Download Rational and Irrational Numbers

Rational and
Irrational Numbers
Honors Math – Grade 8
Get ready for the Lesson
During a 4-month period, an
environmentalist recorded the
following changes to a lake’s
water level:
1
3
4 in.,  2 in, 0.41in., and in.
4
8
These numbers recorded above are
called rational numbers.
A rational number is the quotient of two
a
integers, a and b, written as
b
where
b  0.
Remember:
Natural or Counting Numbers
{1, 2, 3, 4, …}
Whole Numbers
{0, 1, 2, 3, …}
We have already studied
some types of rational
numbers.
Now let’s examine and discuss some
other subsets of rational numbers.
The following types of numbers are rational numbers…
Type of Number
Example Set
Integers – are whole
{…, -3, -2, -1, 0, 1, 2, 3, …}
numbers and their
opposites.
Fractions and mixed
1
3
11
numbers – can be
2

positive or negative
4
8
7
Terminating Decimals have a finite
Decimals – can also
number of digits.
be positive or negative.
0.41, 3.0, -5.7
They are either
Repeating Decimals have a
sequence of one or more digits that
terminating or
repeat indefinitely.
repeating.
2.3 2.333...
Multiplying a number by itself, or raising it to the second
power, is finding the square of that number.
3
square of
8
2
3  3 3  9
 
8 8 64
8
square of 0.5
0.5
 0.5  0.5  0.25
square of  7
 7 
 7  7  49
 42
 4 4  16
opposite of the
square of 4
2
2
Perfect squares, or square numbers, are the square of
natural numbers.
2
1
2
2
2
3
2
4
2
5
2
6
2
7
2
8
1
4
9
 16
 25
 36
 49
 64
The set of perfect squares is {1, 4, 9, 16, 25, 36, 49, 64, 81, …}
A square root of a number is one of two equal
factors of that number.
The positive square root of a number is called the
principal square root. It is indicated with the a
symbol called a radical sign.
The expression under a radical sign is called the
radicand.
The negative square root of a number is indicated
by writing a negative sign if front of the radical.
25  5 5
5
Read as “the principal
square root of 25.”
 25  5
Read as “the negative
square root of 25.”
Find each square root.
 225   (3  5)(3  5)  (3  5)  15
Another way to find the
principal square root of
a perfect square is to
use the prime
factorization.
225  9  25
225  3  3  5  5
Arrange the factors into
groups.
4
9
When the radicand is a fraction,
find the principal square root of
the numerator and the
denominator.
4
4 2


9
9 3
A nonperfect square is a number that is not the square of a natural
number.
To approximate the square of a nonperfect square, find two consecutive
integers that the square root is between.
Between what two consecutive integers is
19 ?
Find two nearby
perfect squares that
19 is between.
1, 4, 9,16, 25, 36, 49,...
List the perfect
squares in order.
19 is
between 16
and 25
THINK
19
16  4
25  5
19 is between 4 and 5.
Square roots of nonperfect square are examples of numbers that are not
rational. Numbers that are not rational are called irrational.
KEY CONCEPT
Irrational Numbers
Irrational numbers are numbers that cannot be expressed in
a
the form b where a and b are integers and b  0 .
The following are examples of irrational numbers.
1. Positive or negative
decimals that are
nonterminating and
nonrepeating or have a
pattern in their digits but
do not repeat exactly.
4.13216582…
-0.505005005…
2. Square roots of
nonperfect squares.
19
 250
42

3. Pi, symbolized by a
Greek letter.
Please note, 3.14 and
22/7 are rational
approximate values.
For each number, list all the terms that apply: whole
number, integer, rational number, and irrational
number.
-321.11
45
Rational
number
Whole number
0.010203…
1.358
Irrational number
Rational
number
Integer
Rational number
22
7
Rational
number
 36
Integer
Rational number
144
Whole number
Integer
Rational number
102
Irrational
number
Find each square root. If the radicand is a
nonperfect square, between with two consecutive
integers would the square root fall?
400  (5  2  2)(5  2  2)
400  40 10
 (5  2  2)
400  8  5  5  2
 20
400  2  2  2  5  5  2
205
144,169, ,196, 225,
205  5  41
This is a nonperfect
square.
Between 14 and 15.
205
Find each square root. If the radicand is a
nonperfect square, between with two consecutive
integers would the square root fall?
49

81
 77
77  7 11
This is a nonperfect
square.
 49 

 

 81 
7

9
 100,  81,  64,  49
Between -9 and -8.
-77