Question

Insert a rational and an irrational number between 2 and 3.

Answer 1

Hint: Find a rational number x between the given two numbers a and b by using \[x=\dfrac{a+b}{2}\]. Also, find the irrational number say y between any two numbers a and b by selecting a number less than b and greater than a which is not terminating and non-repeating.

Complete step-by-step answer:
In this question, we have to insert a rational and an irrational number between 2 and 3. Before proceeding with the question, let us see what rational and irrational numbers are.
Rational Numbers: These are the numbers that can be expressed in the form of \[\dfrac{p}{q}\] where p and q are integers and \[q\ne 0\]. All the whole numbers, natural numbers, fractions \[\left( \text{denominator }\ne 0 \right)\] are rational numbers.
Example: \[0,2,\dfrac{1}{2},1.5,\dfrac{4}{5},etc\]
Irrational Numbers: The numbers which are not rational are irrational numbers when we express these numbers as a decimal form, they go on forever after the decimal and never repeat. Any number of the form \[\sqrt{N}\] is an irrational number given that N must not be a perfect square.
Example: \[\sqrt{2},\sqrt{3},2.1467825556......\]
Now, let us consider our question. Let us find a rational number between 2 and 3. We know that there are infinitely many rational numbers between 2 and 3. We know that a number say x between two numbers ‘a’ and ‘b’ is given by:
\[x=\dfrac{a+b}{2}\]
So, we get a rational number between 2 and 3 as,
\[x=\dfrac{2+3}{2}=\dfrac{5}{2}\]
x = 2.5
As, \[x=\dfrac{5}{2}\] is in the form of \[\dfrac{p}{q}\text{ and }q\ne 0\]. So, this is a rational number.
Let us find an irrational number between 2 and 3. Like a rational number, there are infinitely many irrational numbers between 2 and 3. We know that \[\sqrt{N}\] is an irrational number if N is not a perfect square. So, we can write,
\[2=\sqrt{4}\]
\[3=\sqrt{9}\]
Between \[\sqrt{4}\text{ to }\sqrt{9}\], we can get some irrational numbers like \[\sqrt{5},\sqrt{6},\sqrt{7},\sqrt{8},\sqrt{9},etc.\] as 5, 6, 7, 8 are not perfect squares. Also, we can choose any number say y which is greater than 2 and less than 3 and is non terminating, non-repeating like y = 2.5434212224….
So, we get a rational number between 2 and 3 as 2.5 or \[\dfrac{5}{2}\] and an irrational number between 2 and 3 as 2.5434212224…..

Note: By first looking at the numbers, students can tell if a number is rational or not by examining the digits after the decimal in that number. For rational numbers, either the digits terminate after a certain number or they get repeated while in irrational numbers, digits never get repeated and never get terminated. Also, students are always advised to convert the fractions into decimal form to visualize the magnitude of numbers.