Math: .99repeating = 1?

Factual Questions
61

Agreeing with those who say that you can always subtract one equation from another. Equals subtracted from equals are equal, so to speak.

If you are subtracting equations that are not independent, then you may not be able to find a unique solution (assuming n equations in n unknowns.

But we’re starting with simple math, here.

x = .9999999…(infinte repetition)
10x = 9.99999…

You ask, what if you use 8x instead of 10?
Then you get
8x = 8*(.99999999…) which doesn’t help much. But it’s still a valid equation.

If x = y
Then ax = ay and
Therefore (ax - x) = (ay - y)
Or (a-1)*x = (a-1)*y.

In most instances, this is ho-hum, it won’t help much; but in some cases, it can help solve the system.

This has hit a point of being silly. On the one side, there are mathematicians and the like, citing proof after proof. On the other side, there are people who “don’t believe it” or “don’t like it.” Enough.

62

Sorry if my math is simple and I don’t know big 50 cent words. But I was under the impression that you could multiply one equation by a variable to get a new equation, then find the difference to help you solve a problem. Like this one: Find the sum of all the terms in the series (or is it sequence?) that has 1 as its first term, and every other term 1/2 of the preceding term.

 r= (1 + 1/2 + 1/4 + 1/8 + 1/16 + ...)

2r= (2 + 1 + 1/2 + 1/4 + 1/8 + …)

2r - r = 2

r = 2

If you can’t do this, how else do you solve this problem?

63

I don’t want to give another proof that
1 = 0.9[inf]
We have seen enough proofs of that here, But I want to say something about non-standard analysis. I like non-standard analysis and I think it is very helpful in understanding calculus. However, the standard decimal notation cannot represent infintesimals. The difference between 1 and 0.9[inf] is not infintesimal, it is zero!

I agree with CKDextHavn. This has gotten silly.

Virtually yours,

“Feynman was wrong.
I understand Quantum Physics completely.
Anybody seen my drugs?” - A WallyM7™ .sig

64

23skidoo: Yes, you can multiply and add equations as you’ve suggested, subject to the following TECHNICAL POINT: ONLY if you know that the series converges (to a finite sum.)

For instance, following your example,
x = 1 + 1 + 1 + 1 + 1 …
x = 1 + (1 + 1 + 1 + 1 + …)
Subtracting the two equations:
0 = 1
In this case, you can’t subtract the two equations because the series don’t converge.

65

As has been proven, .99… is equal to one. The reason why some people don’t agree is because it “doesn’t look pretty” as someone wrote earlier (sorry forgot who). This is because, as has also been mentioned, we are using the decimal system which does not handle the infinite very well.

However the naysayers do raise an interesting point that is more philosophy than math.

0.9 is a whole with a tenth missing. add 0.09 and you almost fill the missing tenth but there’s still a hundredth missing. etc, etc. So by adding 9s at the end you are filling the hole in the whole in progressively smaller increments until the whole becomes infinitely small.

So the question is, is an infinitely small hole a hole? I guess the answer is no but I understand why it’s hard to get past.

66

Face it, folks, with our flawed number system, .999… = 1

But it doesn’t look like it should, I know. Then again, you’d think you could divide by zero. Or take zero to the zeroeth power. To me, that last is the biggest offender against our system of numbers, because on one hand, we have a rule saying that anything to the zeroeth power is one, and on the other we have a rule saying that zero to any power is zero. We live in a flawed world.

“Give a man a fire and he’s warm for a day, but set fire to him and he’s warm for the rest of his life.”

–Terry Pratchett

67

Why do you say the number system is flawed?

The number system, independent of our representation of it, works just fine. The number 1 (the multiplicative identity) is uniquely determined. The real number system is a complete ordered field (each of those terms is defined in mathematics) and there are mathematical proofs that it is unique – any other number system is really the same, but might use different representation (for instance, a base 12 or base 8 representation is NOT a different number system.)

The problem is that we have decimal representations of these real numbers. Some of those decimal representations are hard to imagine, like 1/3 being represented by .333333… with the 3 repeated.

One way to deal with those “hard” representations is by thinking of the “true” value as the limit of an infinite sequence:

.3 + .03 + .003 + .0003 + …

That can be a very useful representation, but let’s not pretend that is anything other than 1/3… unless, of course, you STOP somewhere along the way, and then it’s different, it’s no longer the repeating-3.

We can describe this in different ways; we can say the infinite repeating decimal equals 3, we can say the series converges (adds) to 3, we can say that the partial sums converge to 3, we can say that the series IS 3. There is no difference between these various representations of the real number 3, except the way we look at them.

Similarly, we can write
e = 1 + 1/1 + 1/2! + 1/3! + 1/4! +…

That infinite series converges to the unique real number e. That is a representation of a uniquely defined real number e.

And so with .99999… Whether you want to say it “converges to 1” or “partial sums converge to 1” or “is 1”, is saying the same thing. So, have we got this thing moved to a matter of language and taste? I’m really tired of posting on this one.

68

You people miss an important concept here. Doing something an infinite number of times does not mean the same thing as doing it as often as you’d like. 0.9 repeating is not the same as writing a zero with as many nines as you can write as your hand gets tired. It is possible to have a number which is a REAL, RATIONAL, number that cannot be expressed as a decimal since it repeats infinitely. It is not possible to write 1/3 as a decimal without noting that it is 0.3 repeating (which is NOT the same thing as a zero, a decimal point, and writing threes until your hand gets tired). 0.9 repeating is equal to one, zero with a decimal point and as many nines as you feel like writing is never equal to one.

Look at it this way. If x-y=z, and z=0, then x=y. It has to be. If x=1 and y=.9 repeating, what does z equal? It can’t equal anything but zero, since there cannot be a digit following an infinitely repeating digit. .9 repeating equals 1

69

::test post::

70

I think it depends on what you are doing with your numbers.
For most intents and purposes, 0.99… is close enough to 1 to say it equals 1. However, some people do maths (and I won’t go into the uses of it) where they use limits, and a decimal will approach a number but will never equal that number. You can kind of get an idea of that with this teaser…

Set an imaginary finish line somewhere (eg, a point on the wall 5 metres away from you). Find a method of advancing so that even though you always move ahead, you never reach the finish line.

The answer is - always advance by half the remaining distance.

By the time the remaining distance is a millionth of a micrometre, most people would say you’re there. So, 0.9999… can be close enough to 1 that it makes no practical difference, or you can be using a type of maths that says it does make a difference. Use your own judgement.

71

::yawn:: Zeno’s paradox. You can never get from point A to point B, because you must travel one half of the distance in a finite amount of time, the half the remaining distance in a finite amount of time, … The sum of an infinite number of positive numbers is obviously not finite. Therefore movement is not possible. Replace one half in the above with nine tenths, and you have the arguement that 0.9… cannot equal 1.0.

If you believe 0.9… is not equal 1.0, do you also believe that movement is not possible?

Virtually yours,

DrMatrix
“Feynman was wrong.
I understand Quantum Physics completely.
Anybody seen my drugs?” - A WallyM7™ .sig

72

How do you figure from that, that movement is not possible? You can move, you just never reach where you are going.
And no, I said that I don’t believe 0.9… is equal to 1, but it is so close it might make no practical difference, and it depends on the applications of the maths before you can definitively answer. In most applications 0.9… can be considered equal to 1 but technically it is not.
I can get back to you with an equation that when graphed will approach a number (like 1) but never reach it if you like.

73

Dear Lord, here we have a provable mathematical truth (that .999… = 1) and yet people still disbelieving. No wonder that some people doubt less provable stuff, like Evolution or the Existence of God!

OK, one more time. From the top.

If you are talking practical measurements, that you can make with a ruler or some very fine measuring device, then .999… gets as close to 1 as you can measure. If you want to say they are unequal because you stop measuring after (say) a few thousand decimal places, then OK, say that. That’s because you can’t measure any finer.

Similarly, you’ll have to concede that .3333… is never exactly equal to 1/3 (because you stop measuring after a few hundred thousand decimal places).

And you’ll have to admit that pi doesn’t really exist, since 3.14159… is never equal to pi (because you stop measuring after a few thousand decimal places.)

You want to work in the world of finite measurements, that’s fine. Let’s say that we stop everything after a hundred thousand decimal places, OK? That’s all the further we can measure, that’s narrower than the width of an electron. What kind of mathematics do we have, then?

Well, to start, there are only a finite number of numbers between 0 and 1 (just list all those hundred thousand decimal place). And the numbers are discrete. There is no number halfway between (say) .999…[repeat for a hundred thousand places]98 and .999…[repeat for a hundred thousand places]99

((Can I use RFAHTP for “repeat for a hundred thousand places”?))

Just like, if you’re dealing with counting numbers, there is no natural number (integer) between 1 and 2.

It is perfectly acceptable to work in this number system. Notice that .333…RFAHTP…3 added to itself three times will be .999…RFAHTP…9, not the same as 1, so you’ll need to be careful with fractions like 1/3. That number system will not allow you to divide 1 by 3 and remain in the number system; you can only approximate that the answer is something that should exist between between .333…[RFAHTP]3 and .333…[RFAHTP]4.

So, you can’t always remain in that number system when you divide. However, it’s a hundred thousand decimals, for God’s sake, so who cares what happens in the hundred-thouand-and-one-th decimal place? We just drop it from our system.

With me so far? That’s the argument that .999… is not the same as 1.

HOWEVER, if you want to talk about the world of the Real Number system… the mathematical model used by every science, the most common model, the model that gives us the broadest understanding… then you can have infinite decimals, these are well-defined and well-understood.

The Real Number system offers such advantages as:

  • 1/3 added to itself three times gives 1
  • Division of any real number by another (excluding zero) stays within the real number system
  • Pi exists

Note that in the Real Number system, .333… added to itself three times gives .999… but ALSO (in its guise as 1/3) gives 1.

Those two numbers (.999… and 1) are EQUAL. Not just approximate, not just as close as you’d like, but EQUAL, in the same sense that 1 + 1 = 2 or that 1/3 = .33333…

I contend that the Real Number system is a far better system in which to work, even if we cannot actually MEASURE pi exactly.

Are we agreed? Are we done with this? … I know that I am.

74

The point of Xeno’s paradox is that it can be applied with equal (zero) validity to any scale. If you can’t move thirty feet, you can’t move thirty inches, or any distance at all. It doesn’t matter where you intend to go; math can’t read your mind. Xeno’s paradox is a naive way of mixing ostensibly infinite values with finite values.

With a finite speed, you move an infinitely small distance in an infinitely small amount of time. Since measurable amounts of time are just collections of infinitely small amounts of time, and a collection of infinitely small distances is still an infinitely small distance, you can never move. It’s bogus as heck, but it persists.

Here’s the bogus math:
Finite speed = Distance / Time
Time / Infinity = an instant
Distance / Infinity = zero distance
(Time / Infinity) x a large number = a positive non-zero amount of time
(Distance / Infinity) x a large number = zero distance

You see how that isn’t really fair? Xeno seems to be saying that, by taking an instantaneous snapshot of an object with a measurable velocity, you reveal that the velocity is really 0 / 0, which of course is equal to 0! But 0 / 0 is not equal to zero. It is an empty set.

It really makes me want to throw a spear at the guy.

75

::sigh:: In math two quantities are either equal or not. Close enough isn’t good enough. If you start at the point A and place the point B half way to your goal, you will never get half-way there, so you certainly will never get to where you are going. I’d say that pretty much says you can’t go anywhere.

As far as a function that approches 1 but never reaches 1, how about the 1 - 1/10^n. This equals 0.9[n]. But the limit (as n goes to infinity) does reach one. 0.9… is not close enough to 1. It is 1.

Virtually yours,

DrMatrix
“Feynman was wrong.
I understand Quantum Physics completely.
Anybody seen my drugs?” - A WallyM7™ .sig

76

I swear by Georg Cantor and all that’s Holy that I shall not post to this topic again. If I feel like posting I will run into a brick wall until the feeling passes. But don’t worry, I will not get hurt because I will only go 0.9… of the way to the wall.

::THUD::

Virtually yours,

DrMatrix
“Feynman was wrong.
I understand Quantum Physics completely.
Anybody seen my drugs?” - A WallyM7™ .sig

77

Here’s my attempt to formalize what others have been saying as an iteratize proof(many years since i did one, go easy on me).

conjecture. 1 is not equal .999(rep)

principle1. if two numbers are different then there is a measureable difference between them.
reason. if x-y =0 then x=y by definition

principle2. .999(rep) can also be written as 9/10 +9/100 +9/1000 … or 9/10^1 +9/10^2 +9/10^3 …+9/10^k
extrapolation. Therefore there is a measurable difference between 1 and .999(rep)
For any arbitrary number n hypothesized to be the difference between 1 and .999(rep)
we can take the first 1/n terms of the sequence leaving a difference that is smaller then the hypothesized difference.

result: for any n hypothesized to be the difference. we can prove that it is infact not the diference, and the difference is smaller.

Therefore there is no number that can be shown to be the difference, other than 0.
1 and .999(rep) are equal. The conjecture was false.

Do you ever get the feeling that everybody thinks you’re paranoid?

78

I thought CKDextHavn was going to pose the opposite argument. I thought he was going to say for practical purposes, like measuring, you can say 0.999… equals 1.0 I’d agree, for practical real-world applications. Just like we’ll carry out pi only to a few decimal places.

But, technically speaking, we realize such numbers continue ad infinitum. Well, at least I think we’d agree that 0.999… does!
I’d have to take the stand that, in the abstract world of math where lines and planes have no thickness, etc., I’d hope all could accept the abstract concept that 0.999… goes on and on never quite reaching 1.0 Out of respect for those opinions to the contrary, I will even bend and say “IMHO” if that assuages you dis-believers.

Anyway, it all boils down to this old story. A math teacher lined up all the girls of the class along one wall, and all the boys were lined up along the opposing wall facing each other. The math teacher instructed the boys to walk half way across the room. Then half again, and so on…until they kissed the girl awaiting at the opposite wall. The practical boys got to kiss the girl. The theorists are still trying to complete the task today. …And that is the rub, ladies and gentlemen, between practical math and theory.

I will also agree with CKDextHavn that, just like the decimal places over which we argue, we could debate this topic ad infinitum! As for me, I’m going to make a quantum leap of faith and grab for 1.0 cool-one :wink:

In closing, the bright side is, those of us who believe in 0.9999… < 1.0, we shall never age! I’ve never really reached my first birthday! I keep getting closer and closer and closer…

It’s like deja vu all over again; it’s like deja vu all over again!

79

Here’s my attempt to formalize what others have been saying as an iteratize proof(many years since i did one, go easy on me).

conjecture. 1 is not equal .999(rep)

principle1. if two numbers are different then there is a measureable difference between them.
reason. if x-y =0 then x=y by definition

principle2. .999(rep) can also be written as 9/10 +9/100 +9/1000 … or 9/10^1 +9/10^2 +9/10^3 …+9/10^k
extrapolation. Therefore there is a measurable difference between 1 and .999(rep)
For any arbitrary number n hypothesized to be the difference between 1 and .999(rep)
we can take the first 1/n terms of the sequence leaving a difference that is smaller then the hypothesized difference.

result: for any n hypothesized to be the difference. we can prove that it is infact not the diference, and the difference is smaller.

Therefore there is no number that can be shown to be the difference, other than 0.
1 and .999(rep) are equal. The conjecture was false.

Do you ever get the feeling that everybody thinks you’re paranoid?

80

I’m going to point out that the 1 > 0.999… camp, for all their posting, has still never shown that 1 > 0.999…

It is too clear, and so it is hard to see.