.9 rep = 1?

avolkov · New Member Offline Joined: 21 Apr 2008 Posts: 3

Does .9 repeating = 1

i personally have to say this:

in "theoretical math" .9 repeating does = 1

because

1/3 = .3 rep

2/3 = .6 rep

3/3 = .9 rep ... but also 3/3 is known to be one...

but in "real" math...

3/3 is the same as saying 3 divided by 3... known to be one...

and if .9 equals 1... then shouldnt 1 - .9 rep = 0 ??? (to make it easier 1 = 1.0 = 1.0 rep...)

thus shouldnt 1.0 rep - .9 rep = 0???

wouldnt it be something more like

im open to ideas and reasons... so please let me knw wat ure thinking...

**Posted:** Mon Apr 21, 2008 6:49 pm

Nerd_of_the_Ages

Yes it IS equal to zero, by the simple logic of 1 - 1 = 0. And also, I personally find that your proof isn't the one with most depth and gives you most understanding; so here's another proof.

x = 0.9999....
10x = 9.99999.....
10x = 9 + x (make sure you see why; note that we originally assigned x to equal 0.99999... and 9.99999... = 9 + 0.99999....)
9x = 9
x = 9/9
x = 1

Also, what do you mean by "real" and "theoretical?" Could you elaborate more on that? Also, I don't see what you mean by 1 - 0.999... would be 1/infinity. Could elaborate more on that, too?

**Posted:** Mon Apr 21, 2008 7:38 pm

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avolkov · New Member Offline Joined: 21 Apr 2008 Posts: 3

By theoretical math, i mean to point out the type of mathematical thinking which allows for 1 = .9 rep

and "real" math is the customary math we know and love, simple, built on rules... and everything has proof... the dispute between 1 and .9 rep is something a theoretical mathematician would try to answer...

when i said 1/infinity'th, i meant this...

.9 rep is a rational number somewhere on a number line... 1 is a separate rational number somewhere on a number line... they happen to be close... but they are different numbers...

in todays calculations, it seems the difference wouldnt matter at all beween the two, so we consider them the same... but a small difference still exists... tehy are two different rational numbers...

thus... me and my imagination could only show this difference as 1 / infinity'th

kinda hard to grasp... ive never been a good explainer... but try to think about it from a few different perspectives...

**Posted:** Tue Apr 22, 2008 2:52 am

cf249

This question has been discussed several times on this forum (one example is here).

**Posted:** Tue Apr 22, 2008 4:47 am

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Easy solution:

$\frac{1}{3}=.\overline3$
$\frac{1}{3}=.\overline3$
$\frac{1}{3}=.\overline3$
$\frac{1}{3}+\frac{1}{3}+\frac{1}{3}=.\overline3+.\overline3+\overline3$
$1=.\overline9$ .

Another proof:

$x=.\overline9$
$10x=9.\overline9$
$10x=9+x$
$x=1$ .

There you are.

btw

$\frac{1}{2}+\frac{1}{4}+\frac{1}{8}+\frac{1}{16}+\cdots=1-\frac{1}{2^{\infty}}$ ,

but does $\frac{1}{2^{\infty}}$ really matter? depends on your point of view.

**Posted:** Tue Apr 22, 2008 5:57 am

JBL

**Posted:** Tue Apr 22, 2008 6:24 am

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avolkov · New Member Offline Joined: 21 Apr 2008 Posts: 3

But don't you think that .9 rep is a rational number... and so is 1

and on a number line each rational number has 1 and only 1 numerical value...

thus doesnt this mean that .9 and 1 are both different numbers...

and algebra has glitches... which is why i dont believe in the x = .9 rep equation above... its not that its not mathematically incorrect... but just like ext. roots... its a glitch...

mostly because math is a human creation, and its a system... in some cases a flawed system...

and no one has answered me... if wat i said about the number line is true... then what is the difference between 1 and .9 rep...

**Posted:** Tue Apr 22, 2008 7:12 am

cf249

0.9 (repeating) and 1 have the same value. They are the same number.

What do you mean that "algebra has glitches"?

**Posted:** Tue Apr 22, 2008 7:26 am

_________________
Yogi Berra quotes: "It's like déjà vu all over again".
"90 percent of this game is mental, and the other half is physical."
Mod of the future Mafia Game X, where X is an undetermined integer between 46 and 60, inclusive.

JBL

**Posted:** Tue Apr 22, 2008 8:30 am

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mathking123 · Riemann Hypothesis Offline Joined: 09 Feb 2008 Posts: 458

and, btw, algebra has NO glitches lolz.

it's all based on a rigid and accurate system.
it's all rigorous also....exept for the basic axioms(assumptions) of math. everything else is built off of that just like a building....

**Posted:** Tue Apr 22, 2008 10:42 am

t0rajir0u

**Posted:** Tue Apr 22, 2008 10:45 am

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Nerd_of_the_Ages

Wow, JBL and t0rajir0u pretty much said all there is to be said.

avalkov, just make sure to know that just because something is counter-intuitive (or at least, to your intuition) doesn't mean it's wrong or "algebra has glitches" (which I don't think is proper use of the word glitch, anyway).

Also, about you're thing about the number line, on a less technical note, your logic is flawed, since you... well, let's look at it again.

**Posted:** Wed Apr 23, 2008 6:33 pm

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blahblahblah

To be honest none of these proofs seem to make much sense to me.

What is 0.9999...? It seems like it should be the limit of the sequence 0.9,0.99,0.999, whose k-th term is 0.99...9 (with k nines)

This is a monotone increasing bounded sequence whose supremum is 1, so 0.999 ... = 1.

**Posted:** Thu Apr 24, 2008 8:05 pm

PI-Dimension

$\frac{1}{infty}=0$

Guys... keep up with your math

Here is a good book to read about zero and infinity:

Zero: A biography of a dangerous idea.

**Posted:** Fri Apr 25, 2008 1:50 pm

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**Posted:** Fri Apr 25, 2008 2:02 pm

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**Posted:** Sat Apr 26, 2008 4:25 pm

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**Posted:** Sat Apr 26, 2008 8:15 pm

davidyko

My god, this reminds me of the non-rigorous tripe we had to go through with some of binomial_4eva's threads.

**Posted:** Sat Apr 26, 2008 9:38 pm

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**Posted:** Sat Apr 26, 2008 10:16 pm

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JBL

That's not quite accurate -- we can't prove the axioms of arithmetic consistent relying only on the axioms of arithmetic, but there might be some other consistent set of axioms from which we could prove the real numbers consistent. (Of course, we couldn't prove that system consistent from within itself either, and so on.)

But the comment you've quoted is still bogus: if the real numbers are inconsistent, math based on the real numbers isn't flawed "in some cases," it's a completely meaningless exercise, while if the real numbers are consistent then they don't admit any "glitches."

In addition, although I admire your attempt charitably interpret the nonsense of the OP to render it less embarassing, I think it's pretty clear that the OP couldn't possibly have been making the argument you suggest.

**Posted:** Sun Apr 27, 2008 5:42 am

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