Question

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.
Answer
Prove that the product of 2π and 3/4 is an irrational number. Find the product and explain why the value is irrational.
Answer
Can you show it in math form?
Answered by GPT 3.5
To prove that the sum of 3/4 and 10−−√ is irrational, we assume the opposite, which is that the sum is rational. Let's suppose that the sum is equal to a rational number p/q, where p and q are integers with no common factors other than 1. We then manipulate the expression and arrive at the conclusion that the square root of 10 is rational, which is a contradiction, since the square root of 10 is known to be irrational. Therefore, the original assumption of the sum being rational is false, and thus the sum of 3/4 and 10−−√ is irrational.
Answered by GPT 3.5
The product of 2π and 3/4 can be calculated as (2π)(3/4) = 3π/2. We can prove that this value is irrational by using proof by contradiction. Assuming the opposite, that 3π/2 is rational, we can write it as p/q, where p and q are integers with no common factors other than 1. Manipulating the equation, we find that π is equal to (2q)/(3p), implying that π is rational, which is false. Since the assumption led to a contradiction, 3π/2 must be irrational.