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notabot@lemm.ee ⁨6⁩ ⁨months⁩ ago

You can look at multiplication as a shorthand for repeated addition, so, for example:

3x3=0 + 3 + 3 + 3 = 9

In other words we have three lots of three. The zero will be handy later…

Next consider:

-3x3 = 0 + -3 + -3 + -3 = -9

Here we have three lots of minus three. So what happens if we instead have minus three lots of three? Instead of adding the threes, we subtract them:

3x-3 = 0 - 3 - 3 - 3 = -9

Finally, what if we want minus three lots of minus three? Subtracting a negative number is the equivalent of adding the positive value:

-3x-3 = 0 - -3 - -3 - -3 = 0 + 3 + 3 + 3 = 9

Do let me know if some of that isn’t clear.

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bleistift2@feddit.de ⁨6⁩ ⁨months⁩ ago
This was very clear. Now that I see it, I realize it’s the same reasoning why x^(-3) is 1/(x^3):

2 × -3 = -6 1 × -3 = -3 0 × -3 = 0 -1 × -3 = +3
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affiliate@lemmy.world ⁨6⁩ ⁨months⁩ ago
i think this is a really clean explanation of why (-3) * (-3) should equal 9. i wanted to point out that with a little more work, it’s possible to see why (-3) * (-3) must equal 9. and this is basically a consequence of the distributive law:

0 = 0 * (-3) = (3 + -3) * (-3) = 3 * (-3) + (-3) * (-3) = -9 + (-3) * (-3).

the first equality uses 0 * anything = 0. the second equality uses (3 + -3) = 0. the third equality uses the distribute law, and the fourth equality uses 3 * (-3) = -9, which was shown in the previous comment.

so, by adding 9 to both sides, we get:

9 = 9 - 9 + (-3) * (-3).

in other words, 9 = (-3) * (-3). this basically says that if we want the distribute law to be true, then we need to have (-3) * (-3) = 9.

it’s also worth mentioning that this is a specific instance of a proof that shows (-a) * (-b) = a * b is true for arbitrary rings. (a ring is basically a fancy name for a structure with addition and distribute multiplication.) so, any time you want to have any kind of multiplication that satisfies the distribute law, you need (-a) * (-b) = a * b.

in particular, (-A) * (-B) = A * B is also true when A and B are matrices. and you can prove this using the same argument that was used above.
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