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One objection to thinking of real numbers as "infinite decimals" is that it lends itself to thinking of the distinction between rational and irrational numbers as being primarily about whether the decimals repeat or not. This in turn leads to some very problematic misunderstandings, such as the one in the image below:

Yes, you read it right: the author of this book thinks that $8/23$ is an irrational number, because there is no pattern to the digits.

Now it is easy to dismiss this as simple ignorance: of course the digits do repeat, but you have to go further out in the decimal sequence before this becomes visible. But once you notice this, you start to recognize all kinds of problems with the "non-repeating decimal" notion of irrational number: How can you ever tell, by looking at a decimal representation of a real number, whether it is rational or irrational? After all, the most we can ever see is finitely many digits; maybe the repeating portion starts after the part we are looking at? Or maybe what looks like a repeating decimal (say $0.6666dots$) turns out to have a 3 in the thirty-fifth decimal place, and thereafter is non-repeating?

Now obviously these problems can be circumvented by a more precise notion of "rational": A rational number is one that can be expressed as a ratio of integers, an irrational number is one that cannot be. But you would be surprised how resilient the misconception shown in the image above can be. Related errors are pervasive: I am sure I am not the only one who has seen students use a calculator to get a decimal approximation of some irrational number (say for example $log 2$) and several steps later use a "convert to fraction" command on their graphing calculator to express a string of digits as some close-but-not-equal rational number.

If you really want to get students away from these kinds of mistakes, at some point you have to provide them with a notion of "number" that is independent of the decimal representation of that number.

For the same reason that it is incorrect to think of linear transformations as matrices, or to think of real $n$-dimensional vector spaces as just $mathbb{R}^{n}.$

What's so special about base $10$? Why not binary numbers or ternary numbers? In particular, ternary numbers would come in handy for understanding Cantor ternary sets. But apparently we should think in decimals?

This non-canonical choice is unnecessary, aesthetically displeasing, and distracts from certain intuitions to be gained from, say, the geometric view of the reals as points on a line. It's better to think of the real numbers as what they are: A system of things (what are those things? Equivalence classes of sequences of rationals? Points on a line? It doesn't matter) that have certain very nice properties. Just like vectors are just elements of vector spaces: Objects that obey certain rules. Shoving extra stuff in there distracts from the mathematics.

I think that teaching reals as infinite decimals in the first place is a mistake arising out of lack of a better description.

Here is how I present the problem of infinite decimals. How for example does one define the sum of two infinite decimals ? One is supposed to start to add decimals at the right and work to the left, as in elementary school. But there is no right most digit. So how can we define addition ? Well, OK, to salvage the idea one takes all truncations add them and then prove that they stabilize. That is we must prove that they converge. This can be done, but is is kind of messy. How do we then define multiplication ? Same thing but worse. Imagine proving the distributive law. In any case you have arrived by necessity at the concept of a Cauchy sequence.

The other reason is that infinite decimals lacks the geometric intuition necessary for understanding the topology of the reals.

Decimal notation for general real numbers is not universally intuitive. It has a number of problems:

• Arithmetic with nonterminating decimals has unfamiliar complications, because people are used to doing arithmetic from the right end.


• Some people have great difficulty accepting that representation is not unique


• Some people don't grasp the infinite nature, and instead imagine a decimal as simply having a large but finite number of digits — sometimes with the belief that decimals are inherently approximate and cannot represent a number exactly.


• Some people fail to grasp the infinite nature in a different way, only being able to conceptualize a sequence of terminating decimals

It even has severe philosophical problems; e.g. they are ill-suited for various constructive approaches to mathematics. You can't compute even the first digit of $.333ldots + .666ldots$ unless you can prove that a carry does or does not propagate in from the right. (or you can prove it is a special case, such as the numbers actually being $1/3 + 2/3$, rather than some unknown sequence you actually have to generate to find its digits)

There are pedagogical problems as well; the theory behind decimals arises from that of infinite summations, but calculus and analysis courses tend to prefer to start with limits, continuity, and similar topological notions.

Incidentally, I believe I have seen at least one text whose proof that a complete ordered field exists is by showing the arithmetic on decimal numbers is such.

The existing answers are great but have not addressed the following interesting part of the question:

Isn't it good to see how the conventional treatment is connected to, and grows out of, more 'naive' ideas?

Namely, I shall show how the idea of decimal representation leads naturally and inexorably to the idea of equivalence classes of (regular) Cauchy sequences.

First, one needs to be extremely careful because decimal representations of reals are not unique, and it is a chore even to define the basic arithmetic operations on decimals. For instance, to test whether two decimals are equal is no longer obvious, and simply defining away the problem (like stipulating that decimals cannot have endless repeating '9's) does not help! Surprised? Consider $3 times 0.333overline{3}$. Ordinary multiplication yields $0.999overline{9}$, so the easiest way to resolve this issue is to invoke a canonization at the end of each arithmetic operation. Let me spell out this process in full: If the result ends with "$overline{9}$", change all those digits to "$overline{0}$" and add $1$ to the preceding digit, as usual carrying over if needed. Not to forget the numerous cases due to the sign (positive, negative, zero).

But look! The very idea of canonization corresponds to the idea of picking out representatives under an equivalence relation over the decimals. But then a natural question arises: Why does this equivalence relation seem so weird? It considers $0.999overline{9}$ equivalent to $1.000overline{0}$, but why not any other 'endings'? Interestingly, we can understand it better by using an alternative definition that decimals $x,y$ are equivalent iff $x-y$ using ordinary subtraction (without canonization) is $0.overline{0}$. Addition and subtraction here must be defined for non-negative decimals such that we always subtract the smaller from the larger before attaching the correct sign, and then defined for other signs by the usual cases. This definition shows concretely that except for the equivalences involving "$overline{9}$" and "$overline{0}$" described above, any other pair of decimals have a nonzero difference.

But note that in the subtraction algorithm we needed to know which decimal is larger. We could define non-strictly larger by the usual comparison algorithm, but what does this comparison really mean? Actually this is easily answered if one thinks carefully about the meaning of decimals in the first place.

• "$3.cdots$" means some amount in the range from $3$ to $4$.




• "$3.1cdots$" means some amount in the range from $3.1$ to $3.2$.




• "$3.14cdots$" means some amount in the range from $3.14$ to $3.15$.

In short:

• $3.cdots in [3,4]$.




• $3.1cdots in [3.1,3.2]$.




• $3.14cdots in [3.14,3.15]$.

This is precisely what guides us to invent the comparison algorithm. Also, notice that each decimal, being a sequence of digits, corresponds exactly to a sequence of intervals that narrows with each step, such that the $k$-th interval in the sequence has width $10^{-k}$. This very nicely corresponds to one type of (regular) Cauchy sequences!

Observe now that the definition of Cauchy sequence is neater and easier to use simply because it discards the implementation details, which in the case of decimals includes the carry-over parts of the algorithms and the specific base-10 format and the strong convergence guarantee. Otherwise there really is no difference between decimals and Cauchy sequences. Finally, using the equivalence classes as real numbers is just an alternative to picking a canonical representative from each class.

The infinite decimal interpretation of $mathbb{R}$ leads to problems:
$$ 0.49999dots = 0.5
$$
If you are trying to find a bijection $left[0,1right[ to left[0,1right[ times left[0,1right[$, you might try:
$$ 0.abababababdots mapsto (0.aaaaaaadots,0.bbbbbbbdots)
$$
which does not work due to the existence of $0.4999dots$.

One answer to

What is so wrong with thinking of real numbers as infinite decimals?

is that it's somewhat old fashioned. Another is that it doesn't work really well in more advanced courses in analysis. But Courant's calculus, one of the best texts ever, shows this in Section 2 of the first chapter:

That's a screenshot from page 8; you can see it at

https://archive.org/stream/DifferentialIntegralCalculusVolI/Courant-DifferentialIntegralCalculusVolI#page/n23/mode/2up

I think it's actually good to teach people that definition of a real number when they're in university and old enough to be able to understand what they're being taught. After reading https://www.inc.com/bill-murphy-jr/science-says-were-sending-our-kids-to-school-much-too-early-and-that-can-hurt-th.html, I think that kids can learn from school better if they wait until they're older to start grade 1 so I think it's also true that when they're older, they're more able to learn that definition of a real number and the reason the teachers is using that definition. I know people probably sometimes learn something too young and misunderstand what the teacher was trying to teach them then later, they have trouble breaking their old habits of what they thought they were getting taught. For example, those who got taught what a fraction is and how to add, subtract, multiply, and divide fractions at such a young age might think the rational numbers are all the numbers and have trouble later breaking their old habits and learning that irrational numbers exist. Maybe people didn't quite get taught the decimal representation of a real number properly in elementry school. I think they should be a taught a very similar definition in university. I had an intuitive idea of some of its properties of a complete ordered field but did not think of the property of completeness on my own when I was in elementry school. Later, I once tried to figure out what a real number is and then came up with my own definition. I first constructed the dyadic rationals, the terminating decimals in base 2 then for each cut that has no boundary position as a number in that set, invented a number to lie between those cuts and saw that it does something like correspond to base 2 decimal notations that forbid trailing 1's. I don't think that once people are in university, they should be taught to derive properties from the fact that $(mathbb{R}, 1, 0, +, times, leq)$ is a complete ordered field. I know some people are thinking they can't break their old habits of thinking a decimal representation is a real number when actually the real numbers already existed and somebody invented a notation for each of them. I think it's more important to teach them not to make the unjustified assumption that the real numbers with those operations have those properties of a complete ordered field they did think of. Construction from the Dedekind cuts of the terminating decimals actually works well and from that gives and intuitive way of deciding how to represent each of them but that forbids a string of trailing 9's, different from what I was taught in elementry school that 0.999... = 1. They can then be taught what a complete ordered field is with Modern Algebra being a prerequisite to that course and be taught that +, $times$, and $leq$ have been defined and $(mathbb{R}, 1, 0, +, times, leq)$ has been proven in ZF to be a complete ordered field which is unique up to isomorphism, and that it's even easier to show that it's isomorphic to the base 2 construction than that it's unique up to isomorphism. Maybe that course itself could be a prerequisite to an even later course that teaches how to write a formal proof in ZF where one of the test questions asks you to write a formal proof in ZF that there exists a complete ordered field which is unique up to isomorphism and only those who can figure out how to write a complete formal proof will get full marks so that a job will take on average better applicants.