Tlaloc_Temporal

Ah, typo. 1/3 ≠ 0.333…

It is my opinion that repeating decimals cannot properly represent the values we use them for, and I would rather avoid them entirely (kinda like the meme).

Besides, I have never disagreed with the math, just that we go about correcting people poorly. I have used some basic mathematical arguments to try and intimate how basic arithmetic is a limited system, but this has always been about solving the systemic problem of people getting caught by 0.999… = 1. Math proofs won’t add to this conversation, and I think are part of the issue.

Is it possible to have a coversation about math without either fully agreeing or calling the other stupid? Must every argument about even the topic be backed up with proof (a sociological one in this case)? Or did you just want to feel superior?

simply accept that it has to be the case because 0.333… * 3. […] That is a correct mathematical understanding

This is my point, using a simple system (basic arithmetic) properly will give bad answers in specifically this situation. A correct mathematical understanding of arithmetic will lead you to say that something funky is going on with 0.999… , and without a more comprehensive understanding of mathematical systems, the only valid conclusions are that 0.999… doesn’t equal 1, or that basic arithmetic is limited.

So then why does everyone loose their heads when this happens? Thousands of people forcing algebra and limits on anyone they so much as suspect could have a reasonable but flawed conclusion, yet this thread is the first time I’ve seen anyone even try to mention the limitations of arithmetic, and they get stomped on.

Why is basic arithmetic so sacred that it must not be besmirched? Why is it so hard for people to admit that some tools have limits? Why is everyone bringing in so many more advanced systems when my entire argument this whole time is that a simple system has limits?

That’s my whole argument. Firstly, that 0.999… catches people because using arithmetic properly leads to an incorrect understanding of repeating decimals. And secondly, that starting with the limits of arithmetic will increase understand with less frustration than throwing more complicated solutions around.

My argument have never been with the math, only with our perceptions of it and how we go about teaching it.

This isn’t about limits of accuracy

According to who?

According to me, talking about the origin of the 0.999… issue of the original comment, the “conversion of fractions to decimals”, or using basic arithmetic to manipulate values into repeating decimals. This has been my position the entire time. If this was about the limits of accuracy, then it would be impossible to solve the 0.999… = 1 issue. Yet it is possible, our accuracy isn’t limited in this fashion.

You still haven’t shown why you’re limiting yourself to basic arithmetic.

Because that’s where the entire 0.999… = 1 originates. You’ll never even see 0.999… without using basic addition on each digit individually, especially if you use fractions the entire time. Thus 0.999… is an artifact of basic arithmetic, a flaw of that system.

Different systems for different applications.

Then you agree that not every system is applicable everywhere! Even if you use that system perfectly, you’ll still end up with the wrong answer! Thus the issue isn’t someone using the system incorrectly, it’s a limitation of the system that they used. The correct response to this isn’t throwing heaps of other systems at the person, it’s communicating the limit of that system.

If someone is trying to hammer a screw, chastising them for their swinging technique then using your personal impact wrench in front of them isn’t going to help. They’re just going to hit you with the hammer, and continue using the tools they have. Explaining that a hammer can’t do the twisting motion needed for screws, then handing them a screwdriver will get you both much farther.

Limits of accuracy isn’t algebra.

It never was, and neither is the problem we’ve been discussing. You can talk about glue, staples, clamps, rivets, and bolts as much as you like, people with hammers are still going to hit screws.

What do you mean not taught yet?

I mean those more advanced methods are taught after basic arithmetic. There are plenty of adults that operate primarily with 5th grade math, and a scary number of them do finances…

limits of accuracy

This isn’t about limits of accuracy, we’re working with abstract values and ideal systems. Any inaccuracies must be introduced by those systems.

If you think the system isn’t at fault here, please show me how basic arithmetic can make 0.999… into 1. Show me how the carry method deals with Infinity correctly. If every error is just using the system incorrectly, then a correct use of the system must be applicable to everything, right? You shouldn’t need a new system like algebra to be correct, right?

Neither of those examples use the rules of those system though.

Basic arithmetic on decimap notation is performed by adding/subtracting each digit in each place, or multiplying each digit by each digit then adding those sub totals together, or the yet more complicated long division.

Adding (and by extension multiplying) requires the carry operation, because digits only go up to 9. A string of 9s requires starting at the smallest digit. 0.999… has no smallest digit, thus the carry operation fails to roll it over to 1. It’s a bug that requires more comprehensive methods to understand.

Someone using only basic arithmetic on decimal notation will conclude that 0.999… is not 1. Another person using only geocentrism will conclude that some planets follow spiral orbits. Both conclusions are wrong, but the fault lies with the tools, not the people using them.

The system I’m talking about is elementary decimal notation and basic arithmetic. Carry the 1 and all that. Equations and algebra are more advanced and not taught yet.

There is no method by which basic arithmetic and decimal notation can turn 0.999… into 1. All of the carry methods require starting at the smallest digit, and repeating decimals have no smallest digit.

If someone uses these systems as they were taught, they will get told they’re wrong for doing so. If we focus on that person being wrong, then they’re more likely to give up on math entirely, because they’re wrong for doing as they were taught. If we focus on the limitstions of that system, then they have the explanation for the error, and an understanding of why the more complicated system is preferable.

All models are wrong, but some are useful.

Again, I don’t disagree with the math. This has never been about the math. I get that ever model is wrong, but some are useful. Math isn’t taught like that though, and that’s why people get hung up things like this.

Basic decimal notation doesn’t work well with some things, and insinuates incorrect answers. People use the tools they were taught to use. People get told they’re doing it wrong. People give up on math, stop trying to learn, and just go with what they can understand.

If instead we focus on the limitations of some tools and stop hammering people’s faces in with bigger equations and dogma, the world might have more capable people willing to learn.

I don’t really care how many representations a number has, so long as those representations make sense. 2 = 02 = 2.0 = 1+1 = -1+3 = 8/4 = 2x/x. That’s all fine, we can use the basic rules of decimal notation to understand the first three, basic arithmetic to understand the next three, and basic algebra for the last one.

0.999… = 1 requires more advanced algebra in a pointed argument, or limits and infinite series to resolve, as well as disagreeing with the result of basic decimal notation. It’s steeped in misdirection and illusion like a magic trick or a phishing email.

I’m not blaming mathematicians for this, I am blaming teachers (and popular culture) for teaching that tools are inflexible, instead of the limits of those systems.

In this whole thread, I have never disagreed with the math, only it’s systematic perception, yet I have several people auguing about the math with me. It’s as if all math must be regarded as infinitely perfect, and any unbelievers must be cast out to the pyre of harsh correction. It’s the dogmatic rejection I take issue with.

Are those algorithms taught to people in school?

Once again, I have no issue with the math. I just think the commonly taught system of decimal arithmetic is flawed at representing that math. This flaw is why people get hung up on 0.999… = 1.

Oh the fundamental math works fine, it’s the imperfect representation that is infinite decimals that is flawed. Every base has at least one.

Let me restate: I am of the opinion that repeating decimals are imperfect representations of the values we use them to represent. This imperfection only matters in the case of 0.999… , but I still consider it a flaw.

I am also of the opinion that focusing on this flaw rather than the incorrectness of the person using it is a better method of teaching.

I accept that 1/3 is exactly equal to the value typically represented by 0.333… , however I do not agree that 0.333… is a perfect representation of that value. That is what I mean by 1/3 ≠ 0.333… , that repeating decimal is not exactly equal to that value.

Decimals work fine to represent numbers, it’s the decimal system of computing numbers that is flawed. The “carry the 1” system if you prefer. It’s how we’re taught to add/subtract/multiply/divide numbers first, before we learn algebra and limits.

This is the flawed system, there is no method by which 0.999… can become 1 in here. All the logic for that is algebraic or better.

My issue isn’t with 0.999… = 1, nor is it with the inelegance of having multiple represetations of some numbers. My issue lies entirely with people who use algebraic or better logic to fight an elementary arithmetic issue.

People are using the systems they were taught, and those systems are giving an incorrect answer. Instead of telling those people they’re wrong, focus on the flaws of the tools they’re using.

In base 10, if we add 1 and 1, we get the next digit, 2.

In base 2, if we add 1 and 1 there is no 2, thus we increment the next place by 1 getting 10.

We can expand this to numbers with more digits: 111(7) + 1 = 112 = 120 = 200 = 1000

In base 10, with A representing 10 in a single digit: 199 + 1 = 19A = 1A0 = 200

We could do this with larger carryover too: 999 + 111 = AAA = AB0 = B10 = 1110 Different orders are possible here: AAA = 10AA = 10B0 = 1110

The “carry the 1” process only starts when a digit exceeds the existing digits. Thus 192 is not 2Z2, nor is 100 = A0. The whole point of carryover is to keep each digit within the 0-9 range. Furthermore, by only processing individual digits, we can’t start carryover in the middle of a chain. 999 doesn’t carry over to 100-1, and while 0.999 does equal 1 - 0.001, (1-0.001) isn’t a decimal digit. Thus we can’t know if any string of 9s will carry over until we find a digit that is already trying to be greater than 9.

This logic is how basic binary adders work, and some variation of this bitwise logic runs in evey mechanical computer ever made. It works great with integers. It’s when we try to have infinite digits that this method falls apart, and then only in the case of infinite 9s. This is because a carry must start at the smallest digit, and a number with infinite decimals has no smallest digit.

Without changing this logic radically, you can’t fix this flaw. Computers use workarounds to speed up arithmetic functions, like carry-lookahead and carry-save, but they still require the smallest digit to be computed before the result of the operation can be known.

I’m not saying that math works differently is different bases, I’m using different bases exactly because the values don’t change. Using different bases restates the equation without using repeating decimals, thus sidestepping the flaw altogether.

My whole point here is that the decimal system is flawed. It’s still useful, but trying to claim it is perfect leads to a conflict with reality. All models are wrong, but some are useful.

I never commented on the convenience or usefulness of any method, just tried to explain why so many people get stuck on 0.999… = 1 and are so recalcitrant about it.

If you can accept that 1/3 is 0.333… then you can multiply both sides by three and accept that 1 is 0.99999…

This is a workaround of the decimal flaw using algebraic logic. Trying to hold both systems as fully correct leads to a conflic, and reiterating the algebraic logic (or any other proof) is just restating the problem.

The problem goes away easily once we understand the limits of the decimal system, but we need to state that the system is limited! Otherwise we get conflicting answers and nothing makes sense.

Radioactive waste maybe. Fusion plants are likely to create irradiated parts that degrade quickly, similar to fission plants. Fusion fuel on the other hand, is gaseous, and likes to escape. Hydrogen is explosive, while helium-3 is just expensive.

Decimal notation is a number system where fractions are accomodated with more numbers represeting smaller more precise parts. It is an extension of the place value system where very large tallies can be expressed in a much simpler form.

One of the core rules of this system is how to handle values larger than the highest digit, and lower than the smallest. If any place goes above 9, set that place to 0 and increment the next place by 1. If any places goes below 0, increment the place by (10) and decrement the next place by one (this operation uses a non-existent digit, which is also a common sticking point).

This is the decimal system as it is taught originally. One of the consequences of it’s rules is that each digit-wise operation must be performed in order, with a beginning and an end. Thus even getting a repeating decimal is going beyond the system. This is usually taught as special handling, and sometimes as baby’s first limit (each step down results in the same digit, thus it’s that digit all the way down).

The issue happens when digit-wise calculation is applied to infinite decimals. For most operations, it’s fine, but incrementing up can only begin if a digit goes beyong 9, which never happens in the case of 0.999… . Understanding how to resolve this requires ditching the digit-wise method and relearing decimals and a series of terms, and then learning about infinite series. It’s a much more robust and applicable method, but a very different method to what decimals are taught as.

Thus I say that the original bitwise method of decimals has a bug in the case of incrementing infinite sequences. There’s really only one number where this is an issue, but telling people they’re wrong for using the tools as they’ve been taught isn’t helpful. Much better to say that the tool they’re using is limited in this way, then showing the more advanced method.

That’s how we teach Newtonian Gravity and then expand to Relativity. You aren’t wrong for applying newtonian gravity to mercury, but the tool you’re using is limited. All models are wrong, but some are useful.

Any my argument is that 3 ≠ 0.333…

EDIT: 1/3 ≠ 0.333…

We’re taught about the decimal system by manipulating whole number representations of fractions, but when that method fails, we get told that we are wrong.

In chemistry, we’re taught about atoms by manipulating little rings of electrons, and when that system fails to explain bond angles and excitation, we’re told the model is wrong, but still useful.

This is my issue with the debate. Someone uses decimals as they were taught and everyone piles on saying they’re wrong instead of explaining the limitations of systems and why we still use them.

For the record, my favorite demonstration is useing different bases.

In base 10: 1/3 ≈ 0.333… 0.333… × 3 = 0.999…

In base 12: 1/3 = 0.4 0.4 × 3 = 1

The issue only appears if you resort to infinite decimals. If you instead change your base, everything works fine. Of course the only base where every whole fraction fits nicely is unary, and there’s some very good reasons we don’t use tally marks much anymore, and it has nothing to do with math.

Eh, if you need special rules for 0.999… because the special rules for all other repeating decimals failed, I think we should just accept that the system doesn’t work here. We can keep using the workaround, but stop telling people they’re wrong for using the system correctly.

The deeper understanding of numbers where 0.999… = 1 is obvious needs a foundation of much more advanced math than just decimals, at which point decimals stop being a system and are just a quirky representation.

Saying decimals are a perfect system is the issue I have here, and I don’t think this will go away any time soon. Mathematicians like to speak in absolutely terms where everything is either perfect or discarded, yet decimals seem to be too simple and basal to get that treatment. No one seems to be willing to admit the limitations of the system.

Coal has caused more deaths this year than the entire history of nuclear anything has in total. This includes nuclear energy, nuclear research, nuclear medicine, nuclear irradiation (food storage), and too many orphan sources.