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B Geometric meaning of signs in multiplications

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  1. Aug 26, 2018 #1
    What does 1/-1 (one divided by minus one) mean?
    What does -1 X -1 (minus one multiplied by minus one) mean?
    What are the best graphic representations of multiplication and division?
     
  2. jcsd
  3. Aug 26, 2018 #2

    jedishrfu

    Staff: Mentor

    Welcome to PF!

    What is the context of these questions?

    Are you taking a course in Algebra?
     
  4. Aug 26, 2018 #3

    DaveC426913

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    Gold Member

    Came across this cute little example, showing concepts of negatives and positives:

    The enemy (-) of (*) my enemy (-) is my friend (+).
    The friend (+) of (*) my enemy (-) is my enemy (-).
    The enemy (-) of (*) my friend (+) is my enemy (-).
    The friend (+) of (*) my friend (+) is my friend (+).
     
  5. Aug 26, 2018 #4

    DaveC426913

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    Gold Member

    For starters, division is defined as the inverse of multiplication.

    So, 4 x 2 = 8 has the inverse operation of 8 / 2 = 4.
    And 4 x -2 = -8 has the inverse: -8 / -2 = 4.

    Not the best example, but...

    I dove to 60 feet below sea level (-60) in 10 intervals of 6 foot descents (10 x -6 = -60).

    How far from my final depth was I when I was two descents away (i.e. 2 less than)?
    -6 x -2 = 12
    I was 12 feet from my final depth.


    Often, debts and assets.
     
  6. Aug 26, 2018 #5

    verty

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    Homework Helper

    OP, you just have to learn it. -1*-1 = +1. The minuses cancel. 1 / (-1) = -1 because there is only 1 minus. $${1 \over -1} = {-1 \over 1} = -{1 \over 1} = -1$$
     
  7. Aug 26, 2018 #6

    jedishrfu

    Staff: Mentor

    Actually you should write

    ##1/-1 = (1/-1)*(-1/-1) = (1*-1)/(-1*-1) = -1/1 = -1##
     
  8. Aug 26, 2018 #7

    fresh_42

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    2017 Award

    Staff: Mentor

    The best explanation I found was the following:

    ##1 \cdot 1 = 1## is the area of a square with side length ##1##. Now think of an orientation of this area. Then the signs lead to

    upload_2018-8-26_17-3-23.png

    A purely algebraic answer would probably lead a bit far as a few definitions and concepts will be needed.
     
  9. Sep 8, 2018 #8
  10. Oct 11, 2018 #9
    Thank you all guys. Sorry for the late response. I really appreciate your help. I am just trying to "put a face" on a concept that seems vague for me.
     
  11. Oct 11, 2018 #10
    Just trying to understand!
     
  12. Oct 11, 2018 #11
    Thank you! These are really good examples!
     
  13. Oct 11, 2018 #12
    Please share. I think I can handle some algebra.
     
  14. Oct 11, 2018 #13
  15. Oct 11, 2018 #14

    fresh_42

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    Staff: Mentor

    ##\{\,-1,+1\,\}## are the units in the ring of integers, which is the basic domain our number system with characteristic zero is built upon. The integers are the basis. In fact, it are the natural numbers and the integers are already the first step, in which we constructed a group from the monoid ##(\mathbb{N},+)## in order to reverse additions. As it turns out, ##\mathbb{Z}## is also a ring, i.e. allows a distributive multiplication. Now the units of any ring form a multiplicative group, in this case of order ##2##. There is only one neutral element in a group, which we usually write as ##1## in the multiplicative case (##0## in the additive case). This requires to have ##(-1)\cdot 1 = -1## from which all other formulas follow:

    ##(1 / (-1))= 1\cdot (-1)^{-1}=(-1)^{-1}= -1## for otherwise we would have
    ##-1\stackrel{(*)}{=}1\cdot (-1)\stackrel{(**)}{=}(-1)^{-1}\cdot (-1)\stackrel{(***)}{=}1## which cannot be in the case of characteristic zero.
    ##(*)## definition of unique ##1##
    ##(**)## assumption ##(-1)^{-1}= 1##
    ##(***)## definition of unique inverse

    Next we have ##(-1)\cdot (-1)\stackrel{(a)}{=}(-1)\cdot (-1)^{-1}\stackrel{(b)}{=}1##
    ##(a)## by the previous formula
    ##(b)## by the definition of the inverse

    The proofs that units of a ring form a multiplicative group, and that a group has a unique neutral element and unique inverse elements depend a bit on how you define a group (laws vs. solvability of equations).
     
  16. Oct 16, 2018 at 8:15 AM #15
    When We Multiply:
    Example
    plus.gif × plus.gif two positives make a positive: plus.gif 3 × 2 = 6

    minus.gif × minus.gif two negatives make a positive: plus.gif (−3) × (−2) = 6

    minus.gif × plus.gif a negative and a positive
    make a negative: minus.gif (−3) × 2 = −6

    plus.gif × minus.gif a positive and a negative
    make a negative: minus.gif 3 × (−2) = −6
    Yes indeed, two negatives make a positive, and we will explain why, with examples!

    Signs
    Let's talk about signs.

    "+" is the positive sign, "−" is the negative sign.

    When a number has no sign it usually means that it is positive.

    Example: 5 is really +5
     
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