Chapter 1 Flashcards
What is algebra?
Algebra is the language of all advanced mathematics. It provides us with a toolbox for solving more complicated problems.
What is an integer?
An integer is a number without a fractional part.
What is an ellipsis?
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.
What is the ellipsis symbol? What does it usually represent in mathematics?
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.
What is a positive number?
A positive number is a number that is greater than 0.
What is a negative number?
A negative number is a number that is less than 0.
What is a non-negative number?
A non-negative number is a number that is positive or 0
What is a non-positive number?
A non-negative number is a number that is negative or 0
What is a non-zero number?
A number is called non-zero if it is not equal to 0.
Is zero non-positive or non-negative?
Zero is both non-negative and non-positive.
What do the terms whole number, natural number, and counting number refer to?
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,…
What is a variables?
A variable is a quantity which during a calculation is assumed to vary or be capable of varying in value.
How are variables represented in mathematics?
In mathematics variables are usually represented by letters.
What is the commutative property of addition?
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.
What is the associative property of addition?
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.
What is the commutative property of multiplication?
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.
What is the associative property of multiplication?
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)
What is the order of operations in mathematics?
- evaluate expressions inside parentheses
- compute powers
- multiply and divide from left to right
- add and subtract from left to right
What is the term the distributive property the short form of?
The distributive property is the short form of the distributive property of multiplication over division.
What is the distributive property of multiplication over addition?
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
Using the distributive property to rewrite a(b+c) as ab+ac is called…
expanding.
Using the distributive property to rewrite ab+ac as a(b+c) is called…
factoring.
Which variable is the common factor?
ab+ac=a(b+c)
a
What is the negation of a number?
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
What is the negation of a negation?
Let x be any number then
-(-x)=x
The negation of a negation is the original number x.
multiplying by negation
(-x)y=
x(-y)=
-(xy)
-(xy)
negation times negation
(-x)(-y)=
xy
How does negation distribute over addition?
In the negation of a sum:
-(x+y)=(-x)+(-y)
What is subtraction in general terms?
Subtraction in general terms is the addition of a negation.
a-b = a+(-b)
subtracting a number means adding its opposite
subtraction is also sometimes called the difference
What is the difference between negation and subtraction?
Negation takes one number and returns its opposite.
Subtraction takes two numbers and returns their difference.
subtraction of negation
x-(-y)=
=x+y
subtraction from negation
-x-y=
=-(x+y)
negation of subtraction
-(x-y)=
=-x+y
=y-x
Is addition commutative?
yes
commuting the numbers (changing their order) does not change the sum
Is subtraction commutative?
no
commuting the numbers (changing their order) does change the result
Is multiplication commutative?
yes
commuting the numbers (changing their order) does not change the product
Is addition associative?
yes
regrouping (how the numbers are associated) does not change the sum
Is subtraction associative?
no
regrouping (how the numbers are associated) does change the result
Is multiplication associative?
yes
regrouping (how the numbers are associated) does not change the product
What strategies can be used to solve subtraction problems since subtraction is neither commutative nor associative?
- change subtractions to additions since subtraction is defined in terms of addition of negation.
- rearrange the additions using the commutative and associative properties
What is the distributive property of multiplication over subtraction?
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
What is the identity for addition?
The identity for addition is 0.
What is the identity property of addition?
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
What is the identity property of addition also sometimes referred to as?
The additive identity property.
What is the identity for multiplication?
The identity for multiplication is 1.
What is the reciprocal of x?
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
Which numbers can have a reciprocal?
nonzero numbers
What is the reciprocal of zero?
undefined
What strategy can be used to confirm that two numbers are reciprocals of each other?
multiply them and check that their product is 1.
What is the reciprocal of a reciprocal?
the original number x
because the reciprocal is 1/x so the reciprocal of 1/x is 1/(1/x)=x
reciprocal of product
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
Define division as a combination of multiplication and reciprocation
Let a and b be numbers such that b is not 0. Then quotient a/b is defined as
a/b = a (1/b)
What is division in general terms?
division in general terms is the multiplication of a reciprocal
a/b = a (1/b)
How do we divide by a reciprocal?
multiply by the reciprocal of the reciprocal
x / (1/y) = xy
How do we divide into a negation?
let x and y be numbers such that y is nonzero
(-x) / y = - (x/y)
How do we divide by a negation?
let x and y be numbers such that y is nonzero
x / (-y) = - (x/y)
how do we divide a negation by a negation?
let x and y be numbers such that y is nonzero
(-x) / (-y) = x/y
Is division commutative?
no
Is division associative?
no
What strategies can be used to solve division problems since division is neither commutative nor associative?
- change divisions to multiplications since division is defined in terms of multiplication of reciprocal.
- rearrange the multiplications using the commutative and associative properties
Show ‘the reciprocal of a product is the product of reciprcoals’
1/(xy)=(1/x)(1/y)
show ‘cancellation of common factor in division’
(ab) / (ac) = b / c
Does division distribute over addition?
yes and no
(a+b) / c = a/c + b/c
a / (b+c) is NOT neccesarily equal to a/b + a/c
Does division distribute over subtraction?
yes and no
(a-b) / c = a/c - b/c
a / (b-c) is NOT neccesarily equal to a/b - a/c
show that -(-x)=x
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
show that (-1)x=-x
(-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
show why (-x)y=-(xy) and
why x(-y)=-(xy)
(-x)y
=-1(x)(y) multiplying by -1
by associative property
=-xy
same applies
x(-y)
=x (-1)(y)
=-1(xy)
=-xy
show why a negation multiplied by a negation yields a positive result
(-x)(-y)
=-((x)(-y))
=-(-(xy))
=xy
show how negation distributes over addition
-(x+y)
=-1(x+y) distributive property
=(-1x)+(-1y) expanding
=(-x)+(-y)
