[isaac21259]

That's a pretty far departure from the original "multiplication is just repeated addition". Regardless, I don't think any student would find it helpful to hear "Multiplying two real numbers is simply taking the limit of a sequence of multiplications between rational numbers that converge to the two real ones". In my country irrational numbers are introduced two or three years before limits so you couldn't teach it in schools effectively either.

[HidyBush]

Multiplication outside of positive integers is not "repeated addition".

It took us thousands of years to properly define real numbers. High school students can live without a perfect explanation, or we can just teach limits before college since they are the fundamental concept if calculus.

[isaac21259]

To clarify where I live limits are introduced in high school, irrational numbers just much earlier.

[tgv]

It's a kind of generalization of repeated addition. Once you've been taught about π, you know that 2π lies between 23 and 24, without being told about limits. It may not be rigorous, but it's a good start.

You can't teach "the truth" (whatever you hold that to be). It would set back education instead of advancing it. In this case too, perfect is the enemy of good.