21–30 of 54 posts
If we define multiplication as repeated addition, then we define addition as repeated increment, where A + B is: start at zero, increment A times then B times.
Notationwise it would be good to do as some computer languages do and just use a name and wrap everything: (+ (* (^ (_ something D) C) B) A) = imagine this one graphically: ((((something_D)^C) * B) + A). No ambiguity, no question in what order to apply operations.
Earlier quoted context omitted.
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.
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…
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.
I wonder why exponentials are so common in nature/physics but tetration is not
I wonder why exponentials are so common in nature/physics but tetration is not
Earlier quoted context omitted.
To clarify where I live limits are introduced in high school, irrational numbers just much earlier.
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.
[0] https://waitbutwhy.com/2014/11/1000000-grahams-number.html