1.3 Real Numbers [20m]
Have you ever seen tube-wells that help farmers irrigate their farms? In the above photograph, the water is coming from an underground well covered by a cuboidal block through a cylindrical pipe (the tube) and is being stored into a cuboidal tank (reservoir) that has a square top before being let out in the farms.
Now, here is an interesting thing. No one can determine exactly how much water comes out of the tube in a minute. Also, no one can determine exactly the length of the diagonal of the square top of the reservoir. By the end of this lesson, you will know why these quantities are not exactly measurable.
DEFINITION: IRRATIONAL NUMBERS
A number that is NOT a rational number, is called an irrational number.
So, a number is irrational if it cannot be written as:
1. A ratio of an integer to a non-zero integer, OR
2. A terminating decimal number, OR
3. A recurring decimal number.
Example 1: One of the Greek scholars in the Pythagoras' academy discovered that the length of the diagonal of a unit square, , could not be expressed as a ratio of integers. For having discovered irrational numbers, the scholar is believed to be thrown off the ship into the ocean.
Example 2: is another example of an irrational number that you have already used when calculating area or circumference of circles.
DEFINITION: REAL NUMBERS
A number that is either rational or irrational is called a REAL number.
EXERCISE: State whether the following statement are true or false. Justify your answers.
1. Every irrational number is a real number.
2. Every point on the number line is of the form , where m is a natural number.
3. Every real number is an irrational number.
SELF PRACTICE: State whether the following statements are true or false. Give reasons for your answers.
1. Every rational number is a real number.
2. Every point on the number line is of the form , where m is real number.
3. Every real number is a rational number.
EXERCISE: Are the square roots of all positive integers irrational? If not, give an example of the square root of a number that is a rational number.
INTERESTING FACT: The exact VALUE of an irrational number can never be determined. But, its exact POSITION on the number line can still be determined.
QUESTION: Show how can be represented on the number line.
SELF PRACTICE: Represent , , and on the number line.