Question

is 7.328 rational or irrational

Answer 1

[AI-generated answer — Bot GPT 3.5]

7.328 is a rational number.

Answer 2

is 7.32899999999 rational or irrational

Answer 3

[AI-generated answer — Bot GPT 3.5]

7.32899999999 is also a rational number.

Answer 4

Find an irrational number between 7 and 8. Explain your answer.

Answer 5

[AI-generated answer — Bot GPT 3.5]

One irrational number that lies between 7 and 8 is √2 + 7.

To verify its irrationality, we assume the contrary and suppose that √2 + 7 is rational. This would mean that it can be expressed as a fraction p/q, where p and q are integers and q is not zero. Rearranging the equation, we have √2 = p/q - 7.

Squaring both sides, we get 2 = (p^2/q^2) - 14(p/q) + 49. Multiplying by q^2, we have 2q^2 = p^2 - 14pq + 49q^2.

Notice that since p^2 and 49q^2 are perfect squares, the left side is even. Therefore, p^2 - 14pq must also be even. This implies that p is even, so p = 2k for some integer k.

Substituting this into the equation, we have 2q^2 = 4k^2 - 28kq + 49q^2, which simplifies to q^2 = 2k^2 - 14kq + 24q^2.

The right side is even because all its terms are even. Consequently, q^2 must also be even, implying that q is even. However, this contradicts our initial assumption that p/q is a simplified fraction, as both p and q are even.

Therefore, we can conclude that √2 + 7 is irrational.

Answer 6

is 7.68114574787 rational?

Answer 7

[AI-generated answer — Bot GPT 3.5]

No, 7.68114574787 is not a rational number. It appears to be an irrational number.