# Multiplying Integers

- - Pre-Algebra

Recall that the set of counting numbers is closed over multiplication. Thus, the set of positive integers, also known as counting numbers is closed over multiplication. This implies that if two positive integers are multiplied then their product is also a positive. It has been previously known that multiplication is a repeated addition.

It means that $7 \times 3$ can be written into $7+7+7=21$. Similarly, $\left (-7 \right) \times 3$ is the same as $\left(-7 \right)+\left(-7 \right)+\left(-7 \right)=\left(-21 \right)$ and $3 \times \left(-2 \right)$ is equal to $\left(-2 \right)+\left(-2 \right)+\left(-2 \right)=\left(-6 \right)$. From these illustrations, it can be found that the product of any two integers with unlike signs is negative while the product of any two positive integers is of course a positive integer.

Now let us find out the sign of the product of any two negative integers by considering the series of statements below. (See also the signs of the results for dividing integers.)

$1.\;\; -2\left ( 3 \right )=-6 \; \; \; \;$ The product of any two numbers with unlike signs is negative.
$2.\;\; -2\left ( 0 \right )=0 \; \; \; \;$ The product of any number and zero is always 0.
$3.\;\; 3+ \left ( -3 \right )=0 \; \; \; \;$ This statement explains the additive inverse property.
$4.\;\; -2 \left [3+ \left (-3 \right ) \right ]=0 \; \; \; \;$ Substituting 0 in number 2 with the statement in number 3.
$5.\;\; -2\left ( 3 \right ) +\left ( -2 \right )\left ( -3 \right )=0 \;\;\;$ Applying the distributive property over multiplication.
$6.\;\; -6+\left (-2 \right )\left (-3 \right )=0 \;\;\;$ Substituting $-2\left ( 3 \right)$ by -6, refer to statement 1.
$7.\;\; \left ( -2 \right )\left ( -3 \right )$ must be equal to 6 so that equation 6 will be true. Applying additive inverse property
$8.\;\; \therefore \left ( -2 \right )\left ( -3 \right )=6$

As shown above, it can be ascertained that when two negative integers such as -2 and -3 are multiplied the product is a positive integer.

So now we have the rules for multiplying integers:

Rule 1: The product of two integers with like signs is positive.
Rule 2: The product of two integers with unlike signs is negative.

Examples:

Find the product.

$1.\; 2\left ( -3 \right )\left ( 4 \right )=\left ( 2\times -3 \right )\left ( 4 \right )=-6\left ( 4 \right )=-24$
$2.\; \left ( -6 \right )\left ( -4 \right )\left ( -2 \right )=24\left ( -2 \right )=-48$
$3.\; 2\left ( 5 \right )\left ( -3 \right )=10\left ( -3 \right )=-30$
$4.\; \left (-2 \right )\left ( 5 \right )\left ( -3 \right )=-10\left ( -3 \right )=30$
$5.\; \left (-4 \right )\left (-3 \right )=12$