What’s Wrong with “Multiplication Is Repeated Addition”? (2008)
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What’s Wrong with “Multiplication Is Repeated Addition”? (2008)
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Re: What’s Wrong with “Multiplication Is Repeated Addition”? (2008)
#2I 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 creates this thread)Re: What’s Wrong with “Multiplication Is Repeated Addition”? (2008)
#3The teacher mentions juggling units when adding and multiplying, but is 2 cm × 3 cm the same thing as 2 × 3?
Re: What’s Wrong with “Multiplication Is Repeated Addition”? (2008)
#4I like how 3blue1brown presents it in terms of animated number line translations
Re: What’s Wrong with “Multiplication Is Repeated Addition”? (2008)
#5In some settings, "repeated addition" is the only sensible way to think of multiplication. Such as multiplying an elliptic curve point with a scalar.
Re: What’s Wrong with “Multiplication Is Repeated Addition”? (2008)
#6The teacher mentions juggling units when adding and multiplying, but is 2 cm × 3 cm the same thing as 2 × 3?
Both examples use the same operator, but we resolve them to different functions. Mentally, we use the tuple of argument types to select the appropriate function. Addition does not provide equivalent functions for all multiplication cases. Hence, multiplication is not repeated addition.
Re: What’s Wrong with “Multiplication Is Repeated Addition”? (2008)
#7The teacher mentions juggling units when adding and multiplying, but is 2 cm × 3 cm the same thing as 2 × 3?
Arguably, what they're saying is that 2 x 3 should be understood as a unit-less "scaler" 2 that multiplies a number (of things) "3".
Re: What’s Wrong with “Multiplication Is Repeated Addition”? (2008)
#8I don’t think the aruments made here are compelling. Multiplication can be defined in this way. Saying that multiplication is repeated addition is not the same as saying that multiplication IS addition. Addition relates to multiplication as multiplication relates to powers.
Re: What’s Wrong with “Multiplication Is Repeated Addition”? (2008)
#9I 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 addition doesn't prevent you from looking at the special properties of this new operator.
Re: What’s Wrong with “Multiplication Is Repeated Addition”? (2008)
#10Does anyone have an intuition for why Presburger arithmetic is decidable, but once you introduce x, it is not? If + can be rewritten as repeated succ, (which is decidable) and x can be rewritten as repeated +, then it seems logical that x should be decidable, but as any undergraduate CS student knows, Peano arithmetic is undecidable. What is so special about x exactly?