University:
Johns Hopkins University
Course:
AS.110.601 | Algebra I
Academic year:

2024
Views:

134

Pages:

3
Author:

Cristian Santos

Question 1 Consider the following statements: a) The sum of a rational number and a rational number is rational. b) The sum of a rational number and an irrational number is irrational. c) The sum of an irrational number and an irrational number is irrational. d) The product of a rational number and a rational number is rational. e) The product of a rational number and an irrational number is irrational. f) The product of an irrational number and an irrational number is irrational. Answer a) True: The sum of two rational numbers is always rational. Example: 1 2 + 1 4 = 5 4 ( rational ) . 2 1 + 4 1 = 4 5 (rational). b) True: The sum of a rational number and an irrational number is irrational. Example: 1 2 + 2 ( irrational ) . 2 1 + 2 (irrational). c) False: The sum of two irrational numbers can sometimes be rational. Example: ( 1 2 + 2 ) + ( 1 2 − 2 ) = 1 ( rational ) . ( 2 1 + 2 )+( 2 1 − 2 )=1(rational). d) True: The product of two rational numbers is always rational. Example: 1 2 × 1 2 = 1 4 ( rational ) . 2 1 × 2 1 = 4 1 (rational). e) True: The product of a rational number and an irrational number is irrational. Example: 1 2 × 2 = 2 2 ( irrational ) . 2 1 × 2 = 2 2 (irrational). f) False: The product of two irrational numbers can be rational. Example: 3 × 3 = 3 ( rational ) . 3 × 3 =3(rational). Question 2 Find a) A rational number and b) An irrational number between the given pair 3 7 7 3 and 1 6 6 1 . Answer 3 7 ≈ 0.428571 and 1 6 ≈ 0.166666 7 3 ≈0.428571and 6 1 ≈0.166666. a) Rational number: Take any fraction between 3 7 7 3 and 1 6 6 1 , such as 0.3 0.3 or 4 15 15 4 . b) Irrational number: Take any non-repeating, non-terminating decimal between the pair, such as 0.31415926... 0.31415926... (approximately 𝜋 / 10 π/10). Question 3 Find 100 irrational numbers between 0 0 and 1 100 100 1 : Answer Irrational numbers like 𝑒 1 0 3 , 𝑒 1 0 4 , 𝑒 1 0 5 , … , 𝑒 1 0 103 10 3 e , 10 4 e , 10 5 e ,…, 10 103 e are all irrational and between 0 0 and 1 100 100 1 . Find 50 rational numbers between 1 1 and 2 2: Examples include 1.1 , 1.2 , 1.3 , . . . , 1.50 1.1,1.2,1.3,...,1.50. Find 50 irrational numbers between 1 1 and 2 2: Examples include 1 + 1 𝑒 , 1 + 1 𝑒 2 , … , 1 + 1 𝑒 50 1+ e 1 ,1+ e 2 1 ,…,1+ e 50 1 . Question 4 In which set(s) of numbers would you find 80 80 ? Answer Irrational Number: 80 = 4 5 80 =4 5 is irrational because 5 5 is irrational. Real Number: 80 80 is a real number because all irrational numbers are real. Conclusion: 80 80 belongs to the irrational numbers and the real numbers.