There are a number of conceptual errors in your question.

*In mathematics, as far as I know, you can't divide 100% by 3 without having 0,1...% left.*

In mathematics, "100%" means nothing more or less than "100 per 100", namely "$100 / 100 = 1$. So in mathematics you **can** divide $100\%$ by $3$ without having $0.1\%$ left. $100\% / 3 = 1 / 3 = \frac13$.

*Imagine an apple which was cloned two times, so the other 2 are completely equal in 'quality'. The totality of the 3 apples is 100%. Now, you can divide those 3 apples for 3 persons and you will get 100% divided by 3 and none left.*

Here you correctly observe that if you consider a group of 3 identical apples, and count those apples as 100%, then you can divide that group up into 3 (equal) parts and get none left. As explained above, this is not contradictory to "dividing 100% by 3".

*Is this because 1: mathematics is not real 2: there is no 1 or 2, and it's in fact just an invention for measurements? So, in the division of 100% by 3 WITHOUT any left, is NOT accurate?*

No.

The mathematics concerning integers and rationals and real numbers is **very real**, in the sense that using **purely logical reasoning** we can from **mathematical axioms** prove rigorously and unambiguously many statements about integers and rationals and reals that **can be interpreted** as statements about real-world phenomena, and in such a way that these statements can be **empirically verified**. More specifically, this part of mathematics has both **explanatory power** and **predictive power** about the real world, meaning that it provides succinct explanation for phenomena that we have observed and that it allows us to predict future phenomena that we have not yet observed.

For instance, right now the webpages you are reading from https://math.stackexchange.com have been encrypted using RSA encryption, and the decryption process relies on Fermat's little theorem to be true when interpreted as a statement about large positive integers encoded in your computer in binary format (with the addition and multiplication operations defined according to how your computer does them)!

In other words: If Fermat's little theorem (interpreted as a predictive statement about your computer) fails to be true, then you simply would be unable to read these webpages!

It turns out that Fermat's little theorem can be proven in a suitable form in a mathematical theory called PA (First-Order Peano Arithmetic). And so far, every single known theorem of PA has not been found false under the standard interpretation as a real-world statement. So we definitely have good reason to believe that "**mathematics generated by PA is real**".

As for $1$ and $2$ being inventions, the **abstract concept** of the natural numbers as a model of PA (i.e. they satisfy all the axioms of PA) is indeed a human invention, but that does not necessarily imply that this abstract concept has no real-world meaning, as explained above.

To make it clear, 100% can be divided exactly by 3, both in mathematics and in a suitable interpretation of the concepts "100%" and "3" in the real-world.

~ ~ ~

Coming back to your mention of "$0.1\%$", it is actually due to a conceptual mistake. It is a (mathematical) fact that $100$ is not an **integer multiple** of $3$. That is, there is no integer $k$ such that $100 = 3 × k$. However, there **is a rational $r$** such that $100 = 3 × r$. When we write "$100\% / 3$" in mathematics, we mean **exact division**, which in this case yields the rational number $\frac13$ as the answer.

If you punch a basic calculator (which is designed to show you only some numbers in base ten) and ask for `100% / 3`

, then you may get something like `0.333333333`

. Why? It is **not because** the answer is $0.333333333$. Rather, the calculator is **unable to show the answer in base ten**. Furthermore, if you set your calculator to round the answer to 3 decimal places, then you would instead get `0.333`

, **only because** you asked for the answer rounded to 3 decimal places! So the (mathematical) fact that $100\% = 0.333 × 3 + 0.1\%$ does **not** actually have anything to do with the division of $1$ by $3$.

By the way, this phenomenon involving calculators is an artifact of our choice of base system for general daily life, which influences the design of such calculators. This has nothing to do with the mathematical concept of division. If humans had only 3 fingers on each hand, we might have ended up using base six for daily life.

We can in fact show precisely what would have happened if we used base $6$. We (using base ten) write $21_6$ to mean the number represented by the string of symbols "21" in base $6$, which means $2 × 6 + 1 = 13$. Likewise, in base $6$ we would have $1 / 3 = 0.2_6$ because $\frac13 = \frac26$. Note that we also have $\frac13 = 0.333\overline{3}$, where the overlined "$3$" means that "$3$" is repeated forever. There is no contradiction here. Whenever we do not write the subscript on a number, it is **by convention** understood as a base ten number. To make clear, $\frac13 = 0.333\overline{3}_{10} = 0.2_6$.