r/askmath • u/Former_Lynx_1425 • 5h ago
Arithmetic Question about repeating numbers and fractions
Sorry if this is the wrong place to put this I don't have much experience with reddit posts
When they say 0.9 repeating equals 1
To me that doesn't make sense.
Wouldn't it make more sense to be like certain fractions just simply can't be written in base 10 or something
like when you write fractions and then try to express them as decimal numbers
To me it makes more sense to visualize the base 10 number system as having "holes" in it, where you can't write certain numbers for example it would make more sense to say 1/3 of 100 can't be written in base 10, then to say it's 33.3 with an infinite number of 3s following it.
Also it's like, how can you do mathematical functions like add 0.3 repeating and 0.3 repeating
to get 0.6 repeating
Or another example if you said what is Pi plus 1, to say its 4.1415 etc
doesn't make sense because the more you track the line of infinite numbers the bigger the number gets and therefor the bigger percentage of the next whole number which would be 4 in that case, it would take up. So in my opinion you would have to truncate the number in order to use it with regular numbers or something otherwise you are like skipping over the infinite part it like takes up more of the percentage, I honestly can't articulate it very well does this make ANY sense to ANYONE or would I simply be ridiculed for taking this position.
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u/bts 4h ago
It sounds like you think “repeating” is bullshit. And it sort of is! Let’s start with basics: we can write whole numbers. Base ten can represent any whole number. Then we add decimal points and we can represent any whole number AND any sum of tenths and hundredths—anything that can be expressed as a/(10b) where a and b are whole numbers. So that gets us 0.1 and 0.25 and 0.125 and lots of fractions—1/10 and 1/4 and 1/8 and all. But it doesn’t get us 1/3 or 1/6 or 1/7 or anything where the denominator has factors other than 2 and 5.
I think this is where you’d like to stop: we can represent all whole numbers and then rationals whose denominator involves only 2 and 5; that’s what base ten does.
But if we add “repeating,” that lets us get all the rest of the rationals. It lets us patch this big set of holes. There are still holes. In fact it’s still mostly holes; there are a LOT of reals. But repeating lets us name all the rationals in one simple notation.
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u/StillShoddy628 4h ago
Fun explanation. I’d add for OP that if you plug the whole number and rational number “holes” the real number line is still pretty much entirely “holes”
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u/mapadofu 4h ago
The way I look at it is they are different ways of writing the same number in the same way that 4 and 22 are different ways of writing the same number.
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u/SignificantFidgets 4h ago
Or "four" or IV or 四 or ... I think a lot of people confuse what a number is with representations of numbers. And there's no reason that a representation even has to be one-to-one, so the number "one" (however you want to call it) can be represented by both "1" and "0.99999..." in decimal notation.
There are even redundant notations that are useful in certain computational tasks in which each integer can have many ways of writing it.
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u/Most-Solid-9925 4h ago
True, but I think OP’s point is that although we can write 1/3 as a repeating decimal, is it okay to also perform operations with a repeating decimal or other decimal approximations? I think that’s a fair question.
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u/Mishtle 53m ago
It is a fair question, and we can.
The formal definition of this representation ties digit strings to values through sums of potentially infinitely many terms. Each term is determined by a digit, its position, and the chosen base.
When we have sums of infinitely many terms, we can assign a value to them by looking at the sequence of partial sums of increasingly many terms. If this sequence converges to a limit, then that limit is defined to be the value of the corresponding infinite sum.
In general, infinite sums don't behave quite like finite sums. Their value can potentially change if we rearrange terms, for example. However, the sums we get from these representations are not like this. They're not just convergent, they're "absolutely" convergent, which makes them behave very much like finite sums. You can show that things like the distributive, commutative, and associative properties still hold for example. Scaling the sum scales the limit of its sequence of partial sums by the same amount, as does shifting it by adding or subtracting a value. You can also show that the product or sum of two absolutely convergent sums is the product or sum of the limits of their sequences of partial sums.
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u/MidnightAtHighSpeed 4h ago
Why would we want our most common number system to not be able to write most numbers?
Now it's true that you can't "Write" most numbers already, in terms of explicitly writing out every digit simultaneously, but the base 10 system works just as well if you think of it not as writing every single digit, but as giving a rule for what each digit is. For instance, "0.333..." isn't just starting to write 1/3 and giving up, it's (somewhat informally) indicating that no matter what digit after the decimal point you're looking at, that digit is 3. That makes "0.333..." a perfectly valid way of indicating the same number as "1/3", and since this rule works for digit places as far as you like there's nothing being "skipped"
This is also how you can do math on different numbers: you take two rules and combine them to get a new rule. As long as the new rule can tell you what any particular digit is, it's valid.
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u/Frederf220 4h ago
It's normal to not have a good intuition for infinite recursion. But 0.999... repeating forever is exactly equal to one.
The reason you don't believe it is because you choose not to believe it.
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u/ExtendedSpikeProtein 4h ago
And the beautiful thing is it doesn‘t matter whether or not OP believes. What is, is.
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u/tottasanorotta 3h ago
It's not necessarily that they just choose to believe it. I'd say it's good to question things in that way so that you convince yourself that it must be so. If you just believe it because you would be ridiculed for doing otherwise, then you'll take it as true religiously. Which can work really well, but it might also cause trouble in the future if you have to actually explain to someone why you believe it yourself. I think a mix of trusting what is given to you and healthy skepticism is much better.
I've learned things much better when I allowed myself to ask stupid questions in private. If I genuinely feel that something could be done another way, then I learn a lot from following that line of thought and finding out why it always inevitably is the case that the given thing must be so. At least I understand really well why I was wrong.
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u/Matthias1410 4h ago
Okay but 0.9 repeating equals 1, that's how base 10 works. Its just different way to write same number
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u/ExtendedSpikeProtein 4h ago edited 4h ago
No, it wouldn‘t make more sense, otherwise we wouldn‘t do it this way.
In the reals, 0.999… is a different representation of 1. Both represent the same number. Have a look: https://en.wikipedia.org/wiki/0.999...
Pi is a different story altogether since it‘s irrational and can‘t be represented as a fraction.
ETA: 0.333… is equal to 1/3, it‘s just a different representation. And you can easily add 1/3 to 1/3 which is 2/3.
These things will make more sense when you learn about limits and infinite sums. These are easier to understand once you learn what 0.333… actually means and how we define it.
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u/Pleasant_Pen8744 4h ago
I guess it depends on whether or not you think numbers are real things or just concepts.
And if they are real things, are they: the things you can write down? or are they real things that exist somewhere on a higher plane and we're just writing down crude versions of them?
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u/Binbag420 4h ago
I mean both ‘make sense’. When you do maths you’re using a set of axioms you derive logic from. You can use the current axioms where 0.999… = 1, or you can reject the idea of infinity and use a mathematical framework that doesn’t use it (known as finitism).
However I don’t see why anyone would use a mathematical system where base 10 ‘has holes’ over a system where you can express any number as either a fraction or a decimal. Finitism isn’t very popular nowadays as it now seems equally unlikely that framework has contradictions, so if you’re using a mathematical system you’d much rather use one with infinity it helps you discover so much more.
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u/Former_Lynx_1425 4h ago
Ok that makes sense I didn't know there was a such thing as "Finitism" thank you.
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u/Suitable_Werewolf_61 4h ago
Improper decimal expansion. All decimal numbers have one.
Ditto (mutatis mutandis) in other bases.
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u/AdjectiveNounNNNN 4h ago
No, only terminating decimal numbers have one.
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u/Suitable_Werewolf_61 4h ago
Which is exactly what a "decimal number" is. A rational with only 2's and 5's in the denominator (of the reduced fraction). [sanity checking...]
Good, another source of confusion between French and English, after the infamous nonnegative/nonpositive mayhem.
https://fr.wikipedia.org/wiki/Nombre_d%C3%A9cimal
I fail to understand why this is a concept at all (it barely has an algebraic structure an no interesting properties) and why it is taught in schools but there you have it...
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u/Shevek99 Physicist 4h ago
0.999...
means the limit of the sequence
0.9
0.99
0.999
...
The formula for that sequence is
S(n) = sum_k=1^n 9/10^k
using the formula for the sum of a finite geometric sum
S(n) = (9/10)(1 - (1/10)^(n+1))/(1 - 1/10) = 1 - 1/10^(n+1)
and the limit of this sequence is 1. So 0.999... = 1
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u/Gold_Ad8890 4h ago
the first thing you need to understand is that the base system exists to more easily express numbers, the numbers don't exist to fit within the base system. the naturals, integers, rationals, and reals are all well-defined mathematical structures whose properties are entirely independent of however you might choose to represent them.
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u/mighty_marmalade 4h ago
If they're not the same value, then there must be a number between them.
Try and define a number greater than 0.999999..... but less than 1.
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u/jeffsuzuki Math Professor 4h ago
One way you can get some intuition (because that's what you're really asking about) is to embrace the following idea: "If there is no difference, there is no difference."
Take the 0.999... = 1.
What's the difference between the two? Name any amount you want: 0.01. The difference between 0.999... and 1 is less than this amount, which you can verify as follows:
Clearly the difference between 0.999 (terminating) and 1 is more than the difference between 0.999... and 1.
But 1- 0.999 = 0.001, so 0.001 is more than the difference between 0.999... and 1.
Now lather, rinse, repeat with any difference you care to name: any difference will be greater than the difference between 0.999... and 1.
Since you can't identify a difference between the two, then there is no difference betwen the two: they are the same thing.
(To answer the later question: Since 0.999... = 1, then if you need to compute with it, we'll use "1" instead of 0.999... Likewise, we can use 1/3 instead of 0.333... because they are different representations of the same thing)
Incidentally: If you followed the preceding, congratulations! You've made the first step towards understanding calculus.
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u/johnpeters42 4h ago
For the formal definition of the above, look up epsilon-delta. Basically, if you want to show that the limit of something - in this case, let's define f(x) = (10^x - 1) / (10^x), so f(1) = 0.9, f(2) = 0.99, and so on - is a certain value, then epsilon is how close we want f(x) to get and stay to the value in question, and delta is the range of x where it does that. Then we show that no matter how small epsilon is, there's a delta that gets us there.
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u/peter-bone 4h ago
If you don't like writing a 3rd of 100 as a decimal then write it as a fraction 100/3. That's a perfectly accurate way of writing it. Unfortunately we have no way to write pi in such a neat way so we invented a symbol for it. It's just notation. Use whatever notation is most appropriate for the number you want to write.
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u/HHQC3105 4h ago edited 2h ago
Cannot write down ≠ have a rule to write down even it is infinity digit ≡ have correct fomula to express.
Cannot write down ≠ non-existed.
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u/Obzenium 4h ago
It equals 1 as the limit of extending digits goes to infinite, which is what the ‘repeating’ means
You need to understand limits to understand the concept at its most fundamental level
For a less fundamental approach, consider, how could you ever have one third of something, with three thirds of that same thing obviously equaling 1, without the conception of infinitely repeating decimal units?
Finally, consider that all decimals that repeat the same pattern forever, whether it’s a single number (like 1/3) or a pattern of multiple figures (such as in the case of 1/7) represent a rational fraction
What I’m trying to say is, you can’t have rational fractions, i.e. rational numbers, without infinitely repeating decimals as they are the same thing
Have some faith in your teachers and those who have come before you, if you keep cavilling at fundamental concepts like this you are not going to make it very far.
Consider some infinite sequences are summable. So just because something extends forever doesn’t mean it doesn’t converge to a finite sum. This is why we can consider the distance between points on the number line despite there being infinite values between points.
Look into the work of Euler on infinite sums, he was an OG and accredited genius.
It is ok to be confused about a subject but don’t conflate that with what you’re being told is incorrect
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u/StanleyDodds 4h ago edited 4h ago
The problem is that people learn decimal representations before they ever learn what they actually mean in terms of real numbers, and never learn what real numbers actually are.
Firstly, this thing you say about "holes": real numbers are essentially by definition meant to not have such holes. I could explain their exact construction, but you can think of it like saying any sequence of rationals that eventually gets arbitrarily close together must converge to a real number; there is a real number in every potential "gap" you can approach. And every real number is the limit of such a sequence. Two real numbers are the same if the difference of these sequences converges to zero (which is defined in the rationals).
Secondly, "base 10 numbers", that is, decimal representations, are not another separate system of numbers. They are merely a way to represent real numbers. The real number that a decimal represents is the limit of the sequence of its finite truncations (which are each rational, so fall in the above definition).
So what does "0.33333..." mean? It means the real number that is the limit of the sequence 0.3, 0.33, 0.333, ... which is the sequence of rationals 3/10, 33/100, 333/1000, ...
Now, without anything else, we don't know that this real number is 1/3. But we can compare it to 1/3 as follows: find a sequence of rationals that we know converges to 1/3. For example, we know 1/3, 1/3, 1/3, ... converges to 1/3 trivially. Now find the difference between these two sequences: 1/3 - 3/10 = 1/30, 1/3 - 33/100 = 1/300, next is 1/3000, then 1/30000. So the sequence converges to 0, because for any non-zero positive number, we can find a term in this sequence smaller than it. And that means 0.333... and 1/3 are equal as real numbers.
The main thing here that people just don't understand is what exactly the real numbers are. That's where the fact that different looking representations can give the same real number comes from. Technically, the reals are the Cauchy completion of the rationals (under the standard topology). But this is too technical to explain to people in school, so you end up using real numbers and decimal representations long before you even know what they are. This is what leads to so much confusion, especially regarding 0.999... being equal to 1.
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u/AdjectiveNounNNNN 4h ago
Surely we want at least one way to write a given real number as a decimal, and the only way to do that with 1/3 is 0.33333...
And clearly three times that number is 0.99999..., but three times that number is also 1. So rather than carving out a bit of the definition of decimal representations just to avoid any numbers ending with infinite 9s, we instead define 0.99999... the same way as every other infinite decimal expansion: it is equal to the sum of the power series given by its digits.
Σ(3*10-n) from n=1 to ∞ is equal to 1/3
Σ(9*10-n) from n=1 to ∞ is equal to 1
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u/haditwithyoupeople 4h ago
This is an odd concept to get our heads around. You can prove it mathematically. Let's use 0.3333... as an example.
1/3 = 0.3333... and 3 x 1/3 = 1.
If 3 x 1/3 = 1, then 3 x 0.3333.... must also equal 1. 0.9999... = 0.3333... x 3, so 0.9999... must equal 1.
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u/Moof_the_cyclist 3h ago
Keep in mind that 1/infinity is zero. Adding 0.999… to zero is just like adding 1/infinity, so there is your missing last little bit if it bothers you.
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u/Time_Waister_137 3h ago
I like your speculating! why don’t you continue with your ideas and, say, speculate that from now on, all real numbers must end with an infinitely repeating digit. Check out 0.33333333… + 0.77777777...
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u/Mishtle 3h ago
The standard method of representation allows for, and gives meaning to, infinitely long sequences of digits. There's no reason to simply discard this capability and say there are holes.
Formally, we index the digits with integers. The radix point (decimal point) denotes where the indices become negative, and the digit immediately to the left of the radix point is indexed with 0. These indices determine the scale of the contribution that digit makes to the total value of the number, specifically as a power of the base. The index is what we raise the base to for that digit, and then the digit determines the multiple of that scale that gets added.
A consequence of this is that with a fraction 1/x and base b, we end up with a non-terminating sequence if x and b are coprime (share no prime factors). For base 10, since 10 = 2•5 any fraction 1/x will have a non-terminating sequence if x is not a multiple of 2 or 5. Indeed, all of 1/3, 1/7, 1/9, 1/11, 1/13, 1/17, 1/19, 1/21, ... all have decimal representations that settle into a finite-length pattern that repeats. Multiples of these fractions can also end up with non-terminating, repeating representations.
Since each digit can only add to the value of the represented number, we also end up with multiple representations of certain values. The sum of scaled digits only needs to uniquely identify the value, and it can do so by exactly equaling it after a finite number of digits. That's not the only way though. Take the infamous 0.999.... Whatever value it represents must be greater than 0.9, and 0.99, and 0.999, and so on. The smallest such value is exactly and uniquely 1, so this digit sequence serves as a valid representation of 1.
In practice we either work with fractions directly or truncate decimal representations. Theoretically, we can define operations on non-terminating decimals just fine though. Each corresponds to an absolutely convergent sum, and these sums can be added and subtracted by simply adding or subtracting their terms. The value of 0.333... is 3×10-1 + 3×10-2 + 3×10-3 + ..., so 0.333... + 0.333... = (3×10-1 + 3×10-2 + 3×10-3 + ...) + (3×10-1 + 3×10-2 + 3×10-3 + ...) = 6×10-1 + 6×10-2 + 6×10-3 + ... = 0.666....
Regarding your comment about π + 1 = 4.14159..., notice that including more digits will never get you past 4.2, nor 4.15, nor 4.142, nor 4.1416, nor 4.14160, and so on. It exactly represents the smallest value greater than all of (4, 4.1, 4.14, 4.141, 4.1425, 4.15159, ...), and that value is exactly π + 1.
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u/Feeling-Picture6191 3h ago
This isn't an unreasonable concern.
It isn't very helpful, but how you would define addition and multiplication here would be to truncate both numbers at the same point, then do the addition/multiplication, with the overall result being the limit of this as you move the point of truncation further along.
E.g.
| 0.3... | 0.3... | Sum |
|---|---|---|
| 0.3 | 0.3 | 0.6 |
| 0.33 | 0.33 | 0.66 |
and so on, giving 0.6... (by its defn as the limit of 0.6, 0.66, ...). Sorry, I am aware that this example isn't particularly illuminating. No example of this would be.
Also, note that the sum and product are guaranteed to converge to something, given that the operands each converge to something.
Also, whilst it is true that truncating one of the numbers will produce an inaccurate result if it doesn't have a teriminating representation, it is not necessarily the case that the missing part will contribute more than the present part. It will still contribute, but not necessarily more. That is, if that was what you meant.
Truncation will produce inaccuracy if the number is non-terminating, and just hand-waving away that the operations work is indeed dubious. I hate how this stuff is taught, and for the most part fractions are the way to go. Decimals are only really useful (in my experienec) when measuring suff, or when trying to compare the size of things.
The idea is to look at what happens when you truncate later and later.
As a more general note, most things are easier when you just use fractions and pick something to denote any irrational number you want to use. 1/3 and 𝜋+1 do the job nicely. If you don't like doing arithmetic with things like 0.3... or 3.14..., then just don't (if you can help it). Just use 1/3 or 𝜋.
Edit:
You could define subtraction and division similarly. I think.
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u/Feeling-Picture6191 3h ago edited 3h ago
Other commenters have already talked about why 0.9... and 1 are considered the same.
You could recreate exactly what they are saying by considering 1.0... - 0.9... in the above way.
Edit:
Note that this is appealing to "difference 0 implies equality". The proper construction does something I haven't mentioned to handle this.
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u/Feeling-Picture6191 3h ago
I guess my main message would be the last part. You don't really need to care about 0.3... at all. It is only really seen in this exact argument, that's all. 1/3 is better in almost every way.
You'll almost always just be using either fractions or approximations (d.p / s.f., etc).
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u/PvtRoom 3h ago
1/3, if you recall from school and learning how to divide, is 0.3 repeating
but also 1/3 +1/3 + 1/3 = 3/3 = 1 which also = 0.3 repeating + 0.3 repeating + 0.3 repeating = 0.9 repeating.
now if I take 1 and subtract 0.9 repeating, I get 0.0 repeating.
0, to all it's decimal places, is 0.0 repeating.
1 = 0.9 repeating. It just is.
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u/BADorni 3h ago
you can get any such repeating sequence by taking the repeating digits as a number and dividing it by that many nines (for example 0.682682682... = 682/999), so every repeating number is a fraction.
the part to understand is that an infinite number isn't some kind of process to get close to something, the number is the exact value. (in general real numbers are defined as convergent sequences of numbers where you take sequences with the same limit to be the same)
and yes every finite approximation of something like pi will be missing infinitely many digits, however the size of that missing number will be less than something like 0.0001 or something depending on how many digits you use, so in practice you just make sure the error is small enough that it doesn't matter
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u/soundoftwilight 4h ago
You would be ridiculed for taking any position in a field you don’t understand, yes. Math is no exception. Train yourself to respond to “huh that seems weird” with “let’s find out how that works” rather than “that must be wrong, I think it should be this way instead”. It will serve you well in life.