Homework PracticeSign Up
Back to assignmentPrintable assignment view
Print options

Homework Practice

Printable assignment

Real Numbers Practice 3: Irrationality Proofs

Section A: Proof Structure

Question 1mcq
In a proof by contradiction, what is the first step when proving 2\sqrt{2} is irrational?
A.Assume 2\sqrt{2} is irrational.
B.Assume 22 is odd.
C.Assume every integer is prime.
D.Assume 2\sqrt{2} is rational.
Question 2mcq
In the proof that 2\sqrt{2} is irrational, if 2=\fracracpq\sqrt{2}=\fracrac{p}{q} in lowest terms, then p2=2q2p^2=2q^2. What must be true about pp?
A.pp is even.
B.pp is odd.
C.pp is prime.
D.pp is equal to qq.
Question 3mcq
A proof reaches that both pp and qq are even, even though \fracracpq\fracrac{p}{q} was assumed in lowest terms. What is the contradiction?
A.pp and qq are prime.
B.pp is greater than qq.
C.pp and qq have a common factor 22.
D.qq must be zero.
Question 4mcq
To adapt the proof of 2\sqrt{2} for 3\sqrt{3}, which fact is needed?
A.If 22 divides p2p^2, then 33 divides pp.
B.If 33 divides p2p^2, then 33 divides pp.
C.Every multiple of 33 is even.
D.3\sqrt{3} is a whole number.
Question 5mcq
Which conclusion correctly supports the proof that 5\sqrt{5} is irrational?
A.If 55 divides p2p^2, then 55 divides pp.
B.If 55 divides pp, then pp is even.
C.If p2p^2 is odd, then p=5p=5.
D.Every square number is divisible by 55.

Section B: Misconceptions and Reasoning

Question 6mcq
Why is 3+253+2\sqrt{5} irrational?
A.Because the sum of any two numbers is irrational.
B.If it were rational, then 5=\fracracr−32\sqrt{5}=\fracrac{r-3}{2} would be rational, a contradiction.
C.Because 33 is irrational.
D.Because multiplying by 22 always gives an irrational number.
Question 7mcq
If 7−437-4\sqrt{3} were rational, which expression would also become rational?
A.3=7−4r\sqrt{3}=7-4r
B.3=\fracracr−73\sqrt{3}=\fracrac{r-7}{3}
C.3=\fracrac7−r4\sqrt{3}=\fracrac{7-r}{4}
D.3=4(7−r)\sqrt{3}=4(7-r)
Question 8mcq
A student writes: “2=\fracracpq\sqrt{2}=\fracrac{p}{q}, so p2=2q2p^2=2q^2. Hence pp is even. Therefore 2\sqrt{2} is irrational.” What is missing?
A.They must show pp is odd.
B.They must prove 22 is composite.
C.They must find a decimal approximation.
D.They must also show qq is even and contradict lowest terms.
Question 9mcq
Assertion: 3\sqrt{3} is irrational. Reason: If 33 divides p2p^2, then 33 divides pp. Choose the correct option.
A.Both are true, but the reason is unrelated.
B.Both assertion and reason are true, and the reason supports the proof.
C.Assertion is true, but reason is false.
D.Assertion is false, but reason is true.
Question 10mcq
Which option describes the contradiction structure correctly?
A.Assume irrational, derive rational, accept assumption.
B.Estimate decimal value, then conclude.
C.Check only one example and conclude.
D.Assume rational, derive impossible common factor, reject assumption.