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Question about multiplication

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You can declare that \* is the addition operation.   In that case 1A \* 1A = 2A  and 1A \* 2A = 3A  (not sure why you said 4A).

There's nothing wrong with that, but as you said it's irrelevant.  If you said that fish eat sheep and then later told me that you were redefining fish to mean wolf, then fine.   But I'll probably feel less inclined to keep talking to you.
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You’ve already gotten some good answers, but I just want to point out WHY your uncle’s problem makes no sense: what does it mean to multiply an apple by an apple? Like physically, what do you *do* to multiply them? It’s a nonsense question.

The most rudimentary way to understand multiplication is as repeated addition, so for example if a sack of apples has 10 apples and you have 3 sacks, you can find the total number of apples by taking 10+10+10 or 3*10, the units being apples per sack times sack, coming out as apples.

In another example, if we wanted to find the area of a rectangle that’s 2 inches by 6 inches, we could count up the number of 1inx1in squares by doing 6+6 or 2*6, because what we’re counting is 1x1 squares, the units are in square inches.

But it doesn’t make sense in the real world to multiply apples by apples, there’s no corresponding physical situation. If we were talking about an abstract mathematical situation this wouldn’t be a problem, but we’re talking about apples. Your uncle’s “major problem in mathematics” is his misunderstanding of what it means to multiply things.
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While it's certainly possible to define operations in math, it wouldn't be very practical to define a multiplication operation on apples such that one apple times one apple = two apples.
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I'm not going to engage with his actual reasoning, as it's nonsensical and I know exactly what sort of personality you're describing. Instead, I'll leave a quotation:

*"Never argue with idiots. They will drag you down to their level, and then beat you with experience."*

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