exactly the same as my first measurement. I haven’t caught up to the light beam at all: I’m still just as far from the speed of light as I ever was.

Thank-you, Chronos.

BTW, my computer shows ads for a Zeno device, Sylvan Math, and a set of math skills workbooks. I just love how the Google ads work…