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Application to proving irrationality: Some numbers can be proven to be irrational using the Fundamental Theorem of Arithmetic. For example, let’s prove that âˆš3 is irrational. This is a proof by contradiction. Suppose âˆš3 is rational. Let âˆš3 = a/b where a and b are integers (b â‰ 0). Then âˆš3b = a. This is impossible by the Fundamental Theorem of Arithmetic.

Why? Because the prime 3 occurs to an odd power on the right-hand side, but occurs to an even power on the left-hand side (To see this, let a = 3c^2 and b = 3d^2 where c > 0, and b = 3d^2 where d > 0). Then the power of 3 in a is 2c, which is even, and the power of 3 in âˆš3b is 1 + 2d, which is odd.

Homework: Show that âˆš6 is irrational.

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(a) Prove that $\sqrt{3}, \sqrt{5},$ and $\sqrt{6}$ are irrational. Hint: To treat $\sqrt{3},$ for example, use the fact that every integer is of the form $3 n$ or $3 n+1$ or $3 n+2 .$ Why doesn’t this proof work for $\sqrt{4} ?$(b) Prove that $\sqrt[3]{2}$ and $\sqrt[3]{3}$ are irrational.

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DISCOVER “PROVE: Combining Rational and Irrational Numbers Is $\frac{1}{2}+\sqrt{2}$ rational or irrational? Is $\frac{1}{2} \cdot \sqrt{2}$ rational or irrational? Experiment with sums and products of other rational and irrational numbers. Prove the following.(a) The sum of a rational number $r$ and an irrational number $t$ is irrational.(b) The product of a rational number $r$ and an irrationalnumber $t$ is irrational.[Hint: For part (a), suppose that $r+t$ is a rational number $q,$that is, $r+t=q$. Show that this leads to a contradiction.Use similar reasoning for part (b).J

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For each of the following, give a counterexample to show that the statement is false: If a^2 – b^2 > 0, where a and b are real, then a – b > 0. If x and y are irrational and x = y, then xy is irrational.

Proof by Contradiction: Proofs involving irrational numbers and modulus are often best solved by contradiction. Use contradiction to prove each of the following:

i. Prove that if the sum of two numbers, a and b, is irrational, then at least one of the numbers is irrational.

ii. Prove that there is no integer a such that a â‰¡ 2 (mod 6) and a â‰¡ 7 (mod 9).

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Prove or disprove that the product of two irrational numbers is irrational.

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