Let $a=\sqrt{2}$ and $b=\sqrt{3}$.
(a) Find a rational number and an irrational number strictly between $a$ and $b$.
Answer (2)
A rational number between 2 and 3 is 2 3 , and an irrational number is 2.5 .
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Approximate 2 ≈ 1.414 and 3 ≈ 1.732 .
Find a rational number between 1.414 and 1.732 , such as 1.5 = 2 3 .
Find an irrational number between 2 and 3 , such as 2.5 = 2 10 .
Thus, a rational number is 2 3 and an irrational number is 2 10 .
Rational: 2 3 , Irrational: 2 10
Explanation
Problem Analysis and Approximations We are given a =
\[ \sqrt{2} \] and b =
\[ \sqrt{3} \] , and we need to find a rational and an irrational number strictly between a and b . First, let's approximate the values of a and b .
Finding Rational Approximation We know that 2 ≈ 1.414 and 3 ≈ 1.732 . Therefore, we need to find a rational and an irrational number between 1.414 and 1.732 .
Identifying a Rational Number A rational number is a number that can be expressed as a fraction q p , where p and q are integers and q = 0 . A simple rational number between 1.414 and 1.732 is 1.5 , which can be written as 2 3 . So, 1.5 is a rational number between 2 and 3 .
Identifying an Irrational Number An irrational number cannot be expressed as a fraction of two integers. To find an irrational number between 2 and 3 , we can consider a square root of a non-perfect square number between 2 and 3 . For example, 2.5 is between 2 and 3 . So, 2.5 is an irrational number between 2 and 3 . We can rewrite 2.5 as 2 5 = 2 10 .
Alternative Irrational Number Alternatively, we can take the average of 2 and 3 , which is 2 2 + 3 . This is also an irrational number between 2 and 3 .
Final Answer Therefore, a rational number between 2 and 3 is 1.5 = 2 3 , and an irrational number between 2 and 3 is 2.5 = 2 10 .
Examples
Understanding rational and irrational numbers helps in various real-life scenarios. For instance, when designing a bridge, engineers need to use precise measurements. While some measurements can be expressed as rational numbers (like the length of a beam), others might involve irrational numbers (like calculations involving circular shapes or certain material properties). Knowing how to work with both types of numbers ensures the bridge is built safely and accurately. Another example is in finance, where interest rates or investment returns might involve both rational (e.g., 5% interest) and irrational numbers (e.g., growth rates based on continuous compounding).
