A polynomial is a polynomial, a decimal representation of a real number is a decimal representation of a real number, and your representation of complex numbers has funny properties once you begin exponentiating that.
I'd like to see how you'd show kids that: (1+x+x^2)·(1-x^3) is a "repeated addition", both belong to the ring Z[x].
> I'd like to see how you'd show kids that: (1+x+x^2)·(1-x^3) is a "repeated addition", both belong to the ring Z[x].
Is that not just (1 * (1-x^3) + x * (1-x^3) + x^2 * (1-x^3)) ??
The polynomial itself gives us the means at which we logically split up the multiplication into component parts.
Just as 3.14 * 3 == 3 * 3 + 0.1 * 3 + 0.04 * 3, when we move onto polynomials, we do the same exact thing. EDIT: remember, ALL REAL NUMBERS ARE POLYNOMIALS with a base of 10.
Or to put it another way: when x == 10, your polynomial of (1 + x + x^2) * (1-x^3) == 111 * (-999). That is to say: real numbers are simply polynomials where "x" has been defined to be a particular number, instead of an abstract entity. We call that number the radix-base.
If you instead defined the base to be x = 16 (hexadecimal numbers), you'd get 111 * (-FFF), which you'll find will satisfy similar properties. Now leave x-undefined (since it could be 10 or 16), and what do you get?
Polynomial math. Or so called "Carry-less multiplication" (https://en.wikipedia.org/wiki/Carry-less_product). We don't have a ring yet though: we still need to perform a modulus on all those polynomials to return to a proper ring (and if the modulus is irreducable, we have a Galois field). But we can already see how polynomials and the Reals are so closely related.
Precisely, you're not repeating p(x) q(x)-times, you've used that p(x) is a linear combination of monomials and then the distributive property of Z[x].
Now, you could argue that this is exactly a way to "add repeatedly", but at some point pushing analogies stops being helpful to your students.
> Now, you could argue that this is exactly a way to "add repeatedly", but at some point pushing analogies stops being helpful to your students.
That's not what this blogpost is arguing about. This blogpost is arguing that "Multiplication is Repeated Addition" is unhelpful at the elementary school level and stops being true at some point.
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My argument is otherwise. "Multiplication is Repeated Addition" is clearly helpful in grade school. Almost everybody I know has learned Multiplication through that method.
Secondly: I cannot think of a single instance where its not true. Yes, I've had to use linear-combinations to extend it out to polynomials, but clearly the property holds even in polynomial-land.
Its not useful to teach multiplication of polynomials with "Repeated Addition". But the advice is "not wrong", in fact, polynomial multiplication continues to see many similarities with Real and Complex multiplication. Especially if we consider a "Basis" to be analogs to the thing that's repeatedly-added.
I'd like to see how you'd show kids that: (1+x+x^2)·(1-x^3) is a "repeated addition", both belong to the ring Z[x].