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Lesson Explainer: Properties of Addition of Rational Numbers Mathematics • 7th Grade

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 π‘Žπ‘, π‘π‘‘βˆˆβ„š, then π‘Žπ‘+𝑐𝑑=π‘Žπ‘‘π‘π‘‘+𝑏𝑐𝑏𝑑=π‘Žπ‘‘+𝑏𝑐𝑏𝑑.

We can also note that since 𝑏 and 𝑑 are nonzero, then 𝑏𝑑 is also nonzero. So, π‘Žπ‘‘+𝑏𝑐, π‘π‘‘βˆˆβ„€, and 𝑏𝑑≠0. 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 14+12 on the following number line.

We combine the displacements of 14 and 12 from 0 to get the point at a displacement of 34 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 π‘Ž units and then 𝑏 units, this will be the same as moving 𝑏 units and then π‘Ž units. In particular, for any rational numbers π‘Žπ‘ and 𝑐𝑑, π‘Žπ‘+𝑐𝑑=𝑐𝑑+π‘Žπ‘.

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 π‘Žπ‘+ο€»βˆ’π‘Žπ‘ο‡ for a rational number π‘Žπ‘. We know that this will be the point on the number line that is π‘Žπ‘ units from 0 in one direction and π‘Žπ‘ 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 π‘Žπ‘ have an additive inverse βˆ’π‘Žπ‘, 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 π‘Žπ‘, we should have π‘Žπ‘+0=π‘Žπ‘. 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 π‘Žπ‘+0=0+π‘Žπ‘=π‘Žπ‘.

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 π‘Žπ‘, 𝑐𝑑, and 𝑒𝑓, we have

  • π‘Žπ‘+𝑐𝑑=π‘Žπ‘‘+𝑏𝑐𝑏𝑑, which shows that the sum of rational numbers is a rational number (the closure property of the addition of rational numbers),
  • π‘Žπ‘+0=0+π‘Žπ‘=π‘Žπ‘, where 0 is called the additive identity (the additive identity property of the addition of rational numbers),
  • π‘Žπ‘+𝑐𝑑=𝑐𝑑+π‘Žπ‘, which tells us we can reorder the sum of rational numbers (the commutativity property of the sum of rational numbers),
  • ο€»π‘Žπ‘+𝑐𝑑+𝑒𝑓=π‘Žπ‘+𝑐𝑑+𝑒𝑓, which tells us we can evaluate the sum of rational numbers in any order (the associativity property of the addition of rational numbers),
  • βˆ’π‘Žπ‘βˆˆβ„š and π‘Žπ‘+ο€»βˆ’π‘Žπ‘ο‡=0 (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 π‘Žπ‘ is the number whose sum with π‘Žπ‘ makes 0. Thus, the additive inverse of 0.7 is number π‘Ž such that 0.7+π‘Ž=0.

We can solve this for π‘Ž by subtracting 0.7 from both sides. We get π‘Ž=βˆ’0.7.

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 βˆ’0.7.

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 ο€Ό513+34+14 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 ο€Ό513+34+14=513+ο€Ό34+14.

We then note that 34 and 14 have the same denominator, so we can add these fractions together and simplify as follows: 513+ο€Ό34+14=513+ο€Ό3+14=513+44=513+1.

Finally, we can rewrite 1 as 1313 and evaluate to get 513+1=513+1313=5+1313=1813.

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?

  1. 12+23=23+12
  2. 12+ο€Όβˆ’12=0
  3. 23+0=23
  4. 12+23=76
  5. ο€Ό12+23+34=12+ο€Ό23+34

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 π‘Žπ‘, 𝑐𝑑, π‘’π‘“βˆˆβ„š, then ο€»π‘Žπ‘+𝑐𝑑+𝑒𝑓=π‘Žπ‘+𝑐𝑑+𝑒𝑓.

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 π‘Ž=1, 𝑏=2, 𝑐=2, 𝑑=3, 𝑒=3, and 𝑓=4. 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 12+0=12?

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 π‘Žπ‘, we must have π‘Žπ‘+0=π‘Žπ‘, and by applying commutativity, we also have 0+π‘Žπ‘=π‘Žπ‘.

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 13+ο€Όβˆ’13οˆβˆ’1113+1113 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 13+ο€Όβˆ’13οˆβˆ’1113+1113=ο€Ό13+ο€Όβˆ’13+ο€Όβˆ’1113+1113.

We then note that 13 and βˆ’13 and βˆ’1113 and 1113 are additive inverses, so their respective sums are both 0.

Hence, ο€Ό13+ο€Όβˆ’13+ο€Όβˆ’1113+1113=0+0=0.

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 ο€Ό14+34+βˆ’14.

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 ο€Ό14+34+βˆ’14=ο€Ό1+34+βˆ’14=1+3βˆ’14=34.

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 ο€Ό14+34+βˆ’14=βˆ’14+ο€Ό14+34.

Then, by applying the associativity property of the addition of rational numbers, we have βˆ’14+ο€Ό14+34=ο€Όβˆ’14+14+34.

Next, we note that 14 and βˆ’14 are additive inverses. Thus, ο€Όβˆ’14+14+34=0+34.

Finally, we note that 0 is the additive identity, so 0+34=34.

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, π‘Žπ‘+𝑐𝑑=π‘Žπ‘‘+𝑏𝑐𝑏𝑑 for integers π‘Ž, 𝑏, 𝑐, and 𝑑, where 𝑏 and 𝑑 are nonzero.
    • The additive identity property: the rational number 0 acts as an identity of addition. This means that, for any rational number π‘Žπ‘, we must have π‘Žπ‘+0=0+π‘Žπ‘=π‘Žπ‘. 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 π‘Žπ‘ and 𝑐𝑑, we have π‘Žπ‘+𝑐𝑑=𝑐𝑑+π‘Žπ‘.
    • 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 π‘Žπ‘, 𝑐𝑑, and 𝑒𝑓, we have ο€»π‘Žπ‘+𝑐𝑑+𝑒𝑓=π‘Žπ‘+𝑐𝑑+𝑒𝑓.
    • 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 π‘Žπ‘, we must have βˆ’π‘Žπ‘βˆˆβ„š and π‘Žπ‘+ο€»βˆ’π‘Žπ‘ο‡=0.

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