In this explainer, we will learn how to use properties of addition in the set of rational numbers and determine the additive inverse.
We have seen that we can add rational numbers in a few different ways. For example, we can represent rational numbers as points on a number line and add their displacements from 0 to add the numbers together. This gives us a way of adding the numbers together numerically by rewriting the fractions to have the same denominator.
In general, we know that if , , then
We can also note that since and are nonzero, then is also nonzero. So, , , and . Hence, the sum of two rational numbers is a rational number.
This is a useful way to add any two rational numbers. However, we can use both this and the number line example of the addition of rational numbers to justify many useful properties of the addition of rational numbers.
For example, consider on the following number line.
We combine the displacements of and from 0 to get the point at a displacement of from 0 in the positive direction. However, we can notice something interesting: it does not matter in which order we add the displacements, because we will always end up at the same point. We can see this in the diagram.
This result will be true in general; if we move units and then units, this will be the same as moving units and then units. In particular, for any rational numbers and ,
This is known as the commutative property of the addition of rational numbers. It is worth noting that the above explanation is not a proof of this property, but it is a justification of why we define addition in this way.
There are many other properties we can justify by using this same reasoning. For example, consider for a rational number . We know that this will be the point on the number line that is units from 0 in one direction and units in the other direction; these cancel out to give the point of displacement 0 from 0.
This is known as the additive inverse property of the addition of rational numbers. It is worth noting that this property has one extra fact, which is that all rational numbers have an additive inverse , which is itself a rational number.
There is one more result we will show by using a number line. We note that adding 0 to any rational number will not change its displacement from 0 on a number line.
Hence, for any rational number , we should have . This is called the additive identity property of the addition of rational numbers. This is often combined with the commutativity of the addition of rational numbers to say that .
There are more results similar to these properties, which we can justify by considering number lines; however, showing these results is outside the scope of this explainer. Instead, we will state some of these properties as follows.
Property: Properties of the Addition of Rational Numbers
For any rational numbers , , and , we have
- , which shows that the sum of rational numbers is a rational number (the closure property of the addition of rational numbers),
- , where 0 is called the additive identity (the additive identity property of the addition of rational numbers),
- , which tells us we can reorder the sum of rational numbers (the commutativity property of the sum of rational numbers),
- , which tells us we can evaluate the sum of rational numbers in any order (the associativity property of the addition of rational numbers),
- and (the additive inverse property of the addition of rational numbers).
Let’s now see an example of determining the additive inverse of a given rational number.
Example 1: Finding the Additive Inverse of a Given Rational Number
Find the additive inverse of 0.7.
Answer
We recall that the additive inverse of a rational number is the number whose sum with makes 0. Thus, the additive inverse of 0.7 is number such that
We can solve this for by subtracting 0.7 from both sides. We get
It is worth noting that, in general, the additive inverse of a rational number is its negative. So, the additive inverse of 0.7 is .
In our next example, we will use the properties of the addition of rational numbers to simplify an expression involving the sum of rational numbers.
Example 2: Using the Properties of Addition to Simplify a Numerical Expression
Simplify using the properties of addition.
Answer
We could answer this question by writing all of the rational numbers to have equal denominators. However, we are told to simplify the expression using the properties of addition, and the reason for this is that it allows us a shortcut.
We recall that the associativity property for the addition of rational numbers allows us to evaluate this sum in any order. In particular, we have
We then note that and have the same denominator, so we can add these fractions together and simplify as follows:
Finally, we can rewrite 1 as and evaluate to get
In our next example, we will determine which of five given equations represents the associative property of the addition of rational numbers.
Example 3: Identifying the Associative Property with Rational Numbers
Which equation shows the associative property of addition?
Answer
We recall that the associative property of the addition of rational numbers tells us that we can add rational numbers in any order. More formally, this says, if , , , then
It is worth noting that each of the given options is a different property of the addition of rational numbers. Option A is the commutativity property, option B is the additive inverse property, option C is the additive identity property, and option D is an example of the closure property.
We see then that option E is the associativity equation with , , , , , and . Hence, option E shows the associative property of the addition of rational numbers.
In our next example, we will determine which property of the addition of rational numbers is shown by a given equation.
Example 4: Identifying an Additive Property with Rational Numbers
What property of addition is demonstrated by ?
Answer
The equation we are given simplifies the sum of a rational number and 0 to show that addition by 0 leaves the number unchanged. This is why 0 is known as the additive identity; it leaves numbers unchanged under addition. More formally, we say for any rational number , we must have , and by applying commutativity, we also have .
Hence, this equation shows an example of the additive identity property.
In our next example, we will evaluate an expression involving the sum and difference of rational numbers by using the properties of the addition of rational numbers.
Example 5: Using the Properties of Addition to Simplify a Numerical Expression
Find the value of the expression using the properties of addition.
Answer
We first recall that, by applying the associativity property of the addition of rational numbers, we can evaluate this expression in any order. In particular, we have
We then note that and and and are additive inverses, so their respective sums are both 0.
Hence,
In our final example, we will see two different methods for evaluating an expression involving the sum of rational numbers.
Example 6: Using the Properties of Addition of Rational Numbers to Complete a Calculation
Evaluate .
Answer
We can evaluate this expression in several different ways. For example, we can note that the denominators of the three fractions are all equal. This means we can evaluate this sum by adding the numerator. This gives
This is not the only method we could use; we can also use the fact that the addition of rational number is commutative to rewrite the expression as
Then, by applying the associativity property of the addition of rational numbers, we have
Next, we note that and are additive inverses. Thus,
Finally, we note that 0 is the additive identity, so
Let’s finish by recapping some of the important points from this explainer.
Key Points
- The addition of rational numbers has several properties that can be used to simplify expressions, such as the following:
- The closure property: the sum (or difference) of any rational numbers is also rational. In general, for integers , , , and , where and are nonzero.
- The additive identity property: the rational number 0 acts as an identity of addition. This means that, for any rational number , we must have . In other words, addition by 0 leaves rational numbers unchanged.
- The commutativity property: the sum of rational numbers is commutative. This means that we can reorder any sum of rational numbers. In general, for any two rational numbers and , we have .
- The associativity property: the sum of rational numbers is associative. This means we can evaluate the sum of rational numbers in any order. In general, this means for any rational numbers , , and , we have .
- The additive inverse property: all rational numbers have an additive inverse that is also a rational number. In general, this tells us that, for any rational number , we must have and .
