1. Representing Irrational Numbers on the Number Line
   You can represent irrational numbers on the number line by using the Pythagorean theorem to find the hypotenuse of a right triangle.
   
   
   (Example) Representing the irrational numbers and on the number line:
1. On graph paper, draw a right triangle with legs of length along the number line and perpendicular to it.
2. Calculate the length of the hypotenuse of :
   
3. Draw a circle centered at point with radius . The points where the circle intersects the number line, labeled and , correspond to and , respectively.
   
   
2. Real Numbers and the Number Line
1. Every real number corresponds to a unique point on the number line, and each point on the number line corresponds to a real number.
2. There are infinitely many real numbers between any two distinct real numbers.
3. The number line can be completely filled by both rational and irrational numbers, i.e., all real numbers.


3. Properties of Real Numbers
1. There are infinitely many rational numbers between any two distinct rational numbers.
2. There are infinitely many irrational numbers between any two distinct irrational numbers.
3. The number line cannot be completely filled using only rational or only irrational numbers.


4. Comparison of Real Numbers
   To compare the size of real numbers, you can use one of the following three methods:


1. Using the Difference Between Two Numbers
   The relationship between two real numbers and can be determined by the sign of their difference:
1. If , then
2. If , then
3. If , then


2. Using Properties of Inequalities
   (Example) Comparing and :
   Since (because , we have
   
   
3. Using the Values of Square Roots
   (Example) Comparing and :
   Knowing that , we get
   , and .
   Thus,


5. Relationships Among Real Numbers
   Since every real number corresponds to a point on the number line, the following relationships hold, similar to rational numbers:
1. Negative numbers positive numbers.
2. Among positive numbers, the larger the absolute value, the larger the number.
3. Among negative numbers, the larger the absolute value, the smaller the number.