One example of an irrational number between 7 and 8 is the square root of 2.
To determine this, we can start by finding the square of different numbers between 7 and 8:
- 7 squared is 49
- 8 squared is 64
From this, we can see that the square root of 2 lies between the square roots of 49 and 64. The square root of 49 is 7, and the square root of 64 is 8. Since the square root of 2 is between 7 and 8, it satisfies the given condition. Moreover, the square root of 2 is an irrational number because it cannot be expressed as a simple fraction or a terminating or repeating decimal. Its decimal representation goes on forever without repeating, making it irrational.
Find an irrational number between 7 and 8. Explain your answer.
3 answers
simple and smaller
Another example of an irrational number between 7 and 8 is 7.5√2.
To explain this, we can first consider the square of 7.5:
- 7.5 squared is 56.25
Since 56.25 is between 49 and 64, we know that the square root of 56.25 lies between the square roots of 49 and 64, which are 7 and 8 respectively.
To find the square root of 56.25, we can use a calculator or estimation techniques like the Newton-Raphson method. The square root of 56.25 is approximately 7.5.
Since the square root of 56.25 is between 7 and 8, we can express it as 7.5√2. This number is irrational because √2 is irrational, and multiplying an irrational number by a rational number like 7.5 still yields an irrational number.
To explain this, we can first consider the square of 7.5:
- 7.5 squared is 56.25
Since 56.25 is between 49 and 64, we know that the square root of 56.25 lies between the square roots of 49 and 64, which are 7 and 8 respectively.
To find the square root of 56.25, we can use a calculator or estimation techniques like the Newton-Raphson method. The square root of 56.25 is approximately 7.5.
Since the square root of 56.25 is between 7 and 8, we can express it as 7.5√2. This number is irrational because √2 is irrational, and multiplying an irrational number by a rational number like 7.5 still yields an irrational number.