Multiply and Divide Integers

Adding and multiplying integers includes multiplying and dividing positive whole numbers, as well as multiplying and dividing negative numbers. 

All of the normal rules of multiplication and division apply. And these problems are all much easier if you know your multiplication and division facts.

The only complication when you work with integers is remembering how positive and negative numbers work with each other. 

Unlike the rules for adding and subtracting integers, the rules for multiplication and division of integers are simple:

• When you multiply or divide numbers with the same sign (both positive or both negative), the answer is positive.
• When you multipy or divide numbers with different signs (one positive and one negative), the answer is negative. 

Example:

$$\eqalign{^+ 3 \times ^+4&=^+12\\^+ 24 \div^+8&=^+4}$$

These are all positive numbers, so they are just normal multiplication and division facts.  You are multiplying positive numbers by positive numbers and dividing positive numbers by positive numbers, just like you always have.  So, the answers are positive, just as they always have been when you multipy or divide. 

$$\eqalign{^- 3 \times ^-4&=^+12\\^- 24 \div^-8&=^+4}$$

When you multiply or divide two negative numbers, the answer is still positive. So, botton line, if you multiply or divide two integers with the same sign, the answer will be positive. 

$$\eqalign{^+ 3 \times ^-4&=^-12\\^- 3 \times ^+4&=^-12\\^+ 24 \div^-8&=^-4\\^- 24 \div^+8&=^-4}$$

No matter which number is negative, if one number is positive and one negative, the answer is negative. So, when you multiply two integers with different signs, the answer will be negative. 

If you can remember the rule (same=positive; different=negative), you will find it easy to multiply or divide positive and negative numbers. 

These rules apply when you are multiplying two integers, but also when you are multiplying many integers together.

Sometimes students get confused when they have lots of terms to multiply.  To keep it simple, when doing a problem with many integer multiplication or division problems, always multiply or divide two terms at a time, find the answer, decide if the answer is positive or negative, and then put that answer into your big problem to multiply or divide by the next term. Go through the problem two terms by two terms. This way, you can apply the rules to each pair of integers as you multiply or divide them. Once you get used to working with integers, you'll be able to see more quickly whether answers will be positive or negative, but workin through by pairs will always get you to the right answer. 

Example:

$$\eqalign{\mathbf{^-3\times 4}\times9\times^-10\times-11&=&&\qquad\text{Remember, if there is no sign, the number is positive, so 4 is positive here}\\\\^-3\times 4=(^-12) \downarrow&\quad &&\text{Multiply the first two terms and bring -12 into full problem}\\ \\\mathbf{(^-12)\times9}\times^-10\times-11&=\\^-12\times9=(^-108) \downarrow&\quad &&\text{Multiply the next two terms, and bring answer down into rest of problem}\\\\\mathbf{(^-108)\times^-10}\times^-11&=\\^-108\times^-10=(1080)& \quad &&\text{Multiply the next two terms -- note: negative times negative equals positive}\\\\(1080)\times ^-11&=\mathbf{^-11880}&&\text{Multiply the last two terms for final answer}}$$

If you have a long list of terms to multiply, multiply them pair by pair to keep straight the positive and negative signs.