2.4: Properties of the Real Numbers
Overview
- The Closure Properties
- The Commutative Properties
- The Associative Properties
- The Distributive Properties
- The Identity Properties
- The Inverse Properties
A property of a collection of objects is a characteristic that describes the collection. We shall now examine some of the properties of the collection of real numbers. The properties we will examine are expressed in terms of addition and multiplication.
The Closure Properties
If \(a\) and \(b\) are real numbers, then \(a + b\) is a unique real number, and \(a \cdot b\) is a unique real number.
For example, 3 and 11 are real numbers; \(3 + 11 = 14\) and \(3 \cdot 11 = 33\), and both 14 and 33 are real numbers. Although this property seems obvious, some collections are not closed under certain operations. For example,
- The real numbers are not closed under division since, although 5 and 0 are real numbers, \(5/0\) and \(0/0\) are not real numbers.
- The natural numbers are not closed under subtraction since, although 8 is a natural number, \(8 - 8\) is not. (\(8 - 8 = 0\) and 0 is not a natural number).
The Commutative Properties
Let \(a\) and \(b\) represent real numbers.
Commutative Property of Addition:
\(a + b = b + a\)
Commutative Property of Multiplication:
\(a \cdot b = b \cdot a\)
The commutative properties tell us that two numbers can be added or multiplied in any order without affecting the result.
Sample Set A
The following are examples of the commutative properties.
\(3 + 4 = 4 + 3\) Both equal 7
\(5 + x = x + 5\) Both represent the same sum.
\(4 \cdot 8 = 8 \cdot 4\) Both equal 32
\(y7 = 7y\) Both represent the same product
\(5(a + 1) = (a + 1)5\) Both represent the same product
\((x+4)(y+2) = (y+2)(x+4)\) Both represent the same product
Practice Set A
Fill in the ( ) with the proper number or letter so as to make the statement true. Use the commutative properties.
\(6 + 5 + ( ) + 6\)
- Answer
-
5
\(m + 12 = 12 + ( )\)
- Answer
-
\(m\)
\(9 \cdot 7 = ( ) \cdot 9\)
- Answer
-
7
\(6a = a( )\)
- Answer
-
6
\(4(k - 5) = ( )4\)
- Answer
-
\(k - 5\)
\((9a - 1)( ) = (2b + 7)(9a - 1)\)
- Answer
-
\(2b + 7\)
The Associative Properties
Let \(a\), \(b\), and \(c\) represent real numbers.
Associative Property of Addition:
\((a + b) + c = a + (b + c)\)
Associative Property of Multiplication:
\((ab)c = a(bc)\)
The associative properties tell us that we may group together the quantities as we please without affecting the result.
Sample Set B
The following examples show how the associative properties can be used.
\((2 + 6) + 1 = 2 + (6 + 1)\)
\(8 + 1 = 2 + 7\)
\(9 = 9\)
Both equal 9
\((3 + x) + 17 = 3 + (x + 17)\)
Both represent the same sum.
\((2 \cdot 3) \cdot 5 = 2 \cdot (3 \cdot 5)\)
\(6 \cdot 5 = 2 \cdot 15\)
\(30 = 30\)
Both equal 30
\((y)4 = 9(y4)\)
Both represent the same product.
Practice Set B
Fill in the ( ) to make each statement true. Use the associative properties.
\((9 + 2) + 5 = 9 + ( )\)
Solution
\(2 + 5\)
\(x + (5 + y) = ( ) + y\)
Solution
\(x + 5\)
\((11a)6 = 11( )\)
Solution
\(a \cdot 6\)
\([(7m - 2)(m + 3)](m + 4) = (7m - 2)[( ) ( )]\)
Solution
\((m + 3)(m + 4)\)
Sample Set C
Simplify (rearrange into a simpler form): \(5x6b8ac4\).
According to the commutative property of multiplication, we can make a series of consecutive switches and get all the numbers together and all the letters together.
\(5 \cdot 6 \cdot 8 \cdot 4 \cdot x \cdot b \cdot a \cdot c\)
\(960xbac\) Multiply the numbers.
\(960abcx\) By convention, we will, when possible, write all letters in alphabetical order.
Practice Set C
Simplify each of the following quantities.
\(3a7y9d\)
Solution
\(189ady\)
\(6b8acz4\cdot5\)
Solution
\(960abcz\)
\(4p6qr3(a + b)\)
Solution
\(72pqr(a + b)\)
The Distributive Properties
When we were first introduced to multiplication we saw that it was developed as a description for repeated addition.
\(4 + 4 + 4 = 3 \cdot 4\)
Notice that there are three 4's, that is, 4 appears 3 times. Hence, 3 times 4.
We know that algebra is generalized arithmetic. We can now make an important generalization.
When a number \(a\) is added repeatedly \(n\) times, we have:
\(\underbrace{a+a+a+\cdots+a}_{a \text { appears } n \text { times }}\)
Then, using multiplication as a description for repeated addition, we can replace:
\(\underbrace{a+a+a+\cdots+a}_{n \text { times }}\) with \(na\)
For example:
\(x + x + x + x\) can be wrirten as \(4x\) since \(x\) is repeated added \(4\) times.
\(x + x + x + x = 4x\)
\(r + r\) can be written as \(2r\) since \(r\) is repeatedly added \(2\) times.
\(r + r = 2r\)
The distributive property involved both multiplication and addition. Let's rewrite \(4(a + b)\). We proceed by reading \(4(a + b)\) as a multiplication: 4 times the quantity \((a + b)\). This directs us to write:
$$
\begin{aligned}
4(a+b) &=(a+b)+(a+b)+(a+b)+(a+b) \\
&=a+b+a+b+a+b+a+b
\end{aligned}
\]
Now we use the commutative property of addition to collect all the \(a\)'s together and all the \(b\)'s together.
$$
4(a+b)=\underbrace{a+a+a+a}_{4 a^{\prime} s}+\underbrace{b+b+b+b}_{4 b^{\prime} s}
$$
Now, using multiplication as a description for repeated addition, we have
$$
4(a+b)=4 a+4 b
$$
We have distributed the 4 over the sum to both \(a\) and \(b\)
\(4(a+b)=4a+4b\)
\(a(b + c) = a\cdot b + a \cdot c\)
\((b + c)a = a \cdot b + a \cdot c\)
The distributive property is useful when we cannot or do not wish to perform operations inside parentheses.
Sample Set D
Use the distributive property to rewrite each of the following quantities.
\(2(5+7)=2 \cdot 5+2 \cdot 7 \quad \text { Both equal } 24\)
\(6(x + 3) = 6 \cdot x + 6 \cdot 3\) Both represent the same number.
= \(6x + 18\)
\((z + 5)y = zy + 5y = yz + 5y\)
Practice Set D
What property of real numbers justifies:
\(a(b + c) = (b + c)a\)?
- Answer
-
the commutative property of multiplication
Use the distributive property to rewrite each of the following quantities.
\(3(2 + 1)\)
- Answer
-
6 + 3
\((x+6)7\)
- Answer
-
\(7x + 42\)
\(4(a + y)\)
- Answer
-
\(4a + 4y\)
\((9 + 2)a\)
- Answer
-
\(9a + 2a\)
\(a(x + 5)\)
- Answer
-
\(ax + 5a\)
\(1(x + y)\)
- Answer
-
\(x + y\)
The Identity Properties
Additive Identity
The number 0 is called the
additive identity
since when it is added to any real number, it preserves the identity of that number. Zero is the only additive identity.
For example, \(6 + 0 = 6\)
Multiplicative Identity
The number 1 is called the
multiplicative identity
since when it multiplies any real number, it preserves the identity of that number. One is the only multiplicative identity.
For example \(6 \cdot 1 = 6\).
We summarize the identity properties as follows:
If \(a\) is a real number, then:
\(a + 0 = a\) and \(0 + a = a\)
If \(a\) is a real number, then:
\(a \cdot 1 = a\) and \(1 \cdot a = a\)
The Inverse Properties
Additive Inverses
When two numbers are added together and the result is the additive identity, 0, the numbers are called additive inverses of each other. For example, when 3 is added to −3 the result is 0, that is, \(3+(−3)=0\). The numbers 3 and −3 are additive inverses of each other.
Multiplicative Inverses
When two numbers are multiplied together and the result is the multiplicative identity, 1, the numbers are called multiplicative inverses of each other. For example, when \(6\) and \(\dfrac{1}{6}\) are multiplied together, the result is 1, that is, \(6 \cdot 16 = 1\). The numbers 6 and \(\dfrac{1}{6}\) are multiplicative inverses of each other.
We summarize the inverse properties as follows:
The Inverse Properties:
1. If \(a\) is any real number, then there is a unique real number \(-a\), such that
\(a + (-a) = 0\) and \(-a + a = 0\)
The numbers \(a\) and \(-a\) are called
additive inverses
of each other.
2. If \(a\) is any nonzero real number, then there is a unique real number \(\dfrac{1}{a}\) such that
\(a \cdot \dfrac{1}{a} = 1\) and \(\dfrac{1}{a} \cdot a = 1\).
The numbers \(a\) and \(\dfrac{1}{a}\) are called
multiplicative inverses
of each other.
Expanding Quantities:
When we perform operations such as \(6(a + 3) = 6a + 18\), we say we are expanding the quantity \(6(a + 3)\).
Exercises
Use the commutative property of addition and multiplication to write expressions for an equal number for the following problems. You need not perform any calculations.
\(x + 3\)
- Answer
-
\(3+x\)
\(5 + y\)
\(10x\)
- Answer
-
\(x10\)
\(18z\)
\(r6\)
- Answer
-
\(6r\)
\(ax\)
\(xc\)
- Answer
-
\(cx\)
\(7(2 + b)\)
\(6(s + 1)\)
- Answer
-
\((s + 1)6\)
\((8 + a)(x + 6)\)
\((x + 16)(a + 7)\)
- Answer
-
\((a + 7)(x + 16)\)
\((x + y)(x - y)\)
\(0.06m\)
- Answer
-
\(m(0.06)\)
\(x + 3\)
- Answer
-
\(3+x\)
\(5(6h + 1)\)
- Answer
-
\((6h + 1)5\)
\(m(a + 2b)\)
\(k(10a - b)\)
- Answer
-
\((10a - b)k\)
\((21c)(0.008)\)
\((-16)(4)\)
- Answer
-
\((4)(-16)\)
\((5)(b - 6)\)
Simplify using the commutative property of multiplication for the following problems. You need not use the distributive property.
\(9x2y\)
- Answer
-
\(18xy\)
\(5a6b\)
\(2a3b4c\)
- Answer
-
\(24abc\)
\(5x10y5z\)
\(1u3r2z5m1n\)
- Answer
-
\(30mnruz\)
\(6d4e1f2(g + 2h)\)
\((\dfrac{1}{2})d(\dfrac{1}{4})e(\dfrac{1}{2})a\)
- Answer
-
\(\dfrac{1}{16}ade\)
\(3(a + 6)2(a - 9)6b\)
\(1(x + 2y)(6 + z)9(3x + 5y)\)
- Answer
-
\(9(x + 2y)(6 + z)(3x + 5y)\)
For the following problems, use the distributive property to expand the quantities.
\(2(y + 9)\)
\(b(r + 5)\)
- Answer
-
\(br + 5b\)
\(m(u + a)\)
\(k(j + 1)\)
- Answer
-
\(jk + k\)
\(x(2y + 5)\)
\(z(x + 9w)\)
- Answer
-
\(xz + 9wz\)
\((1 + d)e\)
\((8 + 2f)g\)
- Answer
-
\(8g + 2fg\)
\(c(2a + 10b)\)
\(15x(2y + 3z)\)
- Answer
-
\(30xy + 45xz\)
\(8y(12a + b)\)
\(z(x + y + m)\)
- Answer
-
\(xz + yz + mz\)
\((a + 6)(x + y)\)
\((x + 10)(a + b + c)\)
- Answer
-
\(zx + bx + cx + 10a + 10b + 10c\)
\(1(x + y)\)
\(1(a + 16)\)
- Answer
-
\(a + 16\)
\(0.48(0.34a + 0.61)\)
\(21.5(16.2a + 3.8b + 0.7c)\)
- Answer
-
\(348.3a + 81.7b + 15.05c\)
Exercises for Review
Find the value of \(4 \cdot 2 + 5(2 \cdot 4 - 6 \div 3) - 2 \cdot 5\).
Is the statement \(3(5 \cdot 3 - 3 \cdot 5) + 6 \cdot 2 - 3 \cdot 4 < 0\) true or false?
- Answer
-
false
Draw a number line that extends from \(-2\) to \(2\) and place points at all integers between and including \(-2\) and \(3\).
Replace the \(*\) with the appropriate relation symbol \((<,>)\). \(-7 * -3\).
- Answer
-
\(<\)
What whole numbers can replace \(x\) so that the statement \(-2 \le x < 2\) is true?