To find two irrational numbers between [tex]\(\sqrt{2}\)[/tex] and 2, let's go through the steps and reasons for our selections.
1. Identify the Range of Interest:
The problem asks us to find two irrational numbers between [tex]\(\sqrt{2}\)[/tex] (approximately 1.414) and 2.
2. Understand the Properties of Irrational Numbers:
Recall that an irrational number is a number that cannot be expressed as a simple fraction - its decimal expansion goes on forever without repeating.
3. Selection Process:
We need to select numbers that fall strictly within the range [tex]\(\sqrt{2} < x < 2\)[/tex] and are proven to be irrational.
4. Calculation of Midpoint:
To help in identifying our numbers, let's calculate the midpoint between [tex]\(\sqrt{2}\)[/tex] and 2.
- Midpoint formula:
[tex]\[
\text{Midpoint} = \frac{\sqrt{2} + 2}{2}
\][/tex]
By approximately knowing the value:
[tex]\[
\sqrt{2} \approx 1.414
\][/tex]
So,
[tex]\[
\text{Midpoint} \approx \frac{1.414 + 2}{2} = \frac{3.414}{2} = 1.707
\][/tex]
5. Select Irrational Numbers Around the Midpoint:
A good strategy is to select numbers slightly below and above this midpoint to ensure they are within the required limits.
- Let's consider the first irrational number just below the midpoint:
[tex]\[
1.6071067811865474
\][/tex]
This number falls in the range [tex]\(\sqrt{2} < 1.607 < 2\)[/tex].
- Next, consider the second irrational number just above the midpoint:
[tex]\[
1.8071067811865476
\][/tex]
This number also lies in the range [tex]\(\sqrt{2} < 1.807 < 2\)[/tex].
6. Verification:
Both chosen numbers:
[tex]\[
1.6071067811865474 \quad \text{and} \quad 1.8071067811865476
\][/tex]
are indeed irrational because they cannot be expressed as fractions and their decimal expansions are non-repeating and non-terminating.
By carefully selecting these numbers slightly above and below the calculated midpoint, we have found our two irrational numbers. Thus, the two irrational numbers between [tex]\(\sqrt{2}\)[/tex] and 2 are:
[tex]\[
1.6071067811865474 \quad \text{and} \quad 1.8071067811865476
\][/tex]
