 Extremeskins

# Does 0.9999 repeating 9 equal 1?

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Suppose .9999999999 (repeating) is not equal to 1.

Then these are two distinct numbers, call them x and z. Then there exists a y such that x + y = z. i.e. (by the additive inverse property). So since x = .99999999 repeating, y = .00000000000 repeating and finally a 1. How many 0's are there?

If there are finitely many 0's (say n) then y = 10^-n. This contradicts that x has infinitely many nonzero digits since x + 10^-n = z

So y must have an infinite number of 0's before the 1. But if y has an infinite number of 0's then y = 0. Then it follows that x = y, i.e. .999999999 (repeating) equals 1.

This does make sense though, Im going with Im not sure anymore..... :laugh:

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yes it is, if there are truly an INFINITE number of 3's and not just A VERY LARGE NUMBER of 3's. 0.9 with a MILLION BILLION 9s on the end does not equal 1. 0.9 with an infinite number of 9s DOES equal 1.

1/3 cannot be expressed as a decimal. Hence we have the expression in terms of limits and Reiman sums.

lim

n->infinity

n

__

\

/__.3*10^(-k)

k=0

The value that you guys are looking at and deciding it is equal to, is what is referred to in the mathematics world as an asymptote. The function approaches the value but never reaches it.

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0.3 repeating 3 is one-third, right?

and 0.6 repeating 6 is two-thirds, right?

what is 0.3 repeating 3 plus 0.6 repeating 6? 0.9 repeating 9.

what is one third plus two thirds? 1.

no rounding. no calculus limits involved. 0.9 repeating 9 equals 1.

Equal means exact.

0.99999 is not exatcly 1.

Nope never.

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.3 repeating = 1/3

3 x .3 repeating = 3 x 1/3

3 x .3 repeating = 1

.9 repeating = 1

It's true.

Try this on a calculator.

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Suppose .9999999999 (repeating) is not equal to 1.

Then these are two distinct numbers, call them x and z. Then there exists a y such that x + y = z. i.e. (by the additive inverse property). So since x = .99999999 repeating, y = .00000000000 repeating and finally a 1. How many 0's are there?

If there are finitely many 0's (say n) then y = 10^-n. This contradicts that x has infinitely many nonzero digits since x + 10^-n = z

So y must have an infinite number of 0's before the 1. But if y has an infinite number of 0's then y = 0. Then it follows that x = y, i.e. .999999999 (repeating) equals 1.

Shouldn't that say x = z?

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Are we talking 0.999 repeating cents, or 0.999 units of money?

Gaaaaaaaaaaaaaaah!

That's still one of my all-time favorite ES threads.

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Equal means exact.

0.99999 is not exatcly 1.

0.9 with an infinite number of nines is EXACTLY equal to one.

it is also not a decimal.

how 'bout them apples?

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0.9 with an infinite number of nines is EXACTLY equal to one.

it is also not a decimal.

how 'bout them apples?

no, it is NOT equal to one, it never REACHES one. The LIMIT of .9 repeating equals one.

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tell you what.

I'll give you .999999 (repeat) dollars and in exchange you give me \$1 dollar.

We can do this infinity times. ##### Share on other sites

then we will be exchanging one dollar bills forever!

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tell you what.

I'll give you .999999 (repeat) dollars and in exchange you give me \$1 dollar.

We can do this infinity times. that doesn't work, MJ. Money is a discreet value and we're dealing with a continuous value. they don't play nice together.

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then we will be exchanging one dollar bills forever!

Yeah, but you'll have to do it by reaching halfway over to the other guy's dollar bill... then closing the distance by half again... then by half again...

In the limit, we're all dead and the Sun exploded long ago.

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well as much as I'd love to stay and discuss this matter further, I'm off to calculus class where we deal with this exact kind of thing on a daily basis. You know, L'Hopital's rule and what-not.

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I will comment one more time, to say that PokerPacker is correct in this thread, everyone else fails at math.

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that doesn't work, MJ. Money is a discreet value and we're dealing with a continuous value. they don't play nice together.

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As has been mentioned already, 0.3 repeating does not equal 1/3, it approximates 1/3. Just like 0.6 repeating does not equal 2/3 it approximates it and 0.9 repeating approximates 1.

Think of it backwards. If we know 0.9 repeating can never truly equal 1, then 1/3 of 0.9 repeating can never truly equal 1/3.

ie.

0.9999... < 1

0.9999... * (1/3) < 1 * (1/3)

0.3333... < (1/3)

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opinions are like A holes, everyone has one, and most of them stink.

how about we go with the truth on this one instead of peoples wishy washy theoretical and philosophical reasoning.

I bring to you Wikipedia...

It outlines SEVERAL proofs that show that .9 repeating does in fact equal 1. And yes, in HS we DID prove this in algebra class, and possibly again in calculus. You have to understand... .999 repeating does'nt simply keep getting infintismally closer to 1, it has already arrived at that infinite point close to 1. numbers don't grow, they are what they are. .9999 repeating is a FIXED number.

I am actually kind of shocked alot of you would post such definitive "no's" so quickly without even looking into it. Are you all Mathematicians to be so confident? Infinity paradoxes aren't exactly child's play. a simple Googling would have shown you the correct answer.

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aw don't end the fun so soon JMUGator19!

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Hell, I am still trying to figure out why gas is always 9/10ths of a penny.

\$192 9/10...

Bull.

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aw don't end the fun so soon JMUGator19!

hahahaha this could have been a never ending thread.

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I stand by my previous statement, that PokerPacker is correct.

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hahahaha this could have been a never ending thread.

If it was a never ending thread woudl that mean that it equals 1 thread?

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hahahaha this could have been a never ending thread.

Kind of "infinitely repeating".

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I stand by my previous statement, that PokerPacker is correct.

If you mean when he said that .9999 repeating does NOT equal 1, then you are both wrong.

If you mean when he said the limit of .99999 repeating equals 1 the you are both right.

If you mean when he said that money is discreet not continuous, then you are both right.

If you mean when he said he was going to class, then you are probably both right.

Way to look up the issues instead of just picking a random person to agree with. In calculus you learn about limits and series. the purpose of being taught Limits in the calculus curriculum is to teach you how to calculate slopes and derivatives. Once he learns a little more, he will find out that the limit and a derivative and a slope are one and the same. And they do in fact reach absolute values even at a single point.

Heres more proof to others who might have also forgotten algebra but recollect some calculus.

A derivative is the slope of a curve at a point. AKA you find to INFINITESIMALLY close points on a curve and draw a line through them. that gives you a slope. its derivative. Now two points infinitesimally close on a line gives you exactly 1 point. which is why we say find the derivative at X point. Newton discovered the proof that used Limits to prove that you can find a slope at a single point.

All this is no different than .9999999999 being equal to 1. Same concept, same thought process. If you believe the limit of .999999 repeating = 1 then you must believe that .99999999 repeating = 1. You CANNOT believe one without the other. sorry.

enjoy the brainteasers! 