Editing Negative number (section)

==Arithmetic involving negative numbers==
The [[Plus and minus signs|minus sign]] "−" signifies the [[Operator (mathematics)|operator]] for both the binary (two-[[operand]]) [[Operation (mathematics)|operation]] of [[subtraction]] (as in {{math|''y'' − ''z''}}) and the unary (one-operand) operation of [[Additive inverse|negation]] (as in {{math|−''x''}}, or twice in {{math|−(−''x'')}}). A special case of unary negation occurs when it operates on a positive number, in which case the result is a negative number (as in {{math|−5}}).

The ambiguity of the "−" symbol does not generally lead to ambiguity in arithmetical expressions, because the order of operations makes only one interpretation or the other possible for each "−". However, it can lead to confusion and be difficult for a person to understand an expression when operator symbols appear adjacent to one another. A solution can be to parenthesize the unary "−" along with its operand.

For example, the expression {{math|7 + −5}} may be clearer if written {{math|7 + (−5)}} (even though they mean exactly the same thing formally). The [[subtraction]] expression {{math|7 − 5}} is a different expression that doesn't represent the same operations, but it evaluates to the same result.

Sometimes in elementary schools a number may be prefixed by a superscript minus sign or plus sign to explicitly distinguish negative and positive numbers as in<ref>{{cite book|title=Understanding by design|author1=Grant P. Wiggins|author2=Jay McTighe|page=[https://archive.org/details/isbn_9780131950849/page/210 210]|year=2005|publisher=ACSD Publications|isbn=1-4166-0035-3|url-access=registration|url=https://archive.org/details/isbn_9780131950849/page/210}}</ref>
{{block indent | em = 1.5 | text = {{math|<sup>−</sup>2 + <sup>−</sup>5}} &nbsp;gives&nbsp;{{math|<sup>−</sup>7}}. }}

===Addition===
[[File:AdditionRules.svg|right|thumb|A visual representation of the addition of positive and negative numbers. Larger balls represent numbers with greater magnitude.]]

Addition of two negative numbers is very similar to addition of two positive numbers. For example,
{{block indent | em = 1.5 | text = {{math|1=(−3) + (−5)  =  −8}}. }}
The idea is that two debts can be combined into a single debt of greater magnitude.

When adding together a mixture of positive and negative numbers, one can think of the negative numbers as positive quantities being subtracted. For example:
{{block indent | em = 1.5 | text = {{math|1=8 + (−3)  =  8 − 3  =  5}} &nbsp;and&nbsp;{{math|1=(−2) + 7  =  7 − 2  =  5}}. }}
In the first example, a credit of {{math|8}} is combined with a debt of {{math|3}}, which yields a total credit of {{math|5}}. If the negative number has greater magnitude, then the result is negative:
{{block indent | em = 1.5 | text = {{math|1=(−8) + 3  =  3 − 8  =  −5}} &nbsp;and&nbsp;{{math|1=2 + (−7)  =  2 − 7  =  −5}}. }}
Here the credit is less than the debt, so the net result is a debt.

===Subtraction===
As discussed above, it is possible for the subtraction of two non-negative numbers to yield a negative answer:
{{block indent | em = 1.5 | text = {{math|1= 5 − 8  =  −3}} }}
In general, subtraction of a positive number yields the same result as the addition of a negative number of equal magnitude. Thus
{{block indent | em = 1.5 | text = {{math|1= 5 − 8  =  5 + (−8)  =  −3}} }}
and
{{block indent | em = 1.5 | text = {{math|1= (−3) − 5  =  (−3) + (−5)  =  −8}} }}

On the other hand, subtracting a negative number yields the same result as the addition a positive number of equal magnitude. (The idea is that ''losing'' a debt is the same thing as ''gaining'' a credit.) Thus
{{block indent | em = 1.5 | text = {{math|1= 3 − (−5)  =  3 + 5  =  8}} }}
and
{{block indent | em = 1.5 | text = {{math|1= (−5) − (−8)  =  (−5) + 8  =  3}}. }}

===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]].

===Division===
The sign rules for [[Division (mathematics)|division]] are the same as for multiplication.  For example,
{{block indent | em = 1.5 | text = {{math|1=8 ÷ (−2)  =  −4}}, }}
{{block indent | em = 1.5 | text = {{math|1=(−8) ÷ 2  =  −4}}, }}
and
{{block indent | em = 1.5 | text = {{math|1=(−8) ÷ (−2)  =  4}}. }}
If dividend and divisor have the same sign, the result is positive, if they have different signs the result is negative.