Multiplying integers

Objective

• Review over how to add integers
• Review over how to add two digit integers
• Review over how to subtract integers
• Review over how to subtract two digit integers
• What is multiplication?
• Go over how to multiply positive integers
• Go over how to multiply negative integers
• Go over how to multiply positive and negative integers

SO THEN HOW DO WE ADD POSITIVE INTEGERS?

So, to add positive integers, we start with the first number given to us, and count upwards of the second number.

For example:

​

3 + 4

We start with 3

​

1 2 3

And then we count upwards 4 more times:

1. 2 3

4

5

6

7

1

2

3

4

So our answer for this problem would be 7

SO HOW DO WE ADD 2 DIGIT POSITIVE INTEGERS?

Adding two digit positive integers is very similar to adding only one digit positive integers, with a tiny change to it.

However, it’s easier to show rather than explain:

23

+18

​

First, we need to add the numbers to the left (in the one’s digit)

3

8

Now here’s a math trick to help you save some time.

Instead of starting at 3, and counting up 8, let’s make our lives easier and start at the bigger number (8), then count up the smaller number (3).

So:

1 2 3 4 5 6 7 8

9 10 11

11

Now, we add the left side

2

1

So, we start at 2, and count up 1.

1 2

3

3

Finally, we add the left side

+___

1

​

Now, we add the right side

4

SO THEN HOW DO WE ADD NEGATIVE INTEGERS?

The same way we add positive integers, except if the two numbers are negative, the answer will also be negative

So, to add negative integers, we start with the first number given to us, and count upwards of the second number.

For example:

​

-3 + (-4)

We start with -3

​

-1 -2 -3

And then we count upwards 4 more times:

-1 -2 -3

-4

-5

-6

-7

-1

-2

-3

-4

So our answer for this problem would be -7

SO HOW DO WE ADD 2 DIGIT NEGATIVE INTEGERS?

Again, if they are both negative, the same way that we would if they were positive, only now the answer is negative.

Let’s use our first example, but make them negative instead.

-23

+(-18)

​

First, we need to add the numbers to the left (in the one’s digit)

3

8

Now here’s a math trick to help you save some time.

Instead of starting at 3, and counting up 8, let’s make our lives easier and start at the bigger number (8), then count up the smaller number (3).

So:

-1 -2 -3 -4 -5 -6 -7 -8

-9 -10 -11

-11

Now, we add the left side

2

1

So, we start at -2, and count up -1.

-1 -2

-3

-3

Finally, we add the left side

+_____

1

​

Now, we add the right side

-4

ADDING BOTH POSITIVE AND NEGATIVE INTEGERS

So what happens when we want to add both positive and negatives together?

Well, believe it or not, that’s subtraction.

Yep, that’s all adding a positive to a negative is, subtraction.

So let’s look at an example:

-1

+ 5

Now there are a few different ways to look at this problem, but I think the best way is to look at the signs of each of these numbers.

+ 5

-1

We can see that 5 is positive and 1 is negative.

We also know that we can rearrange anything we want with addition, as long as we keep the proper signs.

So, instead of doing: -1 + 5

Let’s switch it and make it:

5 + (-1)

Which is the same as saying:

5 – 1 =

4

SO HOW DO WE KNOW WHICH GETS THE SIGN?

Here’s a trick to remember which number will carry the sign.

The bigger number in the addition problem will always carry the sign.

So, for example:

​

-3 + 7

7 is the bigger number, and we can see that 7 is positive, so the answer will be positive.

= 4

However, if we were to have something like:

​

9 + (-15)

​

We can see that -15 is the bigger number, and since -15 is negative, the answer will be negative.

= -6

SO HOW DO WE SUBTRACT POSITIVE INTEGERS?

We know that subtraction is the opposite of addition, so, we can safely say that subtraction will be starting with the first number, and counting down the second number.

So when it comes to subtraction, one thing that is important to remember is to turn the subtraction into addition.

Now, this may sound strange, but let’s do an example to show you what is meant.

38

- 24

As we can see, we are subtracting 38 from 24, but what we need to see is their signs.

38 is positive since it doesn’t have a sign attached to it.

However, 24 is negative.

Now, even though we don’t technically need to, it’s a good idea to get in the habit of adding their opposite.

38

+(- 24)

As we learned yesterday, now that we are adding, we simply need to subtract them off, and the answer will be the larger number’s sign.

So, starting from the right side:

4

1

8 – 4 = 4

3 – 2 = 1

So our answer is 14.

So, what about subtracting negative integers?

Again, it’s important that we add it’s opposite.

So even though it seems ridiculous, it will help.

So, let’s start with another example:

-16

- 15

Now, we can see this looks pretty ugly, especially since we are taking a negative number subtracting a positive number.

But…..is this really a positive number?

-16

+(-15)

When we change the problem from a subtraction problem, to an addition problem, now we can see that the 15 is actually negative.

And we know how to add two negative numbers together

-6 + (-5) = -11

-11

-1 + (-1) = -2

(-2 )

+____

-31

Subtracting positive and negative integers

So basically, again, as long as you add it’s opposite, you’ll see it becomes easier to solve.

For example:

-21

- (-19)

Now, let’s change the equation by adding its opposite instead, and we’ll see that:

-21

+(+19)

When we change the problem from a subtraction problem, to an addition problem, now we can see this problem becomes 21 - 19.

And we know what the answer it

2

But, since the biggest number (21) is negative, that means our answer should be:

-2

Subtracting positive and negative integers

Last one, and again, as long as you add it’s opposite, you’ll see it becomes easier to solve.

For example:

21

- (-19)

Now, let’s change the equation by adding its opposite instead, and we’ll see that:

21

+(+19)

When we change the problem from a subtraction problem, to an addition problem, now we can see this problem is 21 + 19, or an addition problem between two positive integers.

And we know how to solve this!

Starting from the right side:

10

1 + 9 = 10

2 + 1 = 3

3

+_____

40

What is multiplication?

We know that multiplication is a procedure in math, and we know that we need to memorize our times tables, but what is multiplication?

Well, multiplication is adding the same number, multiple times.

So, instead of writing something like:

We can instead, count the number of 7’s we have here:

And simplify this long equation to just:

7(10) = 70

Rules/Ways to multiply numbers

So how do we actually do the multiplication?

Well, there are a few ways to do it, so here are some ways to multiply:

25

x 12

• Multiplying the bottom right number by the top right number: 2(5)

2

1. When dealing with two or more digit integers, we can multiply them by first:

5

10

• Multiplying the bottom right number by the top left number: 2(20)

• Multiplying the bottom left number by the top right number:10(5)

• Multiplying the bottom left number by the top right number:10(20)

2

40

1

50

200

+_____

300

2. When dealing with two or more digit integers, we can also multiply them by:

72�x 28

• Breaking the numbers into their one and tens digits like so:

72 = 70 + 2

28 = 20 + 8

• Making a box, and putting those numbers in the box:

• Now we multiply each number on the left of the box, by the number on the top:

20 x 70 = 1400

1400

560

40

16

• Lastly, we add all the numbers together:

1400

560

40

+ 16

2016

3. When dealing with two or more digit integers, we can also multiply them by:

43

x 27

• Multiplying the bottom right number by the top right number, then carrying the tens digit (like adding): 7(3) = 21 (but we carry the 2)

7

3

1

2

• Multiplying the bottom right number by the top left number: 7(4) then adding the left over 2: (28 + 2) = 30

4

30

70 x 8 = 560

20 x 2 = 40

8 x 2 = 16

• Now, place a 0 in the ones place, and then multiply the bottom left number by the top right number: 2(3) = 6

2

0

6

• Now, multiply the bottom left number by the top left number: 2(4) = 8

8

• And finally, add everything together

+_____

1161

Multiplying positive integers

So now that we know how to multiply, let’s look at multiplying positive numbers together to see what we get:

So, when we multiply two positive numbers like so:

12�x 7

What we get should be another positive number, since the two numbers are positive:

84

But why?

Why is it, that when we multiply two positive numbers together, we get another positive number?

Well, it goes back to the definition of multiplication.

Remember?

Multiplication is nothing more than adding the same number multiple times.

So, our multiplication problem is an easier way of writing:

And we know, that when we’re adding positive numbers together continuously,

we’re going to get another positive number.

Also, for those who are wondering, our equation doesn’t necessarily have to be just 7 added 12 times.

We could write it as 12 added 7 times:

And we would still get the same answer:

Multiplying negative integers

So then, what would multiplying a negative integer with a positive integer look like?

Well, this is a little more complicated, but let’s go back to the definition of multiplication with a new problem:

(-13)� x (-9)

Now, we know that multiplication is adding the same number multiple times.

However, in this equation, we have two negative numbers.

So what does that tell us?

What it actually tells us, is that we are to subtract the same number multiple times.

And, of course, we know how to do that right?

Even though this is ugly, we know how to change this right?

We are subtracting, so we need to add its opposite.

So our subtraction problem will become:

So, I’m sure you’re wondering why the first -9 became a positive.

The reason why is because we are subtracting all of the -9’s from themselves.

So, in this rare instance, the first -9 is included and changes sign.

And again, we don’t have to subtract (-9) 13 times, we can also subtract (-13) nine times:

Multiplying positive and negative integers

So now that we know how to multiply positive with positive integers together, and negative with negative integers together, what about positive and negative integers?

So, let’s start with an example and see what happens:

19� x (-3)

So, remember, multiplication is adding the same number multiple times

Unless we have a negative.

Then it’s subtracting the same number multiple times.

So in this case, written out, we would have:

But again, we know how to do that!

We just did it!

When we subtract, we have to make sure to add its opposite:

Again, I know you may be wondering why the first 19 is negative,

But this is a rare instance since we need to subtract all of the 19’s together.

So the first 19 would become a negative.

And yet again, we can see that if we switch this, and instead add (-3) 19 times,

(since 19 is positive, so we can add instead of subtract)

That we will get:

So, what’s the general rule then?

Basically, when you are multiplying (and also dividing):

• A positive times a positive will always give you a positive
• A negative times a negative will always give you a positive
• A positive times a negative will always give you a negative

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