The numbers that we use regularly for counting and measuring are called the real numbers. They include zero and the positive and negative whole numbers, the numbers that can be written as fractions, and every number in between.
The real numbers are so familiar to us that we hardly notice that there are certain special properties that they have that involve the addition and multiplication operations. One reason for taking the trouble now to consider these properties is that mathematicians have invented objects other than ordinary numbers that have addition and multiplication operations defined for them but for which the rules for real numbers do not always apply.
This property is called the commutative law of addition.
This property is called the associative law of addition.
The numbers $a$a and $b$b are called additive inverses of one another. We usually write $-a$−a instead of $b$b and we recognise the familiar fact that $a+(-a)=0$a+(−a)=0.
The number $0$0 is called the additive identity element of the real numbers.
The additive identity has the property that for any number $a$a, we have $a+0=a$a+0=a. which explains the use of the term identity.
This property is called the commutative law of multiplication.
This property is called the associative law of multiplication.
The numbers $a$a and $b$b are called multiplicative inverses of one another. We usually write $\frac{1}{a}$1a instead of $b$b and we recognise the familiar fact that $a\times\frac{1}{a}=1$a×1a=1.
The number $1$1 is called the multiplicative identity element of the real numbers.
The multiplicative identity has the property $a\times1=a$a×1=a for every number except zero.
The expression $a(b+c)$a(b+c) means that $b$b and $c$c are to be added and then the result is multiplied by $a$a.
This property is called the distributive law.
[If the positions of the addition and multiplication signs are swapped so that we have$a+(b\times c)$a+(b×c), a similar distributive rule is not true. That is, $a+b\times c$a+b×c is not equal to $(a+b)\times(a+c)$(a+b)×(a+c).
This is one situation in which we must observe the correct 'order of operations' when simplifying expressions. The multiplication must be done first.
Note, however, there exist systems in which both these distributive properties hold, notably in the algebra of sets in which the addition and multiplication operations are replaced by the operations union and intersection.]
Verify the distributive law in the special case where $a=3$a=3, $b=13$b=13 and $c=-1$c=−1. Use the expression $a(b+c)$a(b+c).
The distributive law says $a(b+c)=ab+ac$a(b+c)=ab+ac. In this case, we need to check that $3\times(13+(-1))$3×(13+(−1)) is the same as $3\times13+3\times(-1)$3×13+3×(−1).
$3\times(13+(-1))$3×(13+(−1)) | $=$= | $3\times12$3×12 |
$=$= | $36$36 |
And
$3\times13+3\times(-1)$3×13+3×(−1) | $=$= | $39+(-3)$39+(−3) |
$=$= | $36$36 |
So, the expressions are the same for these numbers.
What is the multiplicative inverse of the number $\sqrt{3}$√3?
We ask what number multiplied by $\sqrt{3}$√3 gives the answer $1$1? We might write $x\sqrt{3}=1$x√3=1 and, on dividing both sides by $\sqrt{3}$√3, we discover that $x=\frac{1}{\sqrt{3}}$x=1√3.
So, $\sqrt{3}$√3 and $\frac{1}{\sqrt{3}}$1√3 are multiplicative inverses of one another.
Taking this a little further, we could use a calculator to find the decimal approximations
$\sqrt{3}\approx1.73205$√3≈1.73205 and
$\frac{1}{\sqrt{3}}\approx0.57735$1√3≈0.57735.
If we now multiply these decimal numbers together, we get $\approx0.999999$≈0.999999.
Which property is demonstrated by the following statement?
$4\times\left(9\times5\right)=\left(4\times9\right)\times5$4×(9×5)=(4×9)×5
Commutative property of multiplication
Associative property of multiplication
Distributive property
Associative property of addition
Commutative property of addition
Consider the following statement: $-6\times5=\editable{}\times\left(-6\right)$−6×5=×(−6)
Which property needs to be used to complete the statement?
Commutative property of multiplication
Commutative property of addition
Associative property of multiplication
Associative property of addition
Distributive property
Hence write the missing value.
$-6\times5=\editable{}\times\left(-6\right)$−6×5=×(−6)
Which property does the following statement demonstrate?
$-9\times\left(2\times3\right)=\left(-9\times2\right)\times3$−9×(2×3)=(−9×2)×3
Distributive property
Identity property
Commutative property
Associative property
Inverse property