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What’s Wrong with “Multiplication Is Repeated Addition”? (2008)

denisegaskins.com

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Re: What’s Wrong with “Multiplication Is Repeated Addition”? (2008)

#111

Post by eigenket »

marcosdumay wrote: post #106
eigenket wrote: post #97

Earlier quoted context omitted.

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, i…

Of course, once you decide to iterate over uncountable sets, infinity starts to appear all the time. This isn't the only case of infinite calculations that can't be computed in practice but that mathematics use all the time anyway; and it reflects quite well the fact that multiplying irrational numbers isn't something that one can do practice. There is no problem with it.

I definitely wasn't trying to indicate that there was a problem with this, just pointing out that the path of using the Surreals (or anything else) to give an "algorithmic" description of real addition or multiplication is probably a bad idea.

Re: What’s Wrong with “Multiplication Is Repeated Addition”? (2008)

#112

Post by shawnz »

scatters wrote: post #99
rustybolt wrote: post #27

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. 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`.

Interesting, what about tetration then?

Re: What’s Wrong with “Multiplication Is Repeated Addition”? (2008)

#113

Post by alisonkisk »

eru wrote: post #95
yakubin wrote: post #37

Earlier quoted context omitted.

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 finit…

Which is a technical way of saying "in real life people use finite rational or algebraic approximations for reals, so uncountability of reals and infinite precision aren't a problem".

Re: What’s Wrong with “Multiplication Is Repeated Addition”? (2008)

#114

Post by bidirectional »

horsawlarway wrote: post #109
midjji wrote: post #101

Earlier quoted context omitted.

Not defending the article, but how would you compute: \pi*\pi using repeated addition?

pi + pi + pi + (.141592... of pi) ~= 9.8696 You need the concept of a ratio, so arguably I'm using multiplication to define multiplication, but you're sort of cheating by asking about a fractional number.

Pi is definitely not a fractional number... I don't think it's cheating at all, multiplication on the naturals is repeated addition, that's not the case for the reals.

Re: What’s Wrong with “Multiplication Is Repeated Addition”? (2008)

#115

Post by alisonkisk »

yakubin wrote: post #54
prionassembly wrote: post #49

Earlier quoted context omitted.

Multiplication on the naturals is repeated addition. You might see the natural numbers as training wheels for higher mathematics, but number theorists might disagree...

Number theorists work with integers, not naturals. And they too will agree with me, as they mostly work with polynomials, which show exactly that. I also didn't say natural numbers are training wheels, just this metaphor, which to use HN lingo: doesn't scale.

Number theorists work with complex numbers and irrational numbers too

https://www.britannica.com/science/Riemann-zeta-function

Re: What’s Wrong with “Multiplication Is Repeated Addition”? (2008)

#116

Post by jordan_curve »

Frost1x wrote: post #86
rustybolt wrote: post #27

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. 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…

>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. Education is probably a special case, at least from my educational experience. There's often multiple ways to abstractly represent a problem in mathematics and find the desired solution. The example you pointed out is such a case. When teaching mathematics, I think the…

To make this more explicit, mathematics (even at the primary school level) is arguably more about “proving” things than it is about computation.

Obviously you’re not writing proofs in 3rd grade, but the emphasis should be on students being able to say why their answer is correct rather than just being able to produce the correct answer.

Re: What’s Wrong with “Multiplication Is Repeated Addition”? (2008)

#117

Post by meepmorp »

eru wrote: post #92
sudosysgen wrote: post #73

Earlier quoted context omitted.

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?

Oh god, no. Not at all, especially in the US.

Re: What’s Wrong with “Multiplication Is Repeated Addition”? (2008)

#118

Post by gugagore »

drewcoo wrote: post #34
Agingcoder wrote: post #9

I don't understand. The author is mixing up physics (dimensional analysis) and maths, and trying to give the multiplicand a special role (I didn't even know there was a distinction between the two terms - to me they are both factors). This might be true in the physical world, but in the world of numbers, I think the distinction is irrelevant. Furthermore, being able to compute/define multiplication through repeated a…

Agreed. Multiplier and multiplicand are different words but the commutative property says their values can swap equivalently, so . . . what was the author's point again?

If, for a moment, you conceptualize of multiplication on non-negative whole numbers as repeated addition, then this is the algorithm:

  procedure product(multiplier, multiplicand)
      acc := 0
      for i = 1 to multiplier
          acc := acc + multiplicand 
      end
      return acc
  end
Swapping the arguments is a different computation. But after thinking about it, you realize that you get the same answer all the same. That's the point of

> (2 rows × 3 chairs/row) is not the same as (3 rows × 2 chairs/row), even though both sets contain 6 chairs.

The point with bringing up dimensional analysis is that the above algorithm doesn't work because what does it mean to do `for i = 1 to 3 chairs/row`? You might think of it like

  procedure product(multiplier, multiplicand)
      acc := "0" # an "absolute" zero that cooperates with any dimension
      each single_multiplier in multiplier
          acc := acc + (multiplicand * single_multiplier)
      end
      return acc
  end
But then what is `(multiplicand * single_multiplier)` ?

Re: What’s Wrong with “Multiplication Is Repeated Addition”? (2008)

#119

Post by eigenket »

midjji wrote: post #101
Agingcoder wrote: post #9

I don't understand. The author is mixing up physics (dimensional analysis) and maths, and trying to give the multiplicand a special role (I didn't even know there was a distinction between the two terms - to me they are both factors). This might be true in the physical world, but in the world of numbers, I think the distinction is irrelevant. Furthermore, being able to compute/define multiplication through repeated a…

Not defending the article, but how would you compute: \pi*\pi using repeated addition?

You first define multiplication of natural numbers to be repeated addition, then define multiplication of rationals in terms of multiplication and addition of naturals, then define multiplication of reals in terms of multiplication of Cauchy sequences of rationals ;)

Re: What’s Wrong with “Multiplication Is Repeated Addition”? (2008)

#120

Post by spekcular »

eigenket wrote: post #28
Garlef wrote: post #18

Earlier quoted context omitted.

Neither is more foundational than the other. So I think the use of "just" is not justified here. https://mathoverflow.net/questions/360578/category-theory-an...

Yeah but your comment didn't use the concept of a "category" at all, whereas you used "set" three times. Its definitely more natural to say these definitions come from set theory.

Yes, there's no need to invoke any category theory concepts here. Everything is happening in the category of sets.
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