[hervature]

Could not disagree more with this take. Multiplication of fractions is simply the division of two whole number multiplications. Which, if you are teaching fractions, division has already been taught. Seems like a contrarian take for the sake of being contrarian rather than based on pedagogy. Glad he wasn’t my teacher as he would have confused me.

[isaac21259]

What about irrational numbers? There's no neat way to view multiplication of two irrational numbers as repeated addition. And even if there were a way I don't think it's a useful way to think or teach after the first couple years because it makes obvious things like √2×√2 = 2 seem weird and mysterious.

[hervature]

Others have handled the appropriate construction of irrational numbers. The claim here is not that every student needs to know irrational numbers are a limiting sequence but this is exactly how mathematicians think of it. So it is strange to hear another mathematician claim that repeated addition is somehow lying. At some point, people kind of just accept that it is just something you punch into your calculator and don't even think about it anymore.

Your example is quite bad because sqrt(2)*sqrt(2) = sqrt(2*2) = sqrt(4) = 2. So repeated addition works fine. Let's focus instead on pi*pi. The way calculators do this is precisely as some type of limiting sequence depending on how much precision you want. Because, one cannot "calculate" pi*pi exactly because it is irrational. So, you have 3*3, then 3.1*3.1, then 3.14*3.14, etc. which are all repeated additions with some division (e.g. 314*314/(100*100)). In reality, when multiplying two irrational numbers, we just use enough decimal points for floating point precision and then chop off any potentially erroneous digits after the multiplication.*

[HidyBush]

Irrational numbers are limits of sequences of rational numbers. Multiplying two real numbers is simply taking the limit of a sequence of multiplications between rational numbers that converge to the two real ones.

[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.

[afiori]

Multiplication is repeated additions is the informal way of stating the distributive property of multiplication and addition.

Probably you were taught how to multiply irrational by the property of powers (a^b * c^b = (a*c)^b etc.).

You were not taught a grand unifying theory of multiplication, you were taught how to manipulate operations to turn them into more useful operations.

Teaching these laws also prepares you for when a and b are just symbolic reals with no structure and those laws are the only thing you can use to manipulate them.

[naniwaduni]

You don't need a rigorous notion of limits to informally notice that irrationals have arbitrarily close rational approximations, e.g. by adding successive digits.

[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.

[orangecat]

I agree with your disagreement. By this standard we shouldn't teach F=ma in introductory physics, and we should require kindergartners to understand the ZFC axioms before we can tell them what "3" is.

[hervature]

That's actually the exact opposite of what I'm saying and the exact approach the author is saying. It is unclear what they are proposing, but it smells awfully similar to jumping to modern understanding mathematics in one shot to avoid "repeatedly lying" to students.

As others have pointed out, repeated addition as multiplication readily extends to rational numbers, then to irrational numbers as limits of rational sequences. This is exactly the progression that is taught in Rudin's analysis book and the way to construct the real numbers. At no point in time do you need to backtrack on repeated addition but you need to introduce new concepts division and limits. This is exactly teaching F=ma and then introducing relativity and quantum as the students gain more depth and break past the classical setting.

[orangecat]

Sorry, I was trying to agree with you and apparently phrased it poorly :)