Subtracting Integers
Objective
Quick Review on adding integers
So before we start subtracting integers, let’s go over how to add them first. �
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:
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