x=.9999…
10x=9.9999…
Subtract x from both sides
9x=9
x=1
There it is, folks.
Unfortunately not an ideal proof.
It makes certain assumptions:
- That a number 0.999… exists and is well-defined
- That multiplication and subtraction for this number work as expected
Similarly, I could prove that the number which consists of infinite 9’s to the left of the decimal separator is equal to -1:
...999.0 = x
...990.0 = 10x
Calculate x - 10x:
x - 10x = ...999.0 - ...990.0
-9x = 9
x = -1
And while this is true for 10-adic numbers, it is certainly not true for the real numbers.
While I agree that my proof is blunt, yours doesn’t prove that .999… is equal to -1. With your assumption, the infinite 9’s behave like they’re finite, adding the 0 to the end, and you forgot to move the decimal point in the beginning of the number when you multiplied by 10.
x=0.999…999
10x=9.999…990 assuming infinite decimals behave like finite ones.
Now x - 10x = 0.999…999 - 9.999…990
-9x = -9.000…009
x = 1.000…001
Thus, adding or subtracting the infinitesimal makes no difference, meaning it behaves like 0.
Edit: Having written all this I realised that you probably meant the infinitely large number consisting of only 9’s, but with infinity you can’t really prove anything like this. You can’t have one infinite number being 10 times larger than another. It’s like assuming division by 0 is well defined.
0a=0b, thus
a=b, meaning of course your …999 can equal -1.
Edit again: what my proof shows is that even if you assume that .000…001≠0, doing regular algebra makes it behave like 0 anyway. Your proof shows that you can’t to regular maths with infinite numbers, which wasn’t in question. Infinity exists, the infinitesimal does not.
Yes, but similar flaws exist for your proof.
The algebraic proof that 0.999… = 1 must first prove why you can assign 0.999… to x.
My “proof” abuses algebraic notation like this - you cannot assign infinity to a variable. After that, regular algebraic rules become meaningless.
The proper proof would use the definition that the value of a limit approaching another value is exactly that value. For any epsilon > 0, 0.999… will be within the epsilon environment of 1 (= the interval 1 ± epsilon), therefore 0.999… is 1.
X=.5555…
10x=5.5555…
Subtract x from both sides.
9x=5
X=1 .5555 must equal 1.
There it isn’t. Because that math is bullshit.
Lol what? How did you conclude that if 9x = 5 then x = 1? Surely you didn’t pass algebra in high school, otherwise you could see that getting x from 9x = 5 requires dividing both sides by 9, which yields x = 5/9, i.e. 0.555... = 5/9 since x = 0.555....
Also, you shouldn’t just use uppercase X in place of lowercase x or vice versa. Case is usually significant for variable names.
Divide 1 by 3: 1÷3=0.3333…
Multiply the result by 3 reverting the operation: 0.3333… x 3 = 0.9999… or just 1
0.9999… = 1
In this context, yes, because of the cancellation on the fractions when you recover.
1/3 x 3 = 1
I would say without the context, there is an infinitesimal difference. The approximation solution above essentially ignores the problem which is more of a functional flaw in base 10 than a real number theory issue
This seems to be conflating 0.333...3 with 0.333... One is infinitesimally close to 1/3, the other is a decimal representation of 1/3. Indeed, if 1-0.999... resulted in anything other than 0, that would necessarily be a number with more significant digits than 0.999... which would mean that the failed to be an infinite repetition.
The context doesn’t make a difference
In base 10 --> 1/3 is 0.333…
In base 12 --> 1/3 is 0.4
But they’re both the same number.
Base 10 simply is not capable of displaying it in a concise format. We could say that this is a notation issue. No notation is perfect. Base 10 has some confusing implications
You’re just rounding up an irrational number. You have a non terminating, non repeating number, that will go on forever, because it can never actually get up to its whole value.
1/3 is a rational number, because it can be depicted by a ratio of two integers. You clearly don’t know what you’re talking about, you’re getting basic algebra level facts wrong. Maybe take a hint and read some real math instead of relying on your bad intuition.
I was taught that if 0.9999… didn’t equal 1 there would have to be a number that exists between the two. Since there isn’t, then 0.9999…=1
Somehow I have the feeling that this is not going to convince people who think that 0.9999… /= 1, but only make them madder.
Personally I like to point to the difference, or rather non-difference, between 0.333… and ⅓, then ask them what multiplying each by 3 is.
I’d just say that not all fractions can be broken down into a proper decimal for a whole number, just like pie never actually ends. We just stop and say it’s close enough to not be important. Need to know about a circle on your whiteboard? 3.14 is accurate enough. Need the entire observable universe measured to within a single atoms worth of accuracy? It only takes 39 digits after the 3.
Pi isn’t a fraction (in the sense of a rational fraction, an algebraic fraction where the numerator and denominator are both polynomials, like a ratio of 2 integers) – it’s an irrational number, i.e. a number with no fractional form; as opposed to rational numbers, which are defined as being able to be expressed as a fraction. Furthermore, π is a transcendental number, meaning it’s never a solution to f(x) = 0, where f(x) is a non-zero finite-degree polynomial expression with rational coefficients. That’s like, literally part of the definition. They cannot be compared to rational numbers like fractions.
Every rational number (and therefore every fraction) can be expressed using either repeating decimals or terminating decimals. Contrastly, irrational numbers only have decimal expansions which are both non-repeating and non-terminating.
Since |r|<1 → ∑[n=1, ∞] arⁿ = ar/(1-r), and 0.999... is equivalent to that sum with a = 9 and r = 1/10 (visually, 0.999... = 9(0.1) + 9(0.01) + 9(0.001) + ...), it’s easy to see after plugging in, 0.999... = ∑[n=1, ∞] 9(1/10)ⁿ = 9(1/10) / (1 - 1/10) = 0.9/0.9 = 1). This was a proof present in Euler’s Elements of Algebra.
There are a lot of concepts in mathematics which do not have good real world analogues.
i, the _imaginary number_for figuring out roots, as one example.
I am fairly certain you cannot actually do the mathematics to predict or approximate the size of an atom or subatomic particle without using complex algebra involving i.
It’s been a while since I watched the entire series Leonard Susskind has up on youtube explaining the basics of the actual math for quantum mechanics, but yeah I am fairly sure it involves complex numbers.
pi isn’t even a fraction. like, it’s actually an important thing that it isn’t
The problem is, that’s exactly what the … is for. It is a little weird to our heads, granted, but it does allow the conversion. 0.33 is not the same thing as 0.333… The first is close to one third. The second one is one third. It’s how we express things as a decimal that don’t cleanly map to base ten. It may look funky, but it works.
Cut a banana into thirds and you lose material from cutting it hence .9999
The explanation I’ve seen is that … is notation for something that can be otherwise represented as sums of infinite series.
In the case of 0.999…, it can be shown to converge toward 1 with the convergence rule for geometric series.
If |r| < 1, then:
ar + ar² + ar³ + … = ar / (1 - r)
Thus:
0.999… = 9(1/10) + 9(1/10)² + 9(1/10)³ + …
= 9(1/10) / (1 - 1/10)
= (9/10) / (9/10)
= 1
Just for fun, let’s try 0.424242…
0.424242… = 42(1/100) + 42(1/100)² + 42(1/100)³
= 42(1/100) / (1 - 1/100)
= (42/100) / (99/100)
= 42/99
= 0.424242…
So there you go, nothing gained from that other than seeing that 0.999… is distinct from other known patterns of repeating numbers after the decimal point.
0.999… / 3 = 0.333… 1 / 3 = 0.333… Ergo 1 = 0.999…
(Or see algebraic proof by @Valthorn@feddit.nu)
If the difference between two numbers is so infinitesimally small they are in essence mathematically equal, then I see no reason to not address then as such.
If you tried to make a plank of wood 0.999…m long (and had the tools to do so), you’d soon find out the universe won’t let you arbitrarily go on to infinity. You’d find that when you got to the planck length, you’d have to either round up the previous digit, resolving to 1, or stop at the last 9.
Except it isn’t infinitesimally smaller at all. 0.999… is exactly 1, not at all less than 1. That’s the power of infinity. If you wanted to make a wooden board exactly 0.999… m long, you would need to make a board exactly 1 m long (which presents its own challenges).
It is mathematically equal to one, but it isn’t physically one. If you wrote out 0.999… out to infinity, it’d never just suddenly round up to 1.
But the point I was trying to make is that I agree with the interpretation of the meme in that the above distinction literally doesn’t matter - you could use either in a calculation and the answer wouldn’t (or at least shouldn’t) change.
That’s pretty much the point I was trying to make in proving how little the difference makes in reality - that the universe wouldn’t let you explore the infinity between the two, so at some point you would have to round to 1m, or go to a number 1x planck length below 1m.
It is physically equal to 1. Infinity goes on forever, and so there is no physical difference.
It’s not that it makes almost no difference. There is no difference because the values are identical.
Any real world implementation of maths (such as the length of an object) would definitely be constricted to real world parameters, and the lowest length you can go to is the Planck length.
But that point wasn’t just to talk about a plank of wood, it was to show how little difference the infinite 9s in 0.999… make.
Afaik, the Planck Length is not a “real-world pixel” in the way that many people think it is. Two lengths can differ by an amount smaller than the Planck Length. The remarkable thing is that it’s impossible to measure anything smaller than that size, so you simply couldn’t tell those two lengths apart. This is also ignoring how you’d create an object with such a precisely defined length in the first place.
Anyways of course the theoretical world of mathematics doesn’t work when you attempt to recreate it in our physical reality, because our reality has fundamental limitations that you’re ignoring when you make that conversion that make the conversion invalid. See for example the Banach-Tarski paradox, which is utter nonsense in physical reality. It’s not a coincidence that that phenomenon also relies heavily on infinities.
In the 0.999… case, the infinite 9s make all the difference. That’s literally the whole point of having an infinite number of them. “Infinity” isn’t (usually) defined as a number; it’s more like a limit or a process. Any very high but finite number of 9s is not 1. There will always be a very small difference. But as soon as there are infinite 9s, that number is 1 (assuming you’re working in the standard mathematical model, of course).
You are right that there’s “something” left behind between 0.999… and 1. Imagine a number line between 0 and 1. Each 9 adds 90% of the remaining number line to the growing number 0.999… as it approaches one. If you pick any point on this number line, after some number of 9s it will be part of the 0.999… region, no matter how close to 1 it is… except for 1 itself. The exact point where 1 is will never be added to the 0.999… fraction. But let’s see how long that 0.999… region now is. It’s exactly 1 unit long, minus a single 0-dimensional point… so still 1-0=1 units long. If you took the 0.999… region and manually added the “1” point back to it, it would stay the exact same length. This is the difference that the infinite 9s make-- only with a truly infinite number of 9s can we find this property.
You’re being downvoted because you’re wrong.
My point was only that .9 repeating is still less than 1 in tangible measurement
If it’s less than 1, then it isn’t repeating indefinitely, which is what the ellipsis indicates.
Mathematics is built on axioms that have nothing to do with numbers yet. That means that things like decimal numbers need definitions. And in the definition of decimals is literally included that if you have only nines at a certain point behind the dot, it is the same as increasing the decimal in front of the first nine by one.
That’s not how it’s defined. 0.99… is the limit of a sequence and it is precisely 1. 0.99… is the summation of infinite number of numbers and we don’t know how to do that if it isn’t defined. (0.9 + 0.09 + 0.009…) It is defined by the limit of the partial sums, 0.9, 0.99, 0.999… The limit of this sequence is 1. Sorry if this came out rude. It is more of a general comment.
That’s not an axiom or definition, it’s a consequence of the axioms that define arithmetic and can therefore be proven.
There are versions of math where that isn’t true, with infinitesimals that are not equal to zero. So I think it is an axium rather than a provable conclusion.
Okay, but it equals one.
2/9 = 0.222… 7/9 = 0.777…
0.222… + 0.777… = 0.999… 2/9 + 7/9 = 1
0.999… = 1
No, it equals 1.
Sure, when you start decoupling the numbers from their actual values. The only thing this proves is that the fraction-to-decimal conversion is inaccurate. Your floating points (and for that matter, our mathematical model) don’t have enough precision to appropriately model what the value of 7/9 actually is. The variation is negligible though, and that’s the core of this, is the variation off what it actually is is so small as to be insignificant and, really undefinable to us - but that doesn’t actually matter in practice, so we just ignore it or convert it. But at the end of the day 0.999… does not equal 1. A number which is not 1 is not equal to 1. That would be absurd. We’re just bad at converting fractions in our current mathematical understanding.
Edit: wow, this has proven HIGHLY unpopular, probably because it’s apparently incorrect. See below for about a dozen people educating me on math I’ve never heard of. The “intuitive” explanation on the Wikipedia page for this makes zero sense to me largely because I don’t understand how and why a repeating decimal can be considered a real number. But I’ll leave that to the math nerds and shut my mouth on the subject.
Similarly, 1/3 = 0.3333…
So 3 times 1/3 = 0.9999… but also 3/3 = 1
Another nice one:
Let x = 0.9999… (multiply both sides by 10)
10x = 9.99999… (substitute 0.9999… = x)
10x = 9 + x (subtract x from both sides)
9x = 9 (divide both sides by 9)
x = 1
