Comment on teachings

bleistift2@feddit.de ⁨8⁩ ⁨months⁩ ago

• notabot@lemm.ee ⁨8⁩ ⁨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 ⁨8⁩ ⁨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 ⁨8⁩ ⁨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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• Davel23@fedia.io ⁨8⁩ ⁨months⁩ ago

  Same reason that a double negative makes a positive.

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• ImplyingImplications@lemmy.ca ⁨8⁩ ⁨months⁩ ago

  Here’s a other example:

  A) -3 × (-3 + 3) = ?

  You can solve this by figuring out the brackets first. -3 × 0 = 0

  You can also solve this using the distributive property of multiplication, rewriting the equation as

  A) -3 × (-3 + 3) = 0 (-3 × -3) + (-3 × 3) = 0 (-3 × -3) - 9 = 0 (-3 × -3) = 9

  If (-3 × -3) didn’t equal 9 then you’d get different answers to equation A depending on what method you used to solve it.

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