The Four Operations

Addition

The addition of two negative numbers is very similar to the addition of two positive numbers. For example:

$(−3) + (−5)  =  −8$

The underlying principle is that two debts—negative numbers— can be combined into a single debt of greater magnitude.

When adding together a mixture of positive and negative numbers, another way to write the negative numbers is as positive quantities being subtracted. For example:

$8 + (−3)  =  8 − 3  =  5$

Here, a credit of 8 is combined with a debt of 3, which yields a total credit of 5. However, if the negative number has greater magnitude, then the result is negative:

$(−8) + 3  =  3 − 8  =  −5 $

Similarly:

$(−2) + 7  =  7 − 2  =  5$

Here, a debt of 2 is combined with a credit of 7. The credit has greater magnitude than the debt, so the result is positive. But if the credit is less than the debt, the result is negative:

$2 + (−7)  =  2 − 7  =  −5$

Subtraction

Subtracting positive numbers from each other can yield a negative answer. For example, subtracting 8 from 5:

$5 − 8  =  −3 $

Subtracting a positive number is generally the same as adding the negative of that number. That is to say:

$5 − 8  =  5 + (−8)  =  −3 $

and

$(−3) − 5  =  (−3) + (−5)  =  −8$

Similarly, subtracting a negative number yields the same result as adding the positive of that number. The idea here  is that losing a debt is the same thing as gaining a credit. Therefore:

$3 − (−5)  =  3 + 5  =  8 $

and

$(−5) − (−8)  =  (−5) + 8  =  3$

Multiplication

When multiplying positive and negative numbers, the sign of the product is determined by the following rules:

• The product of two positive numbers is positive.The product of one positive number and one negative number is negative.
• The product of two negative numbers is positive.

For example:

$(−2) × 3  =  −6 $

This is simply because adding −2 together three times yields −6:

$(−2) × 3  =  (−2) + (−2) + (−2)  =  −6$

However,

$(−2) × (−3)  =  6$

The idea again here 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:

$\left (−2\text{ debts } \right ) \times \left ( −3 \text{ each} \right )  =  +6\text{ credit}$

Division

The sign rules for division are the same as for multiplication. 

• Dividing two positive numbers yields a positive number.
• Dividing one positive number and one negative number yields a negative number.
• Dividing two negative numbers yields a positive number.

If the dividend and the divisor have the same sign, that is to say, the result is always positive. For example:

$8 ÷ (−2)  =  −4$

and

$(−8) ÷ 2  =  −4$

but

$(−8) ÷ (−2)  =  4$.

Additional Considerations

The basic properties of addition (commutative, associative, and distributive) also apply to negative numbers. For example, the following equation demonstrates the distributive property: 

$-3(2 + 5) = (-3)\cdot 2 + (-3)\cdot 5 $