Show an example of an irrational number between 9 and 11.
1. 99
2. 10
3. 20
4. 100
The example of an irrational number between 9 and 11 is 10.
To find an irrational number between 9 and 11, we first need to understand what an irrational number is. An irrational number is a number that cannot be expressed as a fraction or a ratio of two integers. They cannot be written exactly as decimals, and their decimal expansions go on forever without repeating.
Let's check the given options:
1. 99: This is not an irrational number as it can be expressed as the ratio 99/1 or 198/2.
2. 10: This is not an irrational number either, as it can be expressed as the ratio 10/1 or 20/2.
3. 20: Similar to the previous options, this number can be expressed as the ratio 20/1 or 40/2, so it is not irrational.
4. 100: This is not an irrational number because it can be expressed as the ratio 100/1 or 200/2.
None of the given options is an irrational number. To find an irrational number between 9 and 11, we can use a commonly known irrational number, such as the square root of 10.
√10 is an irrational number because when we express it as a decimal, it continues infinitely without repeating. To find the value approximately, we can use a calculator:
√10 ≈ 3.16227766...
Now, we can see that the square root of 10 is between 3 and 4. Therefore, it is larger than 3 but smaller than 4.
Therefore, an example of an irrational number between 9 and 11 is √10, which is approximately 3.16227766.