Download rational numbers

REAL
NUMBERS
If you were asked to count, the numbers you’d
say are called counting numbers. These
numbers can be expressed using set notation.
{1, 2, 3, 4, . . . }
If we include 0 we have
the set of whole numbers.
These are also called
the natural numbers.
{0, 1, 2, 3, 4, . . . }
{ …, -3, -2, -1, 0,1, 2, 3, . . . }
Include the opposites of the whole numbers and
you have the set of integers.
Whole numbers are a subset of integers and
counting numbers are a subset of whole
numbers.
rational numbers
integers
whole numbers
counting
numbers
This means to
divide one integer
by another or
“make a fraction”
If we express a new set of numbers as the quotient of
two integers, we have the set of rational numbers
There are numbers that cannot be expressed as
the quotient of two integers. These are called
irrational numbers.
REAL NUMBERS
rational numbers

integers
irrational
whole numbers
counting
numbers
numbers
2
The rational numbers combined with the irrational
numbers make up the set of real numbers.
The Real Number Line
a < 0 means a is negative
         
a > 0 means a is positive

negative numbers
are to the left of 0
a < b is read "a is
less than b" and
means a is further to
the left on the
number line than b







 
positive numbers
are to the right of 0
These are
called strict
inequalities
a > b is read "a is
greater than b" and
means a is further to
the right on the
number line than b
a  b or a  b are nonstrict inequalities and mean besides
being less than or greater than b, a can equal b.
We can graph inequalities on the number line.
x  1
[
-7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8
squared end bracket
Because the inequality is "greater than or equal to"
the solution can equal the endpoint. A square
bracket means it can equal the endpoint.
Let's look at a different inequality sign.
x  1
)
-7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8
rounded end bracket
Since this says "less than" we make the arrow go the
other way. Since it doesn't say "or equal to" the
solution cannot equal the endpoint. A rounded
bracket means it cannot equal the endpoint.
Translating English to Math
sum of two numbers
a+b
difference between two numbers
a-b
The product of two numbers
ab
the quotient of two numbers
a
b
is
=
ORDER OF OPERATIONS
When there is more than one symbol of operation in an
expression, it is agreed to complete the operations in a certain
order. A mnemonic to help you remember this order is below.
Complete
Apply
multiplication
addition
and
subtraction
and inside
division
from
from
left
lefttotoright
right
Do
anyexponents
simplifying
possible
of
parenthesis
starting with innermost parenthesis and working out
PEMDAS
exponents
– applyand
the exponent
now to to
complete
completeaddition
multiplication
and division,
right
right
parenthesis
– combinesubtraction,
these
first leftleft
10

2324
5
8

6
2  3
332422444
22
23  5  3  2  4
2
PEMDAS
COMMUTATIVE PROPERTY
The operations of both addition and multiplication are commutative
ab ba
3 2  2  3
When adding, you can “commute” or trade the terms places
a b  ba
3 2  2  3
When multiplying, you can “commute” or trade the factors places
ASSOCIATIVE PROPERTY
The operations of both addition and multiplication are associative
a  b  c  a  b  c
1  2  3  1  2  3
When adding, you can “associate” and add any terms first and
then add the other term.
a  b c  a  b  c
2  3  4  2  3  4
When multiplying, you can “associate” and multiply any factors
first and then multiply the other factor.
DISTRIBUTIVE PROPERTY
The operation of multiplication distributes over addition
ab  c  ab  ac
32  4  3  2  3  4
The distributive property also holds for a factor that is multiplied
on the left.
b  ca  b  a  c  a
3  42  3  2  4  2
CAUTION: Remember that the value for a
and/or b could also be positive or negative.
a b  ab
 ab  ab
A positive
times a
negative is
A negative
times a
positive is
NEGATIVE
  a   a
The negative
POSITIVE
of a negative
a a

b b
A negative
divided by a
negative is
NEGATIVE
a
a
a

 
b
b
b
A negative NEGATIVE
A positive
divided by a divided by a
negative or positive is
POSITIVE
EXPONENTS
An exponent
means to
multiply the
base together
that many
times.
5
exponent
2
2  2  2  2  2  32
BASE
Anything (except 0)
to the 0 power is 1
A negative exponent
means the reciporical
(move to the bottom
of a fraction)
a  1 if a  0
0
a
n
1
 n if a  0
a
LAWS OF EXPONENTS
To multiply, if the
bases are the
same, keep the
base and add the
exponents
am
mn

a
an
If you have one
base raised to an
exponent and the
whole thing to
another exponent,
keep the base and
a a a
m
n
mn
To divide, if the
bases are the
same, keep the
base and take top
exponent minus
bottom
a 
m
n
a
mn
If you have a
product to an
exponent, apply
n
the
exponent
ab  a ntob n
each factor
 
If you have a
quotient to an
exponent, apply
the exponent to the
numeratorn and to
n
a
a
 
the denominator

 
b
bn
SQUARE ROOTS
This is called a radical
sign and denotes the
principal square root
which is nonnegative.
a a
2
36  6
This is asking what squared is 36.
Although -6 squared is 36, the radical sign
refers to the nonnegative root.
The absolute value
means we get a
nonnegative result.
SCIENTIFIC NOTATION
Scientific notation takes a number (particularly a large
one with lots of zeros or a small one with lots of zeros
between the decimal point and other digits) and uses
powers of 10 to express it more easily.
2 4 0,0 0 0,0 0 0,0 0 0,0 0 0 x 1014
1 2 3 4 5 6 7 8 9 1 1 12 1 14
3
1
The decimal should be after the0first
nonzero
digit.
To keep the number equal we’d need to multiply
by a power of 10. Count how many decimal
places to get to original decimal point to see what
that power of 10 should be.
2.4 x 1014
If the number was very small we'd do the same thing,
but since we'd be counting in the opposite direction, our
power of 10 would be negative.
0 . 0 0 0 0 0 0 3 6 x 10 -7
7 6 5 4 3 2 1
The decimal should be after the first nonzero digit.
To keep the number equal we’d need to multiply
by a negative power of 10. Count how many
decimal places to get to original decimal point to
see what that power of 10 should be.
3.6 x 10 -7