Prove that the product of 2π and 3/4 is an irrational number. Find the product and explain why the value is irrational. Explain your reasoning in 1–2 sentences.(2 points)
The product of 2π and 3/4 is (2π)(3/4) = (3π)/2. Since π is an irrational number, any multiple of π, such as 3π, is also irrational. Therefore, the product (3π)/2 is irrational.
To prove that the product of 2π and 3/4 is an irrational number, we first need to find the product.
The product of 2π and 3/4 is (2π)(3/4) = 6π/4 = 3π/2.
To prove that 3π/2 is irrational, we need to show that it cannot be expressed as the quotient of two integers.
Assuming π is irrational, which is a well-established mathematical fact, the product of 3 and π is also irrational. Since 3/2 is rational (the quotient of two integers), the product 3π/2 cannot be expressed as the quotient of two integers and is therefore irrational.
To prove that the product of 2π and 3/4 is an irrational number, we first find the product. The product of 2π and 3/4 can be calculated by multiplying the numbers: (2π) * (3/4) = (6π/4) = (3π/2).
Now, to show that this value is irrational, we must provide reasoning. However, the product (3π/2) is not inherently irrational as it is a combination of a rational number (3/2) and an irrational number (π).
Thus, the product (3π/2) is not necessarily irrational on its own, but if we can prove that π is irrational, then we can conclude that the product (3π/2) is also irrational. The proof of the irrationality of π is a separate topic altogether.