Prove that the sum of 3/4
and 10−−√
is irrational. Show your work, and in 1–2 sentences, explain why the sum is an irrational number.
To prove that the sum of 3/4 and 10−−√ is irrational, we assume the sum is rational and express it as a/b where a and b are integers with no common factors. Then, by squaring both sides of the equation (a/b)^2 = (3/4 + 10−−√), we can rearrange it to obtain a quadratic equation. After simplifying, we find that the equation has no rational solutions for a and b, which implies that the sum 3/4 + 10−−√ is irrational.
To prove that the sum of 3/4 and 10−−√ is irrational, we will assume the contrary, which is that the sum is rational, and then show that it leads to a contradiction. Let's say the sum is rational and can be expressed as a/b, where a and b are integers.
Hence, 3/4 + 10−−√ = a/b. Rearranging the equation, we get 10−−√ = (a/b) - 3/4.
Since a/b is rational and 3/4 is rational, their difference (a/b - 3/4) is also rational. Therefore, 10−−√, which is the difference, must be rational.
However, it is known that the square root of any non-perfect square number is an irrational number. Since 10 is not a perfect square, 10−−√ is irrational. Therefore, our assumption that the sum of 3/4 and 10−−√ is rational leads to a contradiction, which proves that the sum is in fact an irrational number.