Let [tex]$a=\sqrt{2}$[/tex] and [tex]$b=\sqrt{3}$[/tex].
(a) Find a rational number and an irrational number strictly between a and b.
(b) Use the average of a and b to justify the denseness property of real numbers.
Answer (2)
A rational number between 2 and 3 is 2 3 and an irrational number is 2.5 . The average 2 2 + 3 lies between them, illustrating the density property of real numbers. This means there are infinitely many numbers between any two real numbers.
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A rational number between 2 and 3 is 2 3 .
An irrational number between 2 and 3 is 2.5 .
The average of 2 and 3 , which is 2 2 + 3 , lies between 2 and 3 .
The average justifies the density property of real numbers, as it demonstrates that there is always another real number between any two given real numbers. $\boxed{\text{See steps above}}.
Explanation
Understanding the Problem We are given a =
\[ \sqrt{2} \] and b =
\[ \sqrt{3} \] . We need to find a rational and an irrational number between a and b , and then use the average of a and b to justify the density property of real numbers.
Finding a Rational Number First, let's find a rational number between a and b . We know that 2 ≈ 1.414 and 3 ≈ 1.732 So, a rational number between them is 1.5 = 2 3 .
Finding an Irrational Number Now, let's find an irrational number between a and b . We can consider a number of the form x where 2 < x < 3 . Let's take x = 2.5 . Then 2.5 is an irrational number between 2 and 3 . We have 2.5 ≈ 1.581 which is between 1.414 and 1.732.
Justifying the Density Property To justify the density property, we need to show that the average of a and b , which is 2 a + b = 2 2 + 3 , is between a and b . We have 2 2 + 3 ≈ 2 1.414 + 1.732 = 2 3.146 = 1.573 Since 1.414 < 1.573 < 1.732 , we have 2 < 2 2 + 3 < 3 .
Density Property Explained The density property of real numbers states that between any two real numbers, there exists another real number. The average of two real numbers is always between them. So, 2 a + b = 2 2 + 3 is a real number between a and b . We can repeat this process to find infinitely many real numbers between a and b .
Final Answer Therefore, a rational number between 2 and 3 is 2 3 , an irrational number between 2 and 3 is 2.5 , and the average 2 2 + 3 justifies the density property of real numbers.
Examples
The density property of real numbers is a fundamental concept in mathematics that has practical applications in various fields. For instance, in computer graphics, when rendering a smooth curve or surface, the algorithm needs to approximate the continuous curve with a series of discrete points. The density property ensures that we can always find more points between any two existing points, allowing us to refine the approximation and achieve a smoother visual representation. Similarly, in physics simulations, the density property allows us to model continuous phenomena, such as fluid flow or heat transfer, with increasing accuracy by discretizing the space into smaller and smaller intervals.
