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 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.
Prove that the product of 2π and 3/4 is an irrational number. Find the product and explain why the value is irrational.
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.
Can you show it in math form?
To prove that the sum of 3/4 and 10−−√ is irrational, we need to assume the opposite, i.e., assume that the sum is rational, and then show a contradiction.
Let's assume that the sum of 3/4 and 10−−√ is rational and can be written as p/q, where p and q are integers with no common factors other than 1.
Then, we have (3/4) + (10−−√) = p/q.
Rearranging the equation, we get 10−−√ = (p/q) - (3/4).
Simplifying, we can write 10−−√ as (p - 3q) / (4q).
Since the square root of 10 is irrational, we know that 10−−√ is also irrational. However, in our equation, we assume that (p - 3q) / (4q) is rational. This leads to a contradiction, as an irrational number cannot be expressed as a ratio of two integers. Therefore, the sum of 3/4 and 10−−√ is an irrational number.