Silhouette wrote:Thank you for asking the first not-profoundly-stupid question in this thread for far too long.

I appreciate the kind words. So now if I were to say "You're all wet," or, "You have no idea what you're talking about," I'd feel guilty. So I'll just point out a few things. I could write a lengthy post but I'll keep this mercifully short and just list some bullet iitems.

* The mathematical real numbers \(\mathbb R\) are not anything remotely like the floating point numbers as described by IEEE-754. Yes I know what that is and they are not the real numbers. You seem to have studied computers and not math. Everything you say about computer arithmetic may be true, but totally irrelevant. The question was not, "Can computer multiplication be reduced to addition?" That's completely different question to whether real number multiplication can be so reduced.

* Likewise your off-topic remarks about engineering math, and all practical operations involving real numbers being reducible to rationals. Of course that's true, but equally irrelevant. We're talking about the mathematical real numbers. You know, the ones that require infinitely much information to represent.

* To take you entirely out of the realm of computers, how do I multiply two noncomputable real numbers? Those are the numbers that can not be approximated by a program or a Turing machine. Most reals are noncomputable, as you can see from noting that there are uncountably many reals but only countably many Turing machines. How do you reduce the multiplication of two noncomputable reals to "repeated addition?" The idea is absurd on its face. You could not reduce a noncomputable real to a rational approximation with any finite amount of computing power or memory no matter how large, if your approximation is required to be computable.

* Your handwaving about "algebra" is nonsense. Yes I'm mindful that you said something nice to me. I regret not being able to respond in kind other than to note that you know a lot about computers and engineering math but sadly nothing about math. Your remarks in this area were vague. What do you mean that "algebra" shows that multiplication is reducible to repeated addition? On the contrary. In algebra, multiplication is a map from pairs of real numbers to real numbers, satisfying the usual field axioms. There's nothing in the field axioms about multiplication being repeated addition.

* Likewise your handwaving that " "i" is just another kind of

anomaly like irrational numbers ..." Please friend, I really appreciate that you complimented me. I'm going to hate myself in the morning for being so churlish as to say this, but you just embarrassed yourself.

It's ok that you studied computers and engineering instead of math. But be humble about what you don't know.

If you'd like me to expand on anything I said, please ask. Like I said I could have written a lot more.

One more point. Here is what you meant to say, if you'd studied math: "Well we can define multiplication as repeated addition in Peano arithmetic; then we can lift multiplication to rationals using the standard field of quotients construction; and then we can lift multiplication to the reals by taking limits of Dedekind cuts of rationals; and in this way, although real number multiplication is NOT in any meaningful sense repeated addition, we can indeed find multiplication defined as repeated addition a long way back in the chain of the construction of the real numbers." If you said that, it would be the right answer. I'll leave the definition of multiplication of complex numbers to you.

Another simple way out would have been to say that we can view multiplication as repeated addition for natural numbers but not for rationals, reals, or complex numbers. That would be sensible. But your changing the subject to computers was wrong, because floats are not real numbers. Maybe they just don't tell the CS students that. But you should realize, there's a LOT they don't tell the CS students about math. They run you through "Discrete math" and consider your mathematical education done. It's a crime.

Ok now I feel terrible. You shouldn't have been so nice to me!!