Number Theory

NZ Level 4

Properties of Real Numbers

Lesson

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.

- If we add two numbers, represented by the letters $a$
`a`and $b$`b`, it makes no difference whether we think of $a+b$`a`+`b`or $b+a$`b`+`a`. The order does not matter.

This property is called the *commutative *law of addition.

- If three numbers $a$
`a`, $b$`b`and $c$`c`are to be added, we can add the first two and then add the third to the result, or we can add the first to the result of adding the second and third. In symbols, $(a+b)+c\equiv a+(b+c)$(`a`+`b`)+`c`≡`a`+(`b`+`c`).

This property is called the *associative *law of addition.

- For every real number $a$
`a`, there exists a number $b$`b`such that $a+b=0$`a`+`b`=0.

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*.

- If we multiply two numbers, $a$
`a`and $b$`b`, it makes no difference whether we think of $a\times b$`a`×`b`or $b\times a$`b`×`a`. The order does not matter.

This property is called the *commutative law of multiplication*.

- If three numbers $a$
`a`, $b$`b`and $c$`c`are to be multiplied, we can multiply the first two and then multiply the result by the third, or we can multiply the first by the result of multiplying the second and third. In symbols, $(ab)c\equiv a(bc)$(`a``b`)`c`≡`a`(`b``c`).

This property is called the *associative law of multiplication*.

- For every real number $a$
`a`, except zero, there exists a number $b$`b`such that $ab=1$`a``b`=1.

The numbers $a$`a` and $b$`b` are called *multiplicative inverses* of one another. We usually write $\frac{1}{a}$1`a` instead of $b$`b` and we recognise the familiar fact that $a\times\frac{1}{a}=1$`a`×1`a`=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`.

- We get the same final result by multiplying $a$
`a`and $b$`b`and also $a$`a`and $c$`c`, and then adding the separate results. In symbols, $a(b+c)\equiv ab+ac$`a`(`b`+`c`)≡`a``b`+`a``c`.

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`)=`a``b`+`a``c`. 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

AAssociative property of multiplication

BDistributive property

CAssociative property of addition

DCommutative property of addition

ECommutative property of multiplication

AAssociative property of multiplication

BDistributive property

CAssociative property of addition

DCommutative property of addition

E

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

ACommutative property of addition

BAssociative property of multiplication

CAssociative property of addition

DDistributive property

ECommutative property of multiplication

ACommutative property of addition

BAssociative property of multiplication

CAssociative property of addition

DDistributive property

EHence 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

AIdentity property

BCommutative property

CAssociative property

DInverse property

EDistributive property

AIdentity property

BCommutative property

CAssociative property

DInverse property

E

Generalise properties of multiplication and division with whole numbers