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If we have a number line of numbers between say -3 and 3 (-3 -2 -1 0 1 2 3) and we multiply by a factor $k$ then the positive values will become $k, 2k, 3k$ respectively with 0 remaining zero. If the same is done to the negative values we have $(k)(-1) , (k)(-2), (k)(-3)$ which are $-k, -2k, -3k$ respectively since the negative implies they are opposite sides of the $0$.

What I fail to understand is if I multiply by $(-1)$ to all the terms, the positives will become negative (by the additive inverse property of numbers). But how do the values of $(-1)(-1), (-2)(-1), (-3)(-1)$ become $1, 2, 3$? What is the intuition behind this?

I am doing this without knowing that the product of two negatives is positive.

J D
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2 Answers2

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From playing around with positive integers, you may knwo that $$ (a+b)\cdot c=a\cdot c+b\cdot c$$ always holds. The definition of negative numbers, which is a deliberate extension of the notion of "number", is done in a manner such that this rule (and other important rules) remains valid. This principle is called "persistence".

Also, $-x$ is by definition a number that, when added to $x$, produces $0$. With this in mind, we have $$ (1+(-1))\cdot 42= 1\cdot 42+(-1)\cdot 42=42+(-1)\cdot 42.$$ Hence the number $(-1)\cdot 42$ fulfills the very condiftion that we use to define $-42$: Adding it to $42$ gives $0$.

By similar reasoning, $$(1+(-1))\cdot(-42) = 1\cdot(-42) + (-1)\cdot (-42)=-42+(-1)\cdot (-42)$$ and hence $(-1)\cdot(-42)$ is the unique number that, when added to $-42$, produces $0$ - just like $42$, thus $(-1)(-42)=42$.

Hagen von Eitzen
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$(-1)(-1)-(-1)(-1)=0$

$(-1)(-1-(-1))=0$

$-1-(-1)=0$

$-(-1)=1$

$(-1)(-1)=1$

Hope this helps.

J Tg
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