Some students really struggle to learn fractions. Not only that but also, once they've mastered an understanding of real numbers, they can learn about fractions so much faster and more efficiently later. Maybe teaching real numbers renders teaching rational numbers totally unnecessary. In addition to that, teaching rational numbers too early might cause some people to form the misconception that rational numbers are the only numbers. I think so because I read on the internet about people who don't understand how irrational numbers exist. This question and this page seem to support my theory. Also when they omit unnecessary material, they can fit in more other material while still moving slowly enough that all the students can actually keep up with learning what school is trying to teach them.
It may seem wierd but it might be better to use the base 2 notation for the fractional part of a real number and the base 10 notation for the integer part because of they way they construct the real numbers. Real numbers could be taught as follows. First we can define a natural number as a finite ordinal number. Next, we invent the negative numbers and then redefine +, $\times$, and $\leq$ on them. Next, since each odd number $x$ is not a solution to $2 \times y = x$ in the integers, we invent a solution to that equation in $I$. I know that's how I say it but for them, it's better not to introduce a variable and just say that none of them get you that number when you multiply 2 by it. Let's call each invented solution a half integer. Each half integer $y$ is still not a solution to $2 \times z = y$ so we can again invent a solution to each of them. Now it's easier to define +, $\times$, and $\leq$ on this system than it is to teach fractions and how to multiply and divide them and determine which of two is greater. Some people may quickly figure out that not all numbers can be gotten by multiplying a number by 3 in this system and get confused but the teacher might just have to explain that that's how the system was defined and that they will later teach them a different system where there is a solution to $3 \times x = 1$. They can later be taught the concept of my definition of a Dedekind cut of the dyadic rationals which is not the real definition and my definition is a subset of them that has the property that it is not empty and its complement is not empty and for any dyadic rational in the subset, all smaller dyadic rationals are in the subset. They might start to notice that for some cuts, that cut has a maximal element and for some cuts, its complement has a minimal element and for some cuts, it has no maximal element nor does its complement have a minimal element. Now there's on obvious one-to-one correspondence from the cuts between 0 and 1 to all the functions from $\mathbb{N}$ to {0, 1}. However, we want to invent a new number for the cut only when there isn't already a maximal element of the cut or a minimal element of its complement. Now this gives an obvious binary notation for the fractional part of each real number but that notation forbids a string of trailing 1's just like some authors forbid a string of trailing 9's.
We can then redefine +, $\times$, and $\leq$ on this system. - and $\div$ in this system are just defined in terms of + and $\times$ in this system. We can show that multiplication can be defined in that way in that system and that in that system, multiplication by any nonzero number is bijective and squaring restricted to the nonnegative numbers is also bijective on the nonnegative numbers. Now that they already know a lot of the laws of real numbers used as some of the defining criteria for a complete ordered field, all we need to do is teach them how to divide any real number by any nonzero real number and then say a real number is defined to be a rational number if and only if for some integer $p$ and nonzero integer $q$, it is $p \div q$. Then they might quickly figure out so many properties of rational numbers that some students are struggling to learn.
