Multiplying and dividing with integers
You also have to pay attention to the signs when you multiply and divide. There are two simple rules to remember:
When you multiply a negative number by a positive number then the product is always negative.
When you multiply two negative numbers or two positive numbers then the product is always positive.
This is similar to the rule for adding and subtracting: two minus signs become a plus, while a plus and a minus become a minus. In multiplication and division, however, you calculate the result as if there were no minus signs and then look at the signs to determine whether your result is positive or negative. Two quick multiplication examples:
$$3\cdot (-4)=-12$$
3 times 4 equals 12. Since there is one positive and one negative number, the product is negative 12.
$$(-3)\cdot (-4)=12$$
Now we have two negative numbers, so the result is positive.
Turning to division, you may recall that you can confirm the answer you get by multiplying the quotient by the denominator. If you answer is correct then the product of these two numbers should be the same as the numerator. For example,
$$\frac{12}{3}=4$$
In order to check whether 4 is the correct answer, we multiply 3 (the denominator) by 4 (the quotient):
$$3\cdot 4=12$$
What happens when you divide two negative numbers? For example,
$$\frac{(-12)}{(-3)}=\: ?$$
For the denominator (-3) to become the numerator (-12), you would have to multiply it by 4, therefore the quotient is 4.
So, the quotient of a negative and a positive number is negative and, correspondingly, the quotient of a positive and a negative number is also negative. We can conclude that:
When you divide a negative number by a positive number then the quotient is negative.
When you divide a positive number by a negative number then the quotient is also negative.
When you divide two negative numbers then the quotient is positive.
The same rules hold true for multiplication.
Video lesson
Calculate the following expressions
$$(-4)\cdot (-12),\: \: \: \: \frac{-12}{3}$$