1. Rational and Irrational Numbers
Honors Math – Grade 8

2. Get ready for the Lesson
During a 4-month period, an environmentalist recorded the following changes to a lake’s water level:
These numbers recorded above are called rational numbers.

3. A rational number is the quotient of two integers, a and b, written as
where
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.

4. Example Set Type of Number
The following types of numbers are rational numbers…
Terminating Decimals have a finite number of digits.
0.41, 3.0, -5.7
Repeating Decimals have a sequence of one or more digits that repeat indefinitely.
Decimals – can also be positive or negative. They are either terminating or repeating.
Fractions and mixed numbers – can be positive or negative
{…, -3, -2, -1, 0, 1, 2, 3, …}
Integers – are whole numbers and their opposites.
Example Set
Type of Number

5. Multiplying a number by itself, or raising it to the second power, is finding the square of that number.

6. Perfect squares, or square numbers, are the square of natural numbers.
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.
Read as “the principal square root of 25.”
Read as “the negative square root of 25.”

7. Find each square root.
Another way to find the principal square root of a perfect square is to use the prime factorization.
When the radicand is a fraction, find the principal square root of the numerator and the denominator.
Arrange the factors into groups.

8. 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 ?
Find two nearby perfect squares that 19 is between.
THINK
List the perfect squares in order.
19 is between 16 and 25 19
is between 4 and 5.

9. The following are examples of irrational numbers.
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
the form where a and b are integers and
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. … …
2. Square roots of nonperfect squares.
3. Pi, symbolized by a Greek letter.
Please note, 3.14 and 22/7 are rational approximate values.

10. -321.11 45 0.010203… Rational number Whole number Rational number
For each number, list all the terms that apply: whole number, integer, rational number, and irrational number. 45 …
Rational number
Whole number
Rational number
Irrational number
Integer
Rational number
Whole number
Irrational number
Rational number
Integer
Integer
Rational number
Rational number

11. 205 This is a nonperfect square. Between 14 and 15.
Find each square root. If the radicand is a nonperfect square, between with two consecutive integers would the square root fall?
205
This is a nonperfect square.
Between 14 and 15.

12. -77 Between -9 and -8. This is a nonperfect square.
Find each square root. If the radicand is a nonperfect square, between with two consecutive integers would the square root fall?
Between -9 and -8.
-77
This is a nonperfect square.