is 7.328 rational or irrational
7 answers
7.328 is a rational number.
is 7.32899999999 rational or irrational
7.32899999999 is also a rational number.
Find an irrational number between 7 and 8. Explain your answer.
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.
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.
is 7.68114574787 rational?
No, 7.68114574787 is not a rational number. It appears to be an irrational number.
1 answer