Put the steps in correct order to prove that 2 is irrational. Rank the options below. Hence, the assumption that 22 is rational is wrong. It is irrational. Since is even, a must be even. Hence, p and q are both even, that is, that 2 is a common divisor of p and q. Thus p³ is even. Since the product of odd numbers is odd, pis even, and hence p = 2s. Suppose that 21/3 or 2 √2 is the rational number p/ where and q are positive integers with no common factors. Substituting into the equation p3= 2q3, 85³ = 29³, which simplifies to 45³ = 9³. This contradicts the choice of p/ q, such that p and q have no common factors. Hence, 2 = p³ / q³, or, equivalently, p³ = 29³. This is obtained by cubing on both sides. + + +
Put the steps in correct order to prove that 2 is irrational. Rank the options below. Hence, the assumption that 22 is rational is wrong. It is irrational. Since is even, a must be even. Hence, p and q are both even, that is, that 2 is a common divisor of p and q. Thus p³ is even. Since the product of odd numbers is odd, pis even, and hence p = 2s. Suppose that 21/3 or 2 √2 is the rational number p/ where and q are positive integers with no common factors. Substituting into the equation p3= 2q3, 85³ = 29³, which simplifies to 45³ = 9³. This contradicts the choice of p/ q, such that p and q have no common factors. Hence, 2 = p³ / q³, or, equivalently, p³ = 29³. This is obtained by cubing on both sides. + + +
A First Course in Probability (10th Edition)
10th Edition
ISBN:9780134753119
Author:Sheldon Ross
Publisher:Sheldon Ross
Chapter1: Combinatorial Analysis
Section: Chapter Questions
Problem 1.1P: a. How many different 7-place license plates are possible if the first 2 places are for letters and...
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Transcribed Image Text:Put the steps in correct order to prove that 2 is irrational.
Rank the options below.
Hence, the assumption that 2 √2 is rational is wrong. It is irrational.
Since q³ is even, a must be even. Hence, p and q are both even, that is, that 2 is a common divisor of p and q.
Thus p³ is even. Since the product of odd numbers is odd, pis even, and hence p = 2s.
Suppose that 21/3 or 22 is the rational number p/ q, where p and q are positive integers with no common factors.
Substituting into the equation p³ = 2q³, 8s³ = 2q³, which simplifies to 4s³ = ³.
This contradicts the choice of p/ q, such that p and q have no common factors.
Hence, 2 = p³/q³, or, equivalently, p³ = 29³. This is obtained by cubing on both sides.
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