Find the approximate value of √27,√30, and √34 on a number line.
Rational & Irrational Numbers
Rational numbers are numbers that can be written as fractions in the form
, where p and q are integers and q 0. All rational numbers can be written as terminating or repeating decimals.
Irrational numbers are non-terminating, non-repeating decimals and cannot be written in the form , where p and q are integers and q 0. The square roots of whole numbers that are not perfect squares are irrational numbers.
Examples of Rational Numbers
Examples of Irrational Numbers
Number Line - Irrational
On a number line, all numbers to the right of zero are positive, and all the numbers to the left of zero are negative.
A number line with the points 0, , , and 5 graphed on it is shown below.
Example 1:
At what position on the number line is the red dot located?
A. B. C. D.
Solution:
The dot is located between 1 and 2.
Therefore, the dot is located between and .
Therefore, the red dot is located at , which is approximately 1.7320508076.
So, the only answer choice that falls in this range is B.
Example 2:
Which point represents the value of ?
A.Point W B.Point X C.Point Y D.Point Z
Solution:
The value of is approximately 2.24.
So, lies between 2 and 3.
On the given number line, point X is the only point that lies between 2 and 3.
Therefore, the point that represents the value of is point X.
Compare Irrational Numbers
Irrational numbers can be compared by using decimals to approximate the value of the irrational numbers.
Example 1:
Identify the missing symbol.
Solution:
First, approximate the irrational numbers as decimals.
Then, compare the approximations.
Since 8.83 > 7.14, .
Example 2:
Identify the missing symbol.
Solution:
First, approximate the irrational numbers as decimals, and simplify both expressions.
Then, compare the approximations.
Since 9.42 < 16.09, .
Irrational Numbers & Approximations
To compare the size of irrational numbers, calculate a rational approximation of the numbers and locate these on a number line.
Example 1:
Use the number line above to correctly compare the following numbers.
Solution:
The numbers should be placed on the number line to determine the order, and then choose the correct comparison. Change each number into a decimal for ease of placement.
Therefore, the correct comparison is shown below.
Example 2:
Use the number line above to order the following numbers from least to greatest.
Solution:
Place the numbers on a number line to determine the order. Change each number into a decimal for ease of placement.
Therefore, the correct order is shown below.
Example 3:
Use the number line above to order the following numbers from least to greatest.
Solution:
Place the numbers on a number line to determine the order. Change each number into a decimal for ease of placement.
Therefore, the correct order is shown below.
Recognizing Rational and Irrational Numbers
Click on the image below to watch a video about how to determine if a number is rational or irrational.
NOTE: All Khan Academy content
available for free at www.khanacademy.org Recognizing Rational and Irrational Numbers - Khan Academy*
*Khan Academy is not affiliated with Study Island.
Irrational Numbers & Approximations
Click on the image below to watch a video about how to find an approximation for non-perfect square roots.
NOTE: All Khan Academy content available for free at www.khanacademy.org Approximating Square Roots - Khan Academy*
*Khan Academy is not affiliated with Study Island.
, where p and q are integers and q 
0. All rational numbers can be written as terminating or repeating decimals.
, 
, and 5 graphed on it is shown below.
and 
.
?
.
.
