0.9999 . . . = 1? It sure seems so.

powernoodle

powernoodle

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My brainiac 14 year old comes to me today and claims that 0.9999 . . . . (to infinity) equals 1.

Now, being old school, where left is not right, and up is not down, I favor the idea that 1 = 1, and that 0.9999 . . . (to infinity) does not equal 1. If it equaled 1, we would call it 1. But we don't.

So my son whips this out:

x = 0.9999... (to infinity)
10x = 9.9999...
10x - x = 9.9999... - 0.9999...
9x = 9
x = 1

Ergo, x = 0.9999 . . . , but x also equals 1.

This really irks me! Is my whole life a lie? Do I not know what I do know?

Its also been 35 years since I had algebra II, so maybe my math isn't working. Thats very possible.

Now, if I smoked weed for about 3 days, and then drank a bottle of Nyquil while making out with a chick on a Tilt-a-Whirl, my head would be spinning just enough for this number stuff to make sense. But I don't smoke weed or pound Nyquil. As for the girl and the Tilt-a-Whirl, I'm not saying.

According to the internet, there are a couple of more ways of demonstrating that 0.999 . . . = 1, but my head would explode if I tried to go there.

My hat is off to you guys who can really embrace this notion that 1 equals a number with nothing but zeroes to the left of the decimal. I'm just not quite there yet.
 
In algebra, .999999... = 1
But, .99999... != 1.0

In calculus, .999999... != 1
In calculus, there is this thing called a differential (often written as dx) which is the smallest possible difference. So, .999999... = 1 - dx.
 
Occam's razor law of physics would say yes .9999 is in fact equal to 1. I am on your side and find it too speculative.
 
I mean that the missing 10 millionth is nothing ti dismiss without some debate.
 
powernoodle said:
My brainiac 14 year old comes to me today and claims that 0.9999 . . . . (to infinity) equals 1.

Now, being old school, where left is not right, and up is not down, I favor the idea that 1 = 1, and that 0.9999 . . . (to infinity) does not equal 1. If it equaled 1, we would call it 1. But we don't.

So my son whips this out:

x = 0.9999... (to infinity)
10x = 9.9999...
Click to expand...
So far, so good.

10x - x = 9.9999... - 0.9999...
Click to expand...
Yes...
9x = 9
Click to expand...
No. 9x != 9. If x=.99999999, then 9 * .9999999999 != 9 but an infinitely small number less than 9. And so...

x = 1
Click to expand...
... does not follow either.

Ergo, x = 0.9999 . . . , but x also equals 1.
Click to expand...
Which Q.E.D. is false.

This really irks me! Is my whole life a lie? Do I not know what I do know?
Click to expand...
It's a sleight of hand. A virtual one. ;)
 
philwar said:
It's a sleight of hand.
Click to expand...

Thats where I come down on the issue. I'm just too stupid to prove it.

Seems to me that the equation introduces whole numbers (by multiplying by 10), then removes the 0.999 . . . , leaving behind the whole number which is then said to equal x. Something like that.

I'm sticking with my theory that 1 cannot equal a number with nothing but zeroes to the left of the decimal point.
 
Saying "to infinity" is a lot like dividing by zero. In his "proof", he multiplies infinity by 10 then subtracts infinity from infinity, which should let you prove just about anything. I've seen a "proof" that 0 = 1, but if you check out all the terms, they did it by dividing by 0.
 
The "proofs" presented here are all false, but that .99999... = 1 is no "new math;" it has always (at least since I was in grade school (and that's getting on always now)) been true. Keep in mind that the math notation ... means that the series goes on forever. So, .99999 != 1, but .99999... = 1. Therefore, .99999 != .99999... . .99999... is a "shorthand;" you're not seeing the whole thing.
 
I don't think that our finite brains can really grasp the concept of infinity. We can apprehend it (restate what it is) but not comprehend it (understand it). So at some point, at least for me, this is fun stuff but ultimately an exercise in futility.
 
None of this matters as nothing exists; just ask Zeno.

But to answer the question anyway, which is greater .999999999999999999 or 1?

If .9999999999999999999999999999999999 < 1, then .9999999999999999999999999 cannot = 1 now can it. The difference may be de minimis but it still exists.

Also, your son's equation is flawed. He subtracts x from the left and should subtract x from the right, however, he instead subtracts .999999 from the right. His equation doesn't make sense, I think he should get some after school help from his instructor.
 
Shaky math is always best counteracted with quotes from science fiction:

"A difference which makes no difference IS no difference."
-Spock
 
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