Download Chapter 2 – Rational Numbers Objectives: compare and order


Chapter
2
–
Rational
Numbers
Objectives:
compare
and
order
rational
numbers
:
solve
problems
with
operations
(+,‐,x,÷)
on
rational
numbers
:
determine
the
square
roots
of
a
perfect
square
rational
numbers
Kinds
of
numbers:
Natural
or
counting
numbers
/N
=
{…1,2,3…}
continues
forever
(infinite
set)
Whole
Numbers
=
/W
{…0,1,2,3…}
(*note
–
each
set
contains
all
the
#’s
of
the
previous
set,
with
one
or
more
additional
numbers.‐
In
this
case
the
0
is
added.)
Integer
Numbers
(or
signed
numbers)
=
/I
{…‐2,
‐1,
0,
1,2…}
(add
the
negative
numbers}
Or
(0,
±1,
±2,
±3…}
Rational
Numbers
(or
fractions)
–
in
the
form
of
a/b
where
a,b∈/I,
b
≠
0
(add
fractions
and
decimals)
means:
is
an
element
of
this
set.
Q
(Quotient)=
{a/b
/
a,b
∈
/I,
b≠0}

such
that/where
This
says:
double
slash
(or
bold)
Q,
is
the
set
of
numbers
that
are
in
the
form
of
a
over
b
(which
means
to
divide)
such
that,
a
and
b
are
integers
(+
or
–
whole
numbers)
where
b
(the
denominator)
does
not
equal
zero.
(We
can’t
divide
by
zero,
it
is
“undefined”
in
math)
A
rational
number
can
be
a
fraction
or
decimal.
In
decimal
form
it
can
be
a:
a) terminating
decimal
eg:
¾
=
.75
(
it
stops
)
b) repeating
decimal
eg:
2/3
=
.666
(
never
stops/continuous
)
To
change
a
terminating
decimal
into
a
faction,
put
it
over
its’
place
value
and
reduce
to
lowest
terms.
Eg:
.85
=
85/100
=
17/20
Equivalent
Numbers
–
have
the
same
value.
Eg:
24/‐4,
‐18/3,
‐12/2,
‐(‐6/‐1)
**All
these
numbers
equal
(‐6)**
To
“order”
rational
numbers
get
a
common
denominator
or
convert
to
an
equivalent
decimal
value
and
compare
numbers.
Eg
#1
3/4
and
2/3
=
9/12
>
8/12
Note
**<
=
is
less
than
(looks
like
a
squished
L)
**
>
=
is
greater
than
(points
right)
Be
careful
with
negative
values.
Note:
‐3/4
=
3/‐4
(all
the
same)
Eg:
order
‐3/4
and
‐2/3
=
‐9/12
<
‐8/12
Use
a
number
line
to
check.
I
I
I
I
I
‐10
‐9
‐8
‐7
‐6
*On
a
number
line,
numbers
get
bigger
as
you
move
to
the
right.
(it
is
more
obvious
on
the
positive
side)
Eg
#2:
Order
with
equivalent
decimals
3/4
and
4/5
3/4
=
.75
4/5
=
.80
is
larger
Again,
be
careful
with
negative
values.
Density
Property:
this
says,
between
any
two
rational
numbers
there
exists
another
rational
number.
Eg
#3:
Find
a
rational
number
between3/
4
and
4/5.
3/4
=
15/20
or
24/40
4/5
=
16/20
or
32/40
As
a
decimal:
3/4
=
.75
4/5
=
.80
one
of
the
numbers
could
be
.76,
.77,
.78
.79
As
a
fraction
this
could
be:
.76
=
76/100
Opposite
Rational
Numbers
–
are
the
same
distance
in
opposite
directions
from
zero.
Ie:)
switch
their
signs.
4/5
⇒
‐4/5
Section
2.2
Problem
Solving
with
Decimals
Remember:
to
subtract
you
can
add
the
opposite
Eg:
‐5.96
–
(‐6.83)
=
‐5.96
+
(+6.83)
=
‐5.93
+
6.83
=
+0.87
On
a
calculator
you
can
use
the
“change
sign”
button
(+/‐),
which
is
not
the
subtract
button.
Eg:
to
do
the
above
question:
(+/‐)
5.96
–
(+/‐)
6.83
=
0.87
‐ The
rules
for
multiply
and
divide
with
integers
1) Same
sign
=
+
2) Different
signs
=
‐
Note:
*Parentheses
(or
brackets),
can
be
used
in
the
place
of
a
multiply
sign.
Eg:
‐3(4.5)
=
‐3
x
4.5
Order
of
Operations:
BEDMAS
Brackets
(do
the
work
inside
of
them
first)
Exponents
Divide/Multiply
(as
they
occur
from
left
to
right)
Add/Subtract
(as
they
occur
from
left
to
right)
Section
2.3
Problem
Solving
with
Rationals
(Fractions)
Remember
the
rules
for
operations
with
fractions
(see
notes
from
days
1
&
2).
When
multiplying
you
can
often
remove
(reduce)
common
factors,
which
allows
you
to
work
with
smaller
numbers.
Eg
#1:
3
x
‐2
=
‐1
4
3
2
If
your
calculator
has
the
fraction
button
(ab/c)
it
will
do
all
of
these
operations
for
you.
(Don’t
forget
BEDMAS)
Eg
#2:
3
x
‐2
=
3
ab/c
4
x
+/‐
2
ab/c
3
=
‐1
r
2
=
‐1
4
3
2
Eg
#3:
6
7
÷
3
=
6
ab/c
7
ab/c
8
÷
3
ab/c
4
=
9
r
1
r
6
or
9
1
8
4
6
Section
2.4
Determining
Square
Roots
of
Rational
Numbers
When
the
square
root
of
a
certain
number
is
multiplied
by
itself
(it
is
squared),
the
product
is
that
certain
number.
Eg
#1:
the
square
root
of
36
is
6,
then
6x6=36
36 = 6 

square
root
radical
number
(radicand)
€
Perfect
squares
have:
1)
a
whole
number
square
root,
or
2)
has
two
equal
rational
factors.
Eg
#2:
36 =
6
∴
36
is
a
perfect
square
.25 =
.5
9 /16 =
¾
or
.75
€
Principal
Roots
are
the
positive
square
root
of
a
number.
€
Note:
36 =
=6
or
‐6
since
‐6
x
‐6
=
36
€
But:
+6
is
the
principal
root.
(Use
for
lengths
of
the
sides
of
squares)
Spare
Roots:
MEMORIZE!!!
€ 12
=
1
112
=
121
22
=
4
122
=
144
2
3 =
9
132
=
169
42
=
16
142
=
196
2
5 =
25
152
=
225
62
=
36
162
=
256
2
7 =
49
172
=
289
82
=
64
182
=
324
2
9 =
81
192
=
361
102
=
100
202
=
400
22
=
the
square
root
400
=
the
perfect
square
Section
2.4
Square
Roots
Part
II
Non
perfect
squares
are
rational
numbers
that
cannot
be
expressed
as
the
product
of
two
equal
(the
same)
rational
factors.
Ex:
You
can’t
multiply
any
rational
number
by
itself
to
get
an
answer
of
3,
1.5,
7/8
…
€
€
This
kind
of
square
root:
1) is
a
non‐repeating,
non‐terminating
decimal.
**This
is
a
third
kind
of
decimal**
.
Ex:
2 =
1.414213562
Π =
3.14592654
2)
its
decimal
value
(can’t
be
written
in
fraction
form)
is
approximate
–
not
exact
because
it
continues
forever.
These
numbers
are
called
Irrationals
(Q)
(ie:
not
rational)
Review
sets
of
numbers:
N
W
I
Q
Q