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A comment on What’s Wrong with “Multiplication Is Repeated Addition”? (2008). Parent comment

Re:What’s Wrong with “Multiplication Is Repeated Addition”? ... by prionassembly on Wednesday March 03, 2021 @09:12AM (#26328504)
Operationally, unsure. But if there was an answer, it would be something that multiplied with itself twice would have be the same as 2 multiplied with itself once.

I think it was Gerolamo Cardano who in a book of worked exercises came up with a problem like "obtain two numbers that add to 10 and multiply to 40" and arrived at the solution had to involve square roots of -1. Then he goes "this makes no sense, but let's calculate nevertheless..."

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Re:What’s Wrong with “Multiplication Is Repeated Addition”? ... by Garlef on Wednesday March 03, 2021 @09:33AM (#26328766)
> it would be something that multiplied with itself twice would have be the same as 2 multiplied with itself once

from what i recall that's the way to go:

You can use this to extend the definition of exponentials as "repeated addition" from allowing only natural numbers expoonents to allowing positive rational numbers. Afterwards you can "complete" this to allow real exponents.

But this requires some hard maths to do rigorously.

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Re:What’s Wrong with “Multiplication Is Repeated Addition”? ... by prionassembly on Wednesday March 03, 2021 @10:04AM (#26329238)
You're absolutely right about rigor: I'm not claiming that Cardano discovered complex analysis or anything...
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