Show an example of an irrational number between 9 and 11.
Explain how.
An example of an irrational number between 9 and 11 is √10.
To explain this, we need to first understand what an irrational number is. An irrational number is a number that cannot be expressed as the quotient (fraction) of two integers, and it cannot be expressed as a repeating or terminating decimal. In simpler terms, an irrational number cannot be written as a simple fraction or as a finite decimal.
√10, which is the square root of 10, is an irrational number. We can see this by attempting to express it as a fraction or a decimal.
If we try to express √10 as a fraction, we cannot find two integers that can be divided to give us exactly √10. Therefore, it cannot be expressed as a simple fraction.
If we try to express √10 as a decimal, we get approximately 3.16227766. However, this decimal representation goes on infinitely without repeating or terminating. Therefore, it cannot be expressed as a finite decimal.
Since √10 cannot be expressed as a fraction or as a finite decimal, it is indeed an irrational number. As √10 is approximately 3.16227766, it falls between 9 and 11.
What is the least integer whose square root is an irrational number between 5 and 7?
Explain how.
The least integer whose square root is an irrational number between 5 and 7 is 6.
To explain this, we need to understand that an irrational number is a number that cannot be expressed as a simple fraction or as a finite decimal. In this case, we are looking for the least integer whose square root falls between 5 and 7.
If we calculate the square root of 6, we get approximately 2.449489742783178. This decimal representation goes on infinitely without repeating or terminating, which confirms that it is an irrational number.
Now, let's examine the integers less than 6. The square root of 4 is 2, which is not between 5 and 7. Similarly, the square root of 5 is approximately 2.23606797749979, which is less than 5. Therefore, the least integer whose square root is an irrational number between 5 and 7 is 6.
What is the least integer whose square root is an irrational number between 5 and 7?
26
25
6
13
The least integer whose square root is an irrational number between 5 and 7 is 6.
To find an example of an irrational number between 9 and 11, we can use the square root of a non-perfect square number.
Let's consider the square root of 10.
To verify that √10 is irrational, we can assume that it is rational and express it as a fraction, p/q, where p and q have no common factors other than 1.
So, let's suppose √10 = p/q, where p and q are integers with no common factors.
Squaring both sides of the equation, we get 10 = (p^2)/(q^2).
Rearranging, we have p^2 = 10q^2.
From this equation, we can see that p^2 must be divisible by 10. In turn, this means that p must be divisible by 10. If p is divisible by 10, it follows that p^2 is divisible by 100 (since 10^2 = 100).
So, we have p^2 = 10q^2, which means p^2 is divisible by 100. However, this contradicts our assumption that p^2 is divisible only by 10.
Therefore, our initial assumption that √10 is rational is false. Hence, √10 is an irrational number.
Since the square root of 10 is between 9 and 11 (√9 = 3 and √11 ≈ 3.316), it serves as an example of an irrational number within that range.
To find an example of an irrational number between 9 and 11, we can use the fact that the square root of any non-perfect square number is an irrational number.
First, we need to find a perfect square number between 9 and 11. The closest perfect square number is 9, which is equal to 3 squared (3^2 = 9).
Next, we need to find a non-perfect square number between 9 and 11. One such number is 10.
Now, we can take the square root of 10 to find an irrational number between 9 and 11. The square root of 10 is approximately 3.16227766.
Therefore, an example of an irrational number between 9 and 11 is √10 (approximately 3.16227766).