Multiplying And Dividing Integers

What Are Integers and Why Do We Multiply/Divide Them?

Multiplying and dividing integers is the process of performing multiplication or division on whole numbers and their opposites, which can be positive, negative, or zero. Integers are all the whole numbers you know, like $0,1,2,3,...$, but they also include their negative counterparts: $...,-3,-2,-1$. Think of them as all the whole steps on an infinitely long number line, going both left and right from zero.

We use them all the time in real life. For example, if the temperature drops $2$ degrees every hour for $3$ hours, you can multiply $3\times (-2)$ to find the total change is $-6$ degrees. Or, if you owe someone $\mathrm{\$}20$ and want to split the debt among $4$ friends, you'd calculate $(-20)÷4$ to see that each person owes $\mathrm{\$}5$.

What Are the Rules for Multiplying Integers?

The best part about multiplying integers is that the rules for the signs are consistent and easy to remember. You first multiply the numbers as you normally would, ignoring the signs. Then, you figure out the sign of the answer using these two key rules:

1. If the signs of the two integers are the same, the result is always positive.
2. If the signs of the two integers are different, the result is always negative.

Let's break that down:

• Positive × Positive = Positive. Example: $5\times 3=15$. You've been doing this for years!
• Negative × Negative = Positive. Example: $(-5)\times (-3)=15$. This might seem strange, but two negatives cancel each other out to make a positive. Think of it as "removing a debt," which is a good thing (a positive).
• Positive × Negative = Negative. Example: $5\times (-3)=-15$.
• Negative × Positive = Negative. Example: $(-5)\times 3=-15$.

Here is a simple table to help you remember:

A great way to summarize this is with a simple phrase:

How Do You Multiply Integers? (Examples)

Let's put the rules into practice with a couple of step-by-step examples.

Are the Rules for Dividing Integers the Same?

Yes, they are! This makes learning much easier. The exact same sign rules that apply to multiplication also apply to division. You just divide the numbers first, then determine the sign of the answer.

1. If the signs of the two integers are the same, the quotient (the answer to a division problem) is positive.
2. If the signs of the two integers are different, the quotient is negative.

Let's see how this works:

• Positive ÷ Positive = Positive. Example: $20÷4=5$.
• Negative ÷ Negative = Positive. Example: $(-20)÷(-4)=5$.
• Positive ÷ Negative = Negative. Example: $20÷(-4)=-5$.
• Negative ÷ Positive = Negative. Example: $(-20)÷4=-5$.

The memory aid works perfectly here too:

This simple rule is the key to mastering both multiplication and division of integers.

How Do You Divide Integers? (Examples)

Let's walk through some division problems to see the rules in action.

What Happens When Zero Is Involved?

Zero is a special integer with its own set of rules. These are very important to remember because they are different from the rules for positive and negative numbers.

Multiplication with Zero

This rule is simple: Any number multiplied by zero is always zero.

It doesn't matter if $n$ is positive or negative. For example, $(-1000)\times 0=0$ and $50\times 0=0$.

Division with Zero

Division with zero is a bit more complex. There are two scenarios:

1. Zero divided by another number: If you have zero items and you divide them among any number of people, each person gets zero. So, zero divided by any non-zero integer is always zero.

For example, $0÷(-12)=0$.

2. Dividing by zero: This is the most important rule of all: You can NEVER divide by zero. It is not possible. The answer is not zero; it is not a large number; it is simply "undefined." Think about it: if $12÷4=3$ means that $4\times 3=12$, what would $12÷0$ be? We would need a number that, when multiplied by $0$, gives $12$. But we already know that any number times $0$ is $0$, not $12$. It's an impossible task!

What Are Some Common Mistakes to Avoid?

When you're first learning, it's easy to make a few common mistakes. Watch out for these pitfalls!

• Confusing Addition/Subtraction Rules: The biggest mistake is mixing up the rules for multiplication/division with the rules for addition/subtraction. For example, $(-5)+(-2)=-7$, but many students mistakenly think it should be positive. Remember, the "two negatives make a positive" rule is only for multiplication and division.
• Forgetting the Sign: Sometimes you'll do the math correctly ($8\times 4=32$) but forget to apply the sign rule. If the problem was $(-8)\times 4$, the answer is $-32$, not $32$. Always double-check the signs before you write your final answer.
• Errors in Long Problems: When you have a chain of multiplications like $(-2)\times (-3)\times (-4)$, it's easy to lose track of the sign. A good trick is to count the number of negative signs. An odd number of negative signs means the final answer is negative. An even number of negative signs means the final answer is positive. In our example, there are three negatives (odd), so the answer will be negative: $6\times (-4)=-24$.
• Dividing by Zero: Writing that $8÷0=0$ is a very common error. Remember, division by zero is always "undefined."

Quick Summary and Reference

Feeling overwhelmed? Don't be! It all boils down to a few key ideas. Use this table as a quick reference guide whenever you're working on problems.

The Core Rules for Signs (Multiplication & Division)

The Rules for Zero

• Any integer multiplied by $0$ is $0$.
• $0$ divided by any non-zero integer is $0$.
• You can never divide by $0$. The answer is undefined.