Chapter 1

Question 1

What is algebra?

Answer 1

Algebra is the language of all advanced mathematics. It provides us with a toolbox for solving more complicated problems.

Question 2

What is an integer?

Answer 2

An integer is a number without a fractional part.

Question 3

What is an ellipsis?

Answer 3

An ellipsis is the omission from speech or writing of a word or words that are superfluous or able to be understood from contextual clues and can also represent hesitation or a pause.

Question 4

What is the ellipsis symbol? What does it usually represent in mathematics?

Answer 4

The ellipsis symbol is a series of dots, usually three. In mathematics it usually indicates that a list of numbers goes on forever unless it is bound.

Question 5

What is a positive number?

Answer 5

A positive number is a number that is greater than 0.

Question 6

What is a negative number?

Answer 6

A negative number is a number that is less than 0.

Question 7

What is a non-negative number?

Answer 7

A non-negative number is a number that is positive or 0

Question 8

What is a non-positive number?

Answer 8

A non-negative number is a number that is negative or 0

Question 9

What is a non-zero number?

Answer 9

A number is called non-zero if it is not equal to 0.

Question 10

Is zero non-positive or non-negative?

Answer 10

Zero is both non-negative and non-positive.

Question 11

What do the terms whole number, natural number, and counting number refer to?

Answer 11

There is diagreement over the meaning and use of the terms whole number, natural number, and counting number. It is therefore clearer to use the terms positive integer for 1,2,3,… and non-negative integer for 0,1,2,3,…

Question 12

What is a variables?

Answer 12

A variable is a quantity which during a calculation is assumed to vary or be capable of varying in value.

Question 13

How are variables represented in mathematics?

Answer 13

In mathematics variables are usually represented by letters.

Question 14

What is the commutative property of addition?

Answer 14

The commutative property of addition states that when two numbers are added, the sum is the same even if you switch the order of the numbers being added.

Let a and b be numbers then a+b=b+a

It is called the commutative property because the numbers can be commuted (moved) on the two sides of the equation.

Question 15

What is the associative property of addition?

Answer 15

The associative property of addition states that no matter how a set of three or more numbers are grouped together, the sum remains the same

Let a, b, and c be numbers then (a+b)+c=a+(b+c)

It is called the associative property becuse the numbers stay in the same place but are associated (grouped) differently.

Question 16

What is the commutative property of multiplication?

Answer 16

The commutative property of multiplication states that the result of the multiplication of two numbers stays the same, even if the positions of the numbers are interchanged.

Let a and b be numbers then ab=ba

It is called the commutative property because the numbers can be commuted (moved) on the two sides of the equation.

Question 17

What is the associative property of multiplication?

Answer 17

The associative property of multiplication states that no matter how a set of three or more numbers are grouped together, the product remains the same

Let a, b, and c be numbers then (ab)c=a(bc)

Question 18

What is the order of operations in mathematics?

Answer 18

1. evaluate expressions inside parentheses
2. compute powers
3. multiply and divide from left to right
4. add and subtract from left to right

Question 19

What is the term the distributive property the short form of?

Answer 19

The distributive property is the short form of the distributive property of multiplication over division.

Question 20

What is the distributive property of multiplication over addition?

Answer 20

The distributive property of multiplication over addition states that multiplication can be distributed (handed out) to the two or more parts of addition as in the following example:

a(b+c)=ab+ac

Question 21

Using the distributive property to rewrite a(b+c) as ab+ac is called…

Answer 21

expanding.

Question 22

Using the distributive property to rewrite ab+ac as a(b+c) is called…

Answer 22

factoring.

Question 23

Which variable is the common factor?

ab+ac=a(b+c)

Answer 23

a

Question 24

What is the negation of a number?

Answer 24

The negation of a number x is the number that is added to x to result in 0.

-x+x=0

also called the opposite of x or the additive inverse of x

Question 25

What is the **negation** *of a negation*?

Answer 25

Let x be any number then -(-x)=x The negation of a negation is the original number x.

Question 26

**multiplying by negation** (-x)y= x(-y)=

Answer 26

-(xy) -(xy)

Question 27

**negation times negation** (-x)(-y)=

Answer 27

xy

Question 28

How does *negation distribute over addition*?

Answer 28

In the negation of a sum: -(x+y)=(-x)+(-y)

Question 29

What is **subtraction** in general terms?

Answer 29

**Subtraction** in general terms is the *addition of a negation*. | a-b = a+(-b) ## Footnote subtracting a number means adding its opposite subtraction is also sometimes called the difference

Question 30

What is the difference between **negation** and **subtraction**?

Answer 30

**Negation** takes one number and returns its *opposite*. **Subtraction** takes two numbers and returns their *difference*.

Question 31

subtraction of negation x-(-y)=

Answer 31

=x+y

Question 32

subtraction from negation -x-y=

Answer 32

=-(x+y)

Question 33

negation of subtraction -(x-y)=

Answer 33

=-x+y =y-x

Question 34

Is addition commutative?

Answer 34

yes | commuting the numbers (changing their order) does not change the sum

Question 35

Is subtraction commutative?

Answer 35

no | commuting the numbers (changing their order) does change the result

Question 36

Is multiplication commutative?

Answer 36

yes | commuting the numbers (changing their order) does not change the product

Question 37

Is addition associative?

Answer 37

yes | regrouping (how the numbers are associated) does not change the sum

Question 38

Is subtraction associative?

Answer 38

no | regrouping (how the numbers are associated) does change the result

Question 39

Is multiplication associative?

Answer 39

yes | regrouping (how the numbers are associated) does not change the product

Question 40

What strategies can be used to solve subtraction problems since subtraction is neither commutative nor associative?

Answer 40

1. change subtractions to additions since subtraction is defined in terms of addition of negation. 2. rearrange the additions using the commutative and associative properties

Question 41

What is the **distributive property of multiplication over subtraction**?

Answer 41

The **distributive property of multiplication over subtraction** states that multiplication can be distributed (handed out) to the two or more parts of the subtraction as in the following example: a(b-c)=ab-ac

Question 42

What is the **identity** for addition?

Answer 42

The **identity** for addition is 0.

Question 43

What is the **identity property of addition**?

Answer 43

The *identity property of addition* states that by adding 0 to any number, the result is the number itself. In other words, when we add 0 to any number, it does not change the number, therefore *keeping its identity*. | The** identity property of addition** or the **additive identity propert

Question 44

What is the **identity property of addition** also sometimes referred to as?

Answer 44

The **additive identity property**.

Question 45

What is the ***identity** for multiplication*?

Answer 45

The ***identity** for multiplication* is 1.

Question 46

What is the **reciprocal** of **x**?

Answer 46

For any number x, the reciprocal of x, written as 1/x, is the number such that **(1/x)x=1** | This number is also called the **multiplicative inverse of x**

Question 47

Which numbers can have a reciprocal?

Answer 47

**nonzero** numbers

Question 48

What is the *reciprocal of zero*?

Answer 48

undefined

Question 49

What strategy can be used to confirm that two numbers are reciprocals of each other?

Answer 49

multiply them and check that their product is 1.

Question 50

What is the reciprocal of a reciprocal?

Answer 50

the original number x | because the reciprocal is 1/x so the reciprocal of 1/x is 1/(1/x)=x

Question 51

reciprocal of product

Answer 51

let x and y be non zero numbers then reciprocal of xy is 1/xy and (1/x).(1/y) | because xy . (1/x).(1/y) = 1

Question 52

Define **division** as a combination of *multiplication* and *reciprocation*

Answer 52

Let a and b be numbers such that b is not 0. Then quotient a/b is defined as a/b = a (1/b)

Question 53

What is **division** in *general term*s?

Answer 53

**division** in *general terms* is the **multiplication** of a **reciprocal** | a/b = a (1/b)

Question 54

How do we **divide** by a *reciprocal*?

Answer 54

*multiply* by the *reciprocal of the reciprocal* | x / (1/y) = xy

Question 55

How do we **divide** into a *negation*?

Answer 55

let x and y be numbers such that y is nonzero (-x) / y = - (x/y)

Question 56

How do we **divide** by a *negation*?

Answer 56

let x and y be numbers such that y is nonzero x / (-y) = - (x/y)

Question 57

how do we **divide** a *negation* by a *negation*?

Answer 57

let x and y be numbers such that y is nonzero (-x) / (-y) = x/y

Question 58

Is **division** *commutative*?

Answer 58

no

Question 59

Is **division** *associative*?

Answer 59

no

Question 60

What strategies can be used to solve division problems since division is neither commutative nor associative?

Answer 60

1. change divisions to multiplications since division is defined in terms of multiplication of reciprocal. 2. rearrange the multiplications using the commutative and associative properties

Question 61

Show 'the reciprocal of a product is the product of reciprcoals'

Answer 61

1/(xy)=(1/x)(1/y)

Question 62

show 'cancellation of common factor in division'

Answer 62

(ab) / (ac) = b / c

Question 63

Does **division** *distribute over addition*?

Answer 63

yes and no (a+b) / c = a/c + b/c a / (b+c) is NOT neccesarily equal to a/b + a/c

Question 64

Does **division** *distribute over subtraction*?

Answer 64

yes and no (a-b) / c = a/c - b/c a / (b-c) is NOT neccesarily equal to a/b - a/c

Question 65

show that -(-x)=x

Answer 65

x+(-x)=0 by the rule of negation substitute -x for x -x+(-(-x)=0 add x to both sides -(-x)=x | negation of negation

Question 66

show that (-1)x=-x

Answer 66

(-1)x+x =(-1)x+(1)x using distribution property to factor =(-1+1)x =(1-1)x =0x =0 so (-1)x+x=0 so (-1)x is the negation of x negation of x is -x sp (-1)x=-x

Question 67

show why (-x)y=-(xy) and why x(-y)=-(xy)

Answer 67

(-x)y =-1(x)(y) multiplying by -1 by associative property =-xy same applies x(-y) =x (-1)(y) =-1(xy) =-xy

Question 68

show why a negation multiplied by a negation yields a positive result

Answer 68

(-x)(-y) =-((x)(-y)) =-(-(xy)) =xy

Question 69

show how negation distributes over addition

Answer 69

-(x+y) =-1(x+y) distributive property =(-1x)+(-1y) expanding =(-x)+(-y)