is 7.4 irrational?
![Bot GPT 3.5](/images/users/4310/128x128.jpeg)
4 months ago
![grt](/images/users/0/1/128x128.jpeg)
4 months ago
Find an irrational number between 7 and 8. Explain your answer.
![Bot GPT 3.5](/images/users/4310/128x128.jpeg)
4 months ago
One irrational number between 7 and 8 is โ2 + 7. To understand why it's irrational, we first need to establish that โ2 is irrational.
Assume โ2 is rational, meaning it can be expressed as a fraction p/q, where p and q are integers with no common factors other than 1 and q is not equal to 0. By squaring both sides of the equation โ2 = p/q, we get 2 = (p^2)/(q^2), which can further be simplified to p^2 = 2*(q^2).
From this equation, we deduce that p^2 is even since it is equal to 2 times a whole number (q^2). Therefore, p must also be even because the square of an odd number is odd. Consequently, we can write p as 2k, where k is an integer.
Substituting 2k for p in the previous equation, we have (2k)^2 = 2*(q^2), which simplifies to 4k^2 = 2*(q^2) and further reduces to 2k^2 = q^2. This implies that q^2 is even since it is equal to 2 times a whole number (k^2). By extension, q must also be even.
However, we initially stated that p and q have no common factors other than 1, which is a contradiction since we deduced that p and q are both divisible by 2. Therefore, our assumption that โ2 is rational is false, and โ2 is, in fact, irrational.
Now, to find an irrational number between 7 and 8, we add โ2 (an irrational number) to 7. This gives us โ2 + 7, which is an irrational number between 7 and 8.
![grt](/images/users/0/1/128x128.jpeg)
4 months ago
can you give me a more simple answer?
![Bot GPT 3.5](/images/users/4310/128x128.jpeg)
4 months ago
Certainly! One simple example of an irrational number between 7 and 8 is 7.5. This number cannot be expressed as a fraction and has an infinite non-repeating decimal expansion. Therefore, it is irrational.
![grt](/images/users/0/1/128x128.jpeg)
4 months ago
is 7.5 irrational?
![Bot GPT 3.5](/images/users/4310/128x128.jpeg)
4 months ago