Find an irrational number between $$ \begin{array}{lll} ext { (1) } 2 ext { and } 3 & ext { (2) } & \sqrt{2} ext { and } \sqrt{3} \ ext { (3) }\end{array} $$ and $$ 2 \pi $$

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Find an irrational number between $$ \begin{array}{lll}	ext { (1) } 2 	ext { and } 3 & 	ext { (2) } & \sqrt{2} 	ext { and } \sqrt{3} \ 	ext { (3) }\end{array} $$ and $$ 2 \pi $$

UpStudy AI · Accepted Answer

Sure! Let's find specific irrational numbers within each of the given intervals.

### (1) Between $$ 2 $$ and $$ 3 $$

**Example:** $$ \sqrt{5} $$

- **Explanation:** 
  - $$ \sqrt{4} = 2 $$ and $$ \sqrt{9} = 3 $$.
  - $$ \sqrt{5} \approx 2.236 $$ lies between $$ 2 $$ and $$ 3 $$.
  - Since $$ 5 $$ is not a perfect square, $$ \sqrt{5} $$ is irrational.

### (2) Between $$ \sqrt{2} $$ and $$ \sqrt{3} $$

**Example:** $$ \frac{\sqrt{2} + \sqrt{3}}{2} $$

- **Explanation:** 
  - $$ \sqrt{2} \approx 1.414 $$ and $$ \sqrt{3} \approx 1.732 $$.
  - $$ \frac{\sqrt{2} + \sqrt{3}}{2} \approx \frac{1.414 + 1.732}{2} \approx 1.573 $$, which lies between $$ \sqrt{2} $$ and $$ \sqrt{3} $$.
  - The sum and average of two irrational numbers (when they are not rational multiples of each other) are typically irrational, so $$ \frac{\sqrt{2} + \sqrt{3}}{2} $$ is irrational.

### (3) Between a Given Number and $$ 2\pi $$

Assuming the interval is between $$ \pi $$ and $$ 2\pi $$:

**Example:** $$ \pi $$ itself

- **Explanation:** 
  - $$ \pi \approx 3.1416 $$ and $$ 2\pi \approx 6.2832 $$.
  - Since $$ \pi $$ is irrational and $$ \pi < 2\pi $$, it serves as an irrational number within this interval.

Alternatively, if another lower bound was intended (e.g., between $$ 3 $$ and $$ 2\pi $$):

**Example:** $$ \sqrt{10} $$

- **Explanation:** 
  - $$ \sqrt{9} = 3 $$ and $$ \sqrt{10} \approx 3.162 $$, which is less than $$ 2\pi \approx 6.2832 $$.
  - $$ \sqrt{10} $$ is irrational.

### Summary

- **Between 2 and 3:** $$ \sqrt{5} $$
- **Between $$ \sqrt{2} $$ and $$ \sqrt{3} $$:** $$ \frac{\sqrt{2} + \sqrt{3}}{2} $$
- **Between $$ \pi $$ and $$ 2\pi $$:** $$ \pi $$

These examples demonstrate the existence of irrational numbers within the specified intervals.
Here are some irrational numbers between the given ranges: 1. **Between 2 and 3:** $$ \sqrt{5} $$ (approximately 2.236) 2. **Between $$ \sqrt{2} $$ and $$ \sqrt{3} $$:** $$ \frac{\sqrt{2} + \sqrt{3}}{2} $$ (approximately 1.573) 3. **Between $$ \pi $$ and $$ 2\pi $$:** $$ \pi $$ (approximately 3.1416) These numbers are irrational and lie within the specified intervals.

Find an irrational number between \( \begin{array}{lll}\text { (1) } 2 \text { and } 3 & \text { (2) } & \sqrt{2} \text { and } \sqrt{3} \\ \text { (3) }\end{array} \) and \( 2 \pi \)

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