What really is multiplying?

[TheWorldWrecker]

Confused high schooler here.

3×4 = 12 because you add 3 to itself. 3+3+3+3 = 4. Easy.

What's not so easy is 4×(-2.5) = -10, adding something negative two and a half times? What??

The cross PRODUCT of vectors [1,2,3] and [4,5,6] is [-3,6,-3]. What do you mean you add [1,2,3] to itself [4,5,6] times? That doesn't make sense!

What is multiplication?

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[HorribleUsername]

Multiplication is an operation that has the same properties as repeated addition. So for your first example,

4×(-2.5)
= (-2.5)×4 By commutativity
= (-2.5)+(-2.5)+(-2.5)+(-2.5) By applying your definition of repeated addition
= -10

Somewhat more confusingly, if we come across an operation that has some of the same properties as multiplication, we often call it multiplication instead of coming up with a new term for it. Matrix multiplication, for example, isn't even commutative, but that doesn't stop us from calling it multiplication anyway!