University:
Stanford University
Course:
MATH 19 | Calculus
Academic year:

2023
Views:

13

Pages:

3
Author:

MathPaladin

Questions and Answers Sheet 1 Irrational Numbers Question #1 Prove or disprove that if a and b are rational numbers, then 𝑎𝑏 is also rational. Answer: Given statement: If a and b are rational numbers, then 𝑎𝑏 is also rational. The given statement is false, which will be disproven using a counterexample. Counterexample 1 1 If a=-1 and 𝑏 = , we then obtain (−1)2 = √−1 which is not rational number (it is called an imaginary 2 number) and thus 𝑎𝑏 is not necessarily rational. Result: 1 Statement is false, one counterexample is for a=-1 and 𝑏 = . 2 Question #2 I'm trying to do this proof by contradiction. I know I have to use a lemma to establish that if x is divisible by 3, then 𝑥 2 is divisible by 3. The lemma is the easy part. Any thoughts? How should I extend the proof for this to the square root of 6? Answer: Say √3 is rational. Then √3 can be represented as So 3 = 𝑎2 𝑏2 𝑎 𝑏 , where a and b have no common factors. and 3𝑏 2 = 𝑎2 . Now 𝑎2 must be divisible by 3, but then so must a (fundamental theorem of arithmetic). So we have 3𝑏 2 = (3𝑘)2 and 3𝑏 2 = 9𝑘 2 or even 𝑏 2 = 3𝑘 2 and now we have a contradiction. What is the contradiction? Question #3 Whats the absolute value of 0 2 ? Answer: Explanation: Absolute value means the distance from zero. 0 is the same as 0. 2 So |0| is just 0. Question #4 31 12 √12 + √31 is irrational? Answer: 31 12 I would assume that √12 + √31 is rational and try to find a contradiction However, I don't know where to start. Can someone give me a tip on how to approach this problem? Question #5 𝑛 Expansion of (1 + √2) I was asked to show that ∀𝑛 ∈ 𝑁 there exist a 𝑝 ∈ 𝑁 ∗ such that 𝑛 (1 + √2) = √𝑝 + √𝑝 − 1 𝑛 I used induction but it wasn't fruitful, so I tried to use the binomial expansion of (1 + √2) lack some insight to go further. but it seems I Any hint is welcomed. Answer: The binomial formula shows you that 𝑛 (1 + √2) = 𝑎𝑛 + 𝑏𝑛√2 for some integers 𝑎𝑛 , 𝑏𝑛 . But, the same binomial formula shows you that (convince yourself of this) 𝑛 (1 − √2) = 𝑎𝑛 − 𝑏𝑛√2 for the same integers 𝑎𝑛 , 𝑏𝑛 . Then comes the hint: Calculate both (𝑎𝑛 + 𝑏𝑛√2 )(𝑎𝑛 − 𝑏𝑛√2 ) and 𝑛 𝑛 𝑛 (1 + √2) (1 − √2) = [(1 + √2)(1 − √2)] and compare. 2 2 2 2 You will get that 𝑎 − 2𝑏 = (−1)𝑛 , so 𝑎 and 2𝑏 differ from each other by one, and p will be the 𝑛 𝑛 𝑛 𝑛 larger of the two. Question #6 3 Process to show that √2 + √3 is irrational 3 How can I prove that the sum √2 + √3 is an irrational number ?? Answer: 3 If 𝑥 = √2 + √3 , then 3 (𝑥 − √2) = 𝑥 3 − 3√2𝑥 2 + 6𝑥 − 2√2 = 3 Thus 𝑥 3 + 6𝑥 − 3 = √2(3𝑥 2 + 2) And 𝑥 3 +6𝑥−3 3𝑥 2 +2 = √2 But if x is a rational, then so is the left hand side of the above equality. However we know √2 is not rational. Contradiction, so x is irrational. Question #7 Is sqrt42 a rational number? Answer: First, calculate 42 as the sum of its prime factors. √42 = √(2 × 3 × 7) There are no perfect squares and all the factors are prime numbers. √2, √3 and √7 are all irrational numbers. √42 cannot be rational.