Multiplying & Dividing Integers Lesson

How to Multiply two Integers

Up to this point, we have only performed the multiplication operation with whole numbers. From our previous studies, we should know that the product of two positive numbers is positive:
(+3) x (+5) = +15
(+8) x (+9) = +72
What happens when we start multiplying with negative numbers? When multiplying with two integers only, the rules are very simple:
The product is positive if the two signs are the same:
(+) x (+) = (+)
(-) x (-) = (+)
The product is negative if the two signs are different:
(+) x (-) = (-)
(-) x (+) = (-)
To multiply two integers together, we multiply absolute values and then attach the correct sign based on the sign rules. Let's take a look at a few examples.
Example 1: Find each product.
(-5) x (-8)

• Multiply the absolute values
• 5 x 8 = 40
• Attach the correct sign based on the sign rules:
  • Both factors (-5) and (-8) are negative
  • (-) x (-) = (+)
• +40

(-5) x (-8) = 40
Example 2: Find each product.
(-12) x (13)

• Multiply the absolute values
• 12 x 13 = 156
• Attach the correct sign based on the sign rules:
  • One factor is negative (-12) and the other is positive (13)
  • (-) x (+) = (-)
• -156

(-12) x (13) = -156
Example 3: Find each product.
(-21) x (18)

• Multiply the absolute values
• 21 x 18 = 378
• Attach the correct sign based on the sign rules:
  • One factor is negative (-21) and the other is positive (18)
  • (-) x (+) = (-)
• -378

(-21) x (18) = -378

Multiplying more than two Integers

What happens when we want to multiply more than two integers? There are two ways to complete this task:

1. We can multiply pairs of numbers until we have our final product. Remember, multiplication is commutative, so we can rearrange the multiplication into whatever order is easiest.
2. We can determine the sign of the product first and then multiply absolute values. This is generally a little faster since you only deal with the sign once:
   • Each pair of negative signs yields a positive. For every 2 negatives, you will get a positive result. This means if there is an even number (2,4,6,8,10,…) of negatives, you will get a positive product.
   • When there is an odd number (1,3,5,7,9,…) of negatives, you will get a negative product.

Let's look at a few examples.
Example 4: Find each product.
(-5) x (4) x (-2) x (-8) x (3)

• If we use the second method, we can count the number of negatives involved. Here we have 3 negative numbers: (-5), (-2), and (-8). Since 3 is an odd number, the result of the multiplication will be negative.
• Now that we know the sign, we can multiply absolute values in any order that we would like.
• 5 x 4 x 2 x 8 x 3 = 20 x 2 x 8 x 3 = 40 x 8 x 3 = 320 x 3 = 960
• Attach the (-) sign to the answer above
• -960

(-5) x (4) x (-2) x (-8) x (3) = -960
Example 5: Find each product.
(-1) x (-6) x (-5) x (4) x (-2)

• If we use the second method, we can count the number of negatives involved. Here we have 4 negative numbers: (-1), (-6), (-5), and (-2). Since 4 is an even number, the result of the multiplication will be positive.
• Now that we know the sign, we can multiply absolute values in any order that we would like.
• 1 x 6 x 5 x 4 x 2 = 6 x 5 x 4 x 2 = 30 x 4 x 2 = 120 x 2 = 240
• The answer will be positive (+)
• +240

(-1) x (-6) x (-5) x (4) x (-2) = 240

How to Divide Integers

Once you understand how to multiply two integers, division will be easy. We divide integers using the same rules as for multiplication.
If the signs are the same, the quotient will be positive:
(+) ÷ (+) = (+)
(-) ÷ (-) = (+)
If the signs are different, the quotient will be negative:
(+) ÷ (-) = (-)
(-) ÷ (+) = (-)
To divide integers, we divide absolute values and then attach the correct sign based on the sign rules. Let's look at a few examples:
Example 6: Find each quotient.
(-18) ÷ (-6)

• Divide the absolute values
• 18 ÷ 6 = 3
• Attach the correct sign based on the sign rules:
  • Both (-18) and (-6) are negative
  • (-) ÷ (-) = (+)
• +3

(-18) ÷ (-6) = 3
Example 7: Find each quotient.
(-28) ÷ (7)

• Divide the absolute values
• 28 ÷ 7 = 4
• Attach the correct sign based on the sign rules:
  • The dividend is negative (-28) and the divisor is positive (7)
  • (-) ÷ (+) = (-)
• -4

(-28) ÷ (7) = -4

Sign Rules for Multiplying & Dividing Integers

To develop an understanding for the sign rules presented earlier, let's start with a simple multiplication problem. Recall that multiplication is a shortcut for repeated addition.
2 x 3 = 3 + 3 = 6
2 x (-3) = -3 + (-3) = -6
This shows us that a positive times a negative will give us a negative. What happens if this is reversed?
-2 x 3 = ?
Recall the commutative property of multiplication allows us to multiply in any order.
-2 x 3 = 3 x (-2)
3 x (-2) = -2 + (-2) + (-2) = -6
This shows us that a negative times a positive will give us a negative as well. What happens when both factors are negative?
For this situation, it's easiest to just think about a pattern.

• 3 x (-2) = -6
• 2 x (-2) = -4
• 1 x (-2) = -2
• 0 x (-2) = 0
• -1 x (-2) = 2
• -2 x (-2) = 4
• -3 x (-2) = 6

Starting with 3 x (-2), which we established is -6, we can see that as we decrease the leftmost factor by 1, the product increases by 2. If we continue this pattern to the point where the leftmost factor is negative, now we see that a negative times a negative results in a positive.
Let's use the sign rules from multiplication to think about the sign rules for division.
12 ÷ 6 = 2
Recall we can find the answer to such a division problem using a related multiplication statement.
? x 6 = 12
Since the ? is 2 here, that is the answer for the division problem.
-12 ÷ 6 = ?
? x 6 = -12
At this point, we know that a positive times a negative or a negative times a positive gives us a negative. So here, we know 6 is positive. So it must be true that ? is -2.
-2 x 6 = -12
-12 ÷ 6 = -2
Using a similar thought process with multiplication, we can show that:
12 ÷ (-6) = -2
-12 ÷ (-6) = 2
We can conclude that a positive divided by a negative or a negative divided by a positive is a negative. Additionally, a negative divided by a negative is a positive.

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