Positive integers you already know as natural numbers, and we covered the addition and subtraction of natural numbers already, so we’ll concentrate on the negative integers instead. There are a few simple rules when it comes to the addition and subtraction of integers, and to change things up a little bit, we’ll present them as in the form of a listicle. So, here go the rules for the addition and subtraction of negative numbers.
1. A minus in front of a number changes the sign of the number.
To get a grasp of this rule, we’ll call a couple of old friends to our aid – the number line and the multiplication of natural numbers. Remember how multiplying a number by the number 1 gives you that same number as a result? Well, putting a minus in front of a number is shorthand for multiplying that number by -1. The distance from the origin point on the number line stays the same, but the minus shifts it to the opposite side of the number line.
So, if we put a minus in front of a positive integer, we’ll get a negative version of that same integer. And if we put the minus in front of a negative integer, we’ll get its positive version as a result.
Using just mathematical language, that means that:
$2 \cdot (-1)=-2$
$-2 \cdot (-1)=2.$
2. If a negative integer is behind an operator, it has to be surrounded by parentheses.
This one is here to avoid confusion, because the minus sign is also the operator for subtraction. If we put two operators next to each other, it is unclear if:
- one of them is a sign, and not an operator
- one of them is a typo, or
- a number or a variable is missing between them.
To make things easier, a rule has been created to put negative integers into brackets (parentheses). That way, everybody knows that the minus is there on purpose and that it is a sign.
For example: $ -3 + (-5) = -8 \Rightarrow – 3 – 5 = -8$
Although mistakes can be avoided during addition and subtraction by using rule number one, this rule will be indispensable during multiplication.
3. Adding two negative integers together will always give you a negative integer as a result.
A negative integer represents the distance from a single point positioned left of the point of origin on the number line to the point of origin itself. When we add two negative integers together, we basically get the sum of their distances. But, since both of them are positioned left of the point of origin on the number line, we keep that direction. Like this:
4. Subtracting a negative integer from another negative integer will only give you a negative integer as a result in some cases.
How come, you ask? Well, remember the first rule – a minus in front of a number changes the sign of the number. That also goes for negative integers. If we put a minus in front of a negative integer, it will turn into a positive integer. And when we add a positive integer to any number, we move to the right on the number line.
So, what happens if the subtrahend (second number) is larger than the minuend (first number)? When it turns into a positive integer, we will move past the point of origin and get a positive integer as a result.
5. Subtracting a positive integer from a negative integer is basically the same as adding together two negative integers, and it will always give you a negative integer as a result as well.
Again, rule number one – the minus in front of a positive integer changes its sign. When it does, we’re basically adding two negative integers together, and we covered that in rule number two.
6. Adding a negative integer to a positive integer is basically the same process as the subtraction of two natural numbers.
This is an easy one. An expression like 5 + (-3) can be easily written as 5 – 3, and the result is the same:
$ 5 + (-3) = 5 – 3 = 2$
The only thing we have to watch out for is if the negative number is larger than the positive number. In that case, the result will be a negative number.
7. The commutative property of addition and the associative property of addition that are valid for natural numbers are valid for integers as well.
The commutative property of addition and the associative property of addition are the same for both natural numbers and integers. Just be careful moving the signs around, and you’ll do fine.
Understanding these rules helps us to solve practical problems. Now we know how to solve the problem from the previous lesson. Let’s repeat the problem:
The air temperature today at noon was 39.2°F and by the evening the air temperature declined by 42.8°F. What was the air temperature in the evening?
Now we know that the temperature in the evening was -3.6°F.
If you want to practice a bit more, we prepared some worksheets for you. You can download them using the links below.
Addition and subtraction of integers worksheets
Two integers (96.0 KiB, 1,389 hits)
Three integers (261.5 KiB, 1,070 hits)
Four integers (325.1 KiB, 1,073 hits)