There's no universal agreement on what a real number is so I will construct a set with operations +, $\times$, and $\leq$ that can be proven to be a complete ordered field but won't bother to give a proof that it's a complete ordered field and will call those the real numbers. I don't know of any really simple way of describing what a real number is so I will give as simple a description as I can. The following is just my description of the whole thing, not a universal one so if you use it for teaching, you should make it clear that you are using your own definitions and explaining how you define them. Maybe if you do a really good job of researching how to teach and teach less material so that you can move onto new material at a slower rate, your students will learn it really well. For example, the course should not teach a not very useful concept and how to make calculations about it because that might make there not be enough time to teach more useful material without teaching it too quickly such as the definition of a real number that I will later give that I think should be taught in full to make other course work later easier and another course could teach people background information from which they will later be able to easily learn so many different previously hard to learn math concepts instead of spending so much time teaching only one of them and then from that, they will be able to figure out how to write a formal proof in ZF that there is a complete ordered field which is unique up to isomorphism when they are asked to do so on a test and should only get full marks if they write the formal proof properly so that a future job can get better people which are those who have shown that they have the ability to figure out how to write a complete formal proof the way they were told to and not just an intuitive proof that they cannot figure out how to break down into a complete formal proof. Maybe you could join a research group and give your ideas on how a text book for that course should be written.
First, we construct the integers with the operations of +, $\times$ and $\leq$. For each integer $x$ that's not a solution to $10 \times y = x$, we can invent a solution, $y$ to the equation. Each of the newly invented numbers $y$ is still not a solution to $10 \times z = y$ so for each of those numbers we can invent a solution. We can keep going for ever. There is only one intuitive way to define addition, multiplication, and inequality on the set of all numbers that can be constructed that way There is also only one intuitive way to define a notation for each of those numbers. Take any subset of that set that's nonempty and its complement is also nonempty and for every member of the set, all smaller members are in the set. Let's say a boundary number of that set is one where all smaller numbers are in the set and all larger numbers are not in the set. Some of those sets have boundary numbers and others don't. For each of those sets that doesn't have a boundary number, we can invent a number that's larger than all numbers in the set and smaller than all numbers in its complement and then call all the numbers we've constructed so far real numbers. Now, there's only one intuitive way to define inequality on that set and the notation for each number in that set and it does not invent a second notation for numbers I already invented a notation for such as 0.999... for 1. Rather we say the notation 0.999... is undefined. Now that the notation and inequality on that set has already been defined, there is exactly one way to define 1, 0, +, $\times$, and $\leq$ on that set that I will call $\mathbb{R}$ such that $(\mathbb{R}, 1, 0, +, \times, \leq)$ is a complete ordered field and + and $\times$ restricted to terminating decimals matches what I previously defined and the multiplicative identity 1 has the notation 1 and the additive identity 0 has the notation 0.
Now the way to add 2 notations is by determining which numbers they represent and figuring out the sum then writing the notation of the sum. It's probably better to start all over and do it how you would if you only began school now and try to figure out how you would do it if you didn't have the past experience of how you were taught to add base 10 notations in grade 1. 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 learn better from school if they start it when they're older. Maybe you were so young when you learned how to add that you misunderstood what they were trying to teach you and then had trouble breaking your old habits of thinking it's a universal rule. I once had an interview at Earls and they said they do their job a totally different way and new people tend to learn it better then people with previous experience because people with previous experience tend to have trouble breaking old habits.
That method of adding does not work on nonterminating real numbers when using my notation. The method was to work on the digits going from right to left but there's no last digit to start on. If we decide to instead use a notation that allows for a string of trailing 9's, then there is another simple way to do it that given any 2 notations can produce a notation that represents the sum or product of the two numbers those are notations for. I won't explain how to do multiplication with it but for addition, you can work from left to right. If the digits of one column add up to less than 9, the column to the left has no carry. If they add up to more than 9, the column to the left has a carry. If they add up to 9, the column to the left has a carry if and only if that column has a carry. For some pairs of positive real numbers, for some columns, to determine whether it has a carry, you have to first determine whether the one to the right has a carry but to do that, you first have to determine whether the one to the right has a carry and that will keep going for ever. If you are told the information that starting from a certain point, the digits in each column always add up to 9, then you know the notation for the sum. Now what about subtraction and division.
You may have learned about fractions in elementry school and misunderstood what you were getting taught and rather than understanding that a fraction is really one notation for division of integers which is defined for all pairs of real numbers whose denominator is nonzero, you instead thought of the rational numbers as being constructed from the integers and thought there were no numbers but rational numbers. Rather, division on $\mathbb{R}$ is defined in terms of multiplication on $\mathbb{R}$ and when you then restrict the already existing multiplication and division operations on $\mathbb{R}$ to $\mathbb{Q}$, it ends up agreeing with the way you learned how to multiply and divide fractions earlier and you must reject the assumption that the rational numbers are the only numbers unless you're fine with using only the rational numbers and saying there is no solution to the equation $x^2 = 2$. We can still define the property of being a rational number such that either its square is less than 2 or it is negative and then define $\sqrt{2}$ to mean that property.