Editing Negative number (section)

===Multiplication===
[[File:Multiplication of Positive and Negative Numbers.svg|thumb|A multiplication by a negative number can be seen as a change of direction of the [[Vector (mathematics and physics)|vector]] of [[Magnitude (mathematics)|magnitude]] equal to the [[absolute value]] of the product of the factors.]]
When multiplying numbers, the magnitude of the product is always just the product of the two magnitudes. The [[sign (mathematics)|sign]] of the product is determined by the following rules:
* The product of one positive number and one negative number is negative.
* The product of two negative numbers is positive.
Thus
{{block indent | em = 1.5 | text = {{math|1= (−2) × 3  =  −6}} }}
and
{{block indent | em = 1.5 | text = {{math|1= (−2) × (−3)  =  6}}. }}
The reason behind the first example is simple: adding three {{math|−2}}'s together yields {{math|−6}}:
{{block indent | em = 1.5 | text = {{math|1= (−2) × 3  =  (−2) + (−2) + (−2)  =  −6}}. }}
The reasoning behind the second example is more complicated. The idea again is that losing a debt is the same thing as gaining a credit. In this case, losing two debts of three each is the same as gaining a credit of six:
{{block indent | em = 1.5 | text = {{math| (−2}} debts {{math|) × (−3}} each{{math|1=)  =  +6}} credit. }}
The convention that a product of two negative numbers is positive is also necessary for multiplication to follow the [[distributive law]].  In this case, we know that
{{block indent | em = 1.5 | text = {{math|1= (−2) × (−3)  +  2 × (−3)  =  (−2 + 2) × (−3)  =  0 × (−3)  =  0}}. }}
Since {{math|1=2 × (−3) = −6}}, the product {{math|(−2) × (−3)}} must equal {{math|6}}.

These rules lead to another (equivalent) rule—the sign of any product ''a'' × ''b'' depends on the sign of ''a'' as follows:
* if ''a'' is positive, then the sign of ''a'' × ''b'' is the same as the sign of ''b'', and
* if ''a'' is negative, then the sign of ''a'' × ''b'' is the opposite of the sign of ''b''.
The justification for why the product of two negative numbers is a positive number can be observed in the analysis of [[complex numbers]].