A bit meandering and muddled in some respects, but agree with the gist: Multiplication is not just repeated addition.
What’s Wrong with “Multiplication Is Repeated Addition”? (2008)
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Re: What’s Wrong with “Multiplication Is Repeated Addition”? (2008)
#92Earlier quoted context omitted.
> These articles bring back memories of teachers telling me I was wrong, because I didn't use the quadratic formula, but completed the square instead. That's hilarious, since the quadratic formula is just what you get by solving ax^2 + bx + c = 0 by completing the square. In fact, I have a high school textbook that makes that derivation.
We did that derivation in class, actually! I was so lucky to have a great math teacher in high school.
That's not the norm?
Re: What’s Wrong with “Multiplication Is Repeated Addition”? (2008)
#93I just found this interesting article. If you are interested, it is a review of the following "conversation of articles" (which are reachable from the OP but I guess it's useful enough to list): - https://www.maa.org/external_archive/devlin/devlin_09_07.html - https://www.maa.org/external_archive/devlin/devlin_06_08.html - https://denisegaskins.com/2008/07/01/if-it-aint-repeated-addition/ - (the link I posted that cr…
Clickable:
- https://www.maa.org/external_archive/devlin/devlin_09_07.htm...
- https://www.maa.org/external_archive/devlin/devlin_06_08.htm...
- https://denisegaskins.com/2008/07/01/if-it-aint-repeated-add...
(Sorry, it just bothers me when people don't use clickable links here.)
Re: What’s Wrong with “Multiplication Is Repeated Addition”? (2008)
#94Earlier quoted context omitted.
Algorithms are not finite. E.g. Newton's method.
First sentence on Wikipedia[1]: > In mathematics and computer science, an algorithm (/ˈælɡərɪðəm/ (About this soundlisten)) is a finite sequence of well-defined, computer-implementable instructions, typically to solve a class of problems or to perform a computation. Newton's method is finite too. You perform finitely many iterations. It doesn't calculate roots. It calculates their approximations. [1]: https://en.wiki…
If you use a termination condition that has to do with convergence of iterates instead of a number of iterations (often the case), then you generally don't know beforehand the length of the finite sequences. Maybe you know a bound, but in general you might not even have that.
In an important sense, it only becomes a finite algorithm. It isn't one. You cannot write the finite sequence of instructions down. It's got loops.
To your point about approximations vs not, if you have an algorithm that, for any desired approximation accuracy can compute the square root to that accuracy in a finite number of steps, then that process is as much "the square root" as anything involving the real numbers.
Re: What’s Wrong with “Multiplication Is Repeated Addition”? (2008)
#95Earlier quoted context omitted.
If you define the reals axiomatically, you still need an existence proof. Which will involve addition and multiplication algorithms.
Not algorithms. There will be infinite addition involved, and algorithms are finite. Thinking of multiplication as repeated addition also won't explain anything about it. It's a separate operation. Deal with it. For similar reasons, you can't calculate x-th power of a number, when x is irrational, by decomposing it into exponentiation and roots. This metaphor is just training wheels. At some point you should lose it.…
> There will be infinite addition involved, and algorithms are finite.
People are just bit sloppy, and say algorithm when they mean something slightly different.
See https://stackoverflow.com/questions/28841260/what-is-the-dif... and https://en.wikipedia.org/wiki/Corecursion
Basically, you don't want an 'algorithm' here to produce the whole number.
All you need is some scheme that will produce the next digit in finite time (and the next one and the next one etc).
Re: What’s Wrong with “Multiplication Is Repeated Addition”? (2008)
#96Earlier quoted context omitted.
If we are just looking at numbers with no units attached there is no difference. As soon as there are units, a difference arises. I will collect 12 apples 5 times. The 12 apples are the multiplicand and the 5 times is the multiplier. The answer retains the units from the multiplicand.
Still don't see the difference. Density times volume is mass. Volume times density is mass. Distance times force is work/energy... "5 times I will connect 12 apples"
You're correct. The GP in incorrect about their dimensional analysis. 5 apples 12 times yields apples because it's apples times a dimensionless scalar (count). Newtons times meters is always Newton-meters, never Newtons or meters. Units are never magically dropped in dimensional analysis.
Re: What’s Wrong with “Multiplication Is Repeated Addition”? (2008)
#97Earlier quoted context omitted.
> You can definitely see multiplication as repeated addition. Only for rational numbers. Doesn't work for real and complex numbers. > It's even a useful thing to do, how else would you define multiplication? Axiomatically, not algorithmically.
Multiplication of the Surreals is a recursive operation using sums (addition and subtraction on the left and right sets). Since the Reals are a strict subfield of the Surreals one can define multiplication of the reals using only the same recursive formula and restricting both operands to be Reals.
Its probably worth emphasising that the recursion you need to "construct" the Surreals is infinite, in other words this does not give a reasonable algorithm to (for example) add two real numbers, you need S_omega in order to have even all rational numbers.
The construction is rather involved but if we're only interested in the reals for now you can think of it as defining a real number by a set of rational numbers, in particular define a particular "real number" to be the set of all rational numbers less than it, for example sqrt(2) is defined to be the set of all rationals p/q such that p^2/q^2 In general there is no sensible algorithm to do anything in the real numbers, since most real numbers aren't even computable (there is no way to represent an arbitrary real number on a Turing machine).
Re: What’s Wrong with “Multiplication Is Repeated Addition”? (2008)
#98Earlier quoted context omitted.
They exist to distinguish the element being operated on (the LHS) and what it is operated on by (the RHS). Total technicality, but I could see myself using the term multiplicand/multiplier in my code if I had to implement e.g. a stack-based parser for arithmetic expressions.
> to distinguish the element being operated on (the LHS) I'm not sure what you mean, they are both operated on, it is a binary operator and commutative, there is literally no difference.
It distinguishes these two rectangles.
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**Re: What’s Wrong with “Multiplication Is Repeated Addition”? (2008)
#99These articles bring back memories of teachers telling me I was wrong, because I didn't use the quadratic formula, but completed the square instead. You can definitely see multiplication as repeated addition. It's even a useful thing to do, how else would you define multiplication? This is the whole idea that made mathematics great. You define more complicated operations in terms of familiar operations. Then, if you…
> how else would you define multiplication?
In Church arithmetic, multiplication is defined as function composition; that is, the dot operator `.` in Haskell-like languages. Addition is considerably more complex; it is defined as `lift (.)`. Exponentiation is even simpler; it is function application, so the `id` function - except that we write exponentiation backwards, so actually `flip id`.
Re: What’s Wrong with “Multiplication Is Repeated Addition”? (2008)
#100Earlier quoted context omitted.
> You can definitely see multiplication as repeated addition. Only for rational numbers. Doesn't work for real and complex numbers. > It's even a useful thing to do, how else would you define multiplication? Axiomatically, not algorithmically.
If you define the reals axiomatically, you still need an existence proof. Which will involve addition and multiplication algorithms.
Well, no, not really. The standard definition of the reals is as the unique nontrivial totally-ordered, Dedekind-complete, Archimedean field up to isomorphism.
So what you would really need is a uniqueness proof, with addition and multiplications "provided" by the hypothesis.