2.04 Properties of operations with rational numbers
Introduction
We have learned the properties of operations with integers in order to rewrite and manipulate expressions to make evaluation easier.
We will now extend our previous understanding of properties of operations to add, subtract, multiply and divide rational numbers.
Properties of operations with rational numbers
The box below summarizes some of the properties of real numbers and gives an example of each.
| Property | Symbols | Example |
|---|---|---|
| \text{Commutative property of addition} | a+b=b+a | \dfrac{1}{2} + \dfrac{1}{4}=\dfrac{1}{4}+\dfrac{1}{2} |
| \text{Commutative property of} \\ \text{multiplication} | a \times b=b \times a | \dfrac{1}{2} \times \dfrac{1}{4}=\dfrac{1}{4} \times \dfrac{1}{2} |
| \text{Associative property of addition} | a+(b+c) = \\ (a+b)+c | \dfrac{1}{2}+\left(\dfrac{1}{4}+\dfrac{1}{3}\right)= \left(\dfrac{1}{2}+\dfrac{1}{4}\right)+\dfrac{1}{3} |
| \text{Associative property of}\\ \text{multiplication} | a \times (b\times c)= \\ (a\times b) \times c | \dfrac{1}{2} \times \left(\dfrac{1}{4} \times \dfrac{1}{3}\right)=\left(\dfrac{1}{2} \times \dfrac{1}{4}\right) \times \dfrac{1}{3} |
| \text{Distributive property} | a\times (b+c)= \\ a \times b+a \times c \\ \text{or} \\ a \times (b-c)= \\ a\times b-a \times c | \dfrac{1}{2} \times \left(\dfrac{1}{4}+\dfrac{1}{3} \right)=\dfrac{1}{2} \times \dfrac{1}{4}+\dfrac{1}{2} \times \dfrac{1}{3} \\ \text{or} \\ \dfrac{1}{2} \times \left(\dfrac{1}{4}-\dfrac{1}{3}\right)=\dfrac{1}{2} \times \dfrac{1}{4}-\dfrac{1}{2}\times\dfrac{1}{3} |
| \text{Identity property of addition} | a+0=a | \dfrac{1}{2}+0=\dfrac{1}{2} |
| \text{Identity property of multiplication} | a \times 1=a | \dfrac{1}{2} \times 1=\dfrac{1}{2} |
| \text{Inverse property of addition} | a+(-a)=0 | \dfrac{1}{2}+\left(-\dfrac{1}{2}\right)=0 |
| \text{Inverse property of multiplication} | a \times \dfrac{1}{a}=1, \, a\neq 0 | 2 \times \dfrac{1}{2}=1 |
We know that every quotient of integers (with a non-zero divisor) is a rational number. If p and q are integers, then –\left(\dfrac{p}{q}\right)= \dfrac{-p}{q} = \dfrac{p}{-q}.
The properties above can also be applied to rational numbers in fraction form or decimal form.
An important property that is very helpful in working with rational numbers especially with fractions is the inverse property of multiplication.
We should note that dividing by a number is equivalent to multiplying by its multiplicative inverse (reciprocal). For this reason, we can rewrite a division expression as multiplication.
For example: 4 \div \dfrac{3}{7} = 4\times \dfrac{7}{3}
All of these properties can be applied to help us evaluate expressions involving fractions and decimals more easily.
Examples
Example 1
Consider the expression -14 \dfrac{3}{5}+4-\dfrac{2}{5}.
Complete the following work with properties or statements as reasoning in each row:
| 1 | \displaystyle -14 \dfrac{3}{5}+4-\dfrac{2}{5} | \displaystyle = | \displaystyle -14 \dfrac{3}{5}+4+\left(-\dfrac{2}{5}\right) | Subtracting a number is the same as adding its inverse |
| 2 | \displaystyle = | \displaystyle -14 +\left(-\dfrac{3}{5}\right)+4+\left(-\dfrac{2}{5}\right) | ⬚ | |
| 3 | \displaystyle = | \displaystyle -14 +4+\left(-\dfrac{2}{5}\right)+\left(-\dfrac{3}{5}\right) | ⬚ | |
| 4 | \displaystyle = | \displaystyle -10 -1 | ⬚ | |
| 5 | \displaystyle = | \displaystyle -11 | Evaluate |
Example 2
Simplify the following expression. Show your work and write the properties used as explanation for each row.
1\div \dfrac{3}{4}\times (-2.5)\div \dfrac{2}{3}
| Property | Symbols |
|---|---|
| \text{Commutative property of addition} | a+b=b+a |
| \text{Commutative property of multiplication} | a \times b=b \times a |
| \text{Associative property of addition} | a+(b+c) = (a+b)+c |
| \text{Associative property of multiplication} | a \times (b\times c)= (a\times b) \times c |
| \text{Distributive property} | a\times (b+c)= a \times b+a \times c \\ \text{or} \\ a \times (b-c)= a\times b-a \times c |
| \text{Identity property of addition} | a+0=a |
| \text{Identity property of multiplication} | a \times 1=a |
| \text{Inverse property of addition} | a+(-a)=0 |
| \text{Inverse property of multiplication} | a \times \dfrac{1}{a}=1 |
Subtracting a number is the same as adding its additive inverse.
The opposite of a sum is the sum of its opposite.
Dividing by a number is equivalent to multiplying by its multiplicative inverse (reciprocal).
We can rewrite a division expression as multiplication.