Given that is irrational for all primes and also suppose that 3721 is a prime. Can you conclude that is an irrational number? Is your conclusion correct? Why or why not?

Understand the Question Deductive Reasoning: If we accept the given premise that is irrational for all primes and assume that is a prime, logically following the premise leads to the deduction that is irrational. Validity vs. Truth: A logically valid deduction leads to a false conclusion if the underlying premise is false. To evaluate whether the conclusion is actually correct, we check whether is truly a prime number. Step 1 · Logical Conclusion from the Premise Given the statement: " is irrational for all primes " Under the assumption/hypothesis that is a prime number, by direct logical deduction: Step 2 · Verify if 3721 is Prime Check whether is a prime number by calculating its square root: Since , has factors other than and itself (). Therefore, is a composite number, not a prime number. Since , it is a rational number. Answer > Can you conclude that is irrational? Yes, based on the given assumption that is prime. Is your conclusion correct? No. Why or why not? The conclusion is incorrect because the premise that is prime is false. Since is composite, , which is rational. Common Mistakes Confusing Validity with Fact: Assuming a logical conclusion must be true in reality even when based on a false premise (hypothesis). Prime Factor Check: Overlooking that is a perfect square () and mistaking it for a prime number.

Appendix 1: Proofs in Mathematics | A1.2

Question 7

Given that $\sqrt{p}$ is irrational for all primes $p$ and also suppose that 3721 is a prime. Can you conclude that $\sqrt{3721}$ is an irrational number? Is your conclusion correct? Why or why not?

Solution

Understand the Question

Deductive Reasoning: If we accept the given premise that $\sqrt{p}$ is irrational for all primes $p$ and assume that $3721$ is a prime, logically following the premise leads to the deduction that $\sqrt{3721}$ is irrational.
Validity vs. Truth: A logically valid deduction leads to a false conclusion if the underlying premise is false.
To evaluate whether the conclusion is actually correct, we check whether $3721$ is truly a prime number.

Step 1 · Logical Conclusion from the Premise

Given the statement: " $\sqrt{p}$ is irrational for all primes $p$ "

Under the assumption/hypothesis that $3721$ is a prime number, by direct logical deduction: $\sqrt{3721} \text{ is an irrational number}$

Step 2 · Verify if 3721 is Prime

Check whether $3721$ is a prime number by calculating its square root:

\begin{aligned} 61 \times 61 &= 3721 \\[0.6em] \sqrt{3721} &= 61 \end{aligned}

Since $3721 = 61^2$ , $3721$ has factors other than $1$ and itself ( $1, 61, 3721$ ). Therefore, $3721$ is a composite number, not a prime number.

Since $\sqrt{3721} = 61 = \dfrac{61}{1}$ , it is a rational number.

Answer

Can you conclude that $\sqrt{3721}$ is irrational? Yes, based on the given assumption that $3721$ is prime.

Is your conclusion correct? No.
Why or why not? The conclusion is incorrect because the premise that $3721$ is prime is false. Since $3721 = 61^2$ is composite, $\sqrt{3721} = 61$ , which is rational.

Common Mistakes

Confusing Validity with Fact: Assuming a logical conclusion must be true in reality even when based on a false premise (hypothesis).
Prime Factor Check: Overlooking that $3721$ is a perfect square ( $61^2$ ) and mistaking it for a prime number.

Question 7

More questions in A1.2