Let $a=\sqrt{2}$ and $b=\sqrt{3}$.
(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 1.5 (or 2 3 ), and an irrational number is 2.5 . The average 2 2 + 3 is between these two numbers, illustrating the denseness property of real numbers, which states that between any two distinct real numbers, there is always another real number. This property can be applied repeatedly to find infinitely many numbers between any two real values.
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Find rational number: Approximate 2 ≈ 1.414 and 3 ≈ 1.732 , then 1.5 = 2 3 is a rational number between them.
Find irrational number: 2.5 is an irrational number between 2 and 3 .
Average of a and b : 2 a + b is between a and b .
Denseness property: Repeat the averaging process to show infinitely many real numbers exist between any two distinct real numbers. The final answer is: 1.5 , 2.5
Explanation
Understanding the Problem We are given two numbers, $a =
\sqrt{2} an d b = \sqrt{3} , an d w e n ee d t o f in d a r a t i o na l an d ani rr a t i o na l n u mb er b e tw ee n t h e m . T h e n , w es h o u l d u se t h e a v er a g eo f a an d b$ to justify the denseness property of real numbers.
Approximating the Square Roots First, let's find decimal approximations of a and b to help us find a rational number between them. We know that 2 ≈ 1.414 and 3 ≈ 1.732 .
Finding a Rational Number (a) A rational number between a and b is a number that can be expressed as a fraction q p , where p and q are integers and q = 0 . Since 1.414 < 1.5 < 1.732 , we can choose 1.5 = 2 3 as a rational number between 2 and 3 .
Finding an Irrational Number To find an irrational number between a and b , we can consider a number of the form x , where x is a non-square rational number between 2 and 3. For example, we can take x = 2.5 = 2 5 . Then, 2 < 2.5 < 3 , so 2.5 is an irrational number between a and b .
Average of a and b (b) The average of a and b is 2 a + b = 2 2 + 3 . Since a < b , we have a + a < a + b < b + b , which means 2 a < a + b < 2 b . Dividing by 2, we get a < 2 a + b < b . This shows that the average of two real numbers is always between them.
Denseness Property of Real Numbers Since a and b are real numbers, 2 a + b is also a real number between a and b . We can repeat this process to find a real number between a and 2 a + b , and between 2 a + b and b . For example, we can find the average of a and 2 a + b , which is 2 a + 2 a + b = 4 3 a + b . This number is between a and 2 a + b , and therefore also between a and b . We can continue this process indefinitely, which shows that between any two real numbers, there is another real number. This justifies the denseness property of real numbers.
Final Answer In summary:
(a) A rational number between 2 and 3 is 2 3 = 1.5 , and an irrational number between 2 and 3 is 2.5 .
(b) The average of 2 and 3 , which is 2 2 + 3 , lies between 2 and 3 . We can repeat this averaging process indefinitely, demonstrating that there are infinitely many real numbers between any two distinct real numbers, thus illustrating the denseness property of real numbers.
Examples
The denseness property of real numbers is crucial in many areas of mathematics and physics. For example, when designing a sensor that measures temperature, engineers need to ensure that the sensor can detect arbitrarily small changes in temperature. This is because temperature, like real numbers, is dense. Another example is in computer graphics, where curves and surfaces are often approximated by a series of line segments or triangles. The more segments or triangles used, the better the approximation. The denseness property ensures that we can always find a finer approximation, making the image more realistic. In essence, the denseness property allows for continuous refinement and arbitrarily precise measurements or representations.
