Math 10 Lesson 1 Answers Lesson Questions Question 1 When we calculate the radical, radicals that are rational numbers result in a rational number while radicals that are irrational result in an irrational number. 0. 0.89897...irrational rational rational 6 8 rational 7 16 16 1 1 rational 0.7... irrational 9 9 7 0. 0. rational 1 1.16... irrational 0.6 0.6 rational 1.17008... irratio nal Question Compute the following numbers and classify them as natural, whole, integer, rational, and/or irrational: 16 rational, integer, whole and natural 0.107... irrational 16 16 rational 81 81 Question Which numbers below belong to each set: natural, whole, integer, rational, and/or irrational? rational 0.17 rational 6 rational, integer 17 rational, integer, whole, natural irrational 6 irrational 1 rational 8 rational, integer, whole, natural 11 11 rational, integer, whole, natural 6.1111... rational Dr. Ron Licht 1 www.structuredindependentlearning.com
Question Classify each of the following numbers as rational or irrational. Provide an explanation. Number Rational or irrational Explanation 0 rational whole number π irrational non-repeating number 6 6 rational whole number.8 irrational non-repeating number.8 rational terminating decimal 99 1 rational division of two integers 00.6067... irrational non-repeating number 6. rational repeating number 7 rational whole number Question Which of the following numbers are irrational. Provide an explanation. Number Irrational (yes or no) Explanation 1.70... yes non-terminating decimal 6 6 100 10.898979... yes non-terminating decimal 6 1 6 6 1 1 1. no terminating decimal 0.866... yes non-terminating decimal 6 8 Dr. Ron Licht www.structuredindependentlearning.com
Question 6 Use a number line to order the following numbers from least to greatest 1.1 1.6 6 1.8 11. 8.8 8.8 1.6 1.1 6 1.8 11. -6 - - 0 Question 7 is not a real number. It is impossible to find a root value that, when multiplied by itself, results in a negative number (i.e. = and = ). (However, the idea of the square root of a negative number eventually led to a whole new branch of mathematics called Complex Numbers.) Dr. Ron Licht www.structuredindependentlearning.com
Nasty question of the day For a right angle triangle, the lengths of the sides must obey Pythagorus equation c = a + b (a) All sides have rational number lengths. There are an infinite number of triangles like this (Google whole number right triangles): 16 9 1 1 1 1 169 1 169 169 (b) Exactly sides have rational number lengths. There are an infinite number of triangles like this: 1 c c 9 c 1 c 1 (c) Exactly 1 side has a rational number length. There are an infinite number of triangles like this: 7 11 11 7 11 7 11 11 (d) No sides have rational number lengths. There are an infinite number of triangles like this: 8 7 1 1 8 7 1 8 7 1 1 Dr. Ron Licht www.structuredindependentlearning.com
Assignment 1. a) The square root of 8 is between the root of () and the root of 9 (). Since 8 is close to 9, we try.9 and.8 and find that.8 is the best answer b) The cube root of 9 is between the cube root of 8 () and the cube root of 7 (). Since 9 is very close to 8 we try.1 and. and find that.1 is the best answer c) 1.8 d).6. a) The calculator returns an error message; the square of a real number will always be positive. b) Any non-zero even index c) i) Any odd index ii) Any even index. a) As written the number 1.7 8 71 is rational since it terminates. b) The root of 10 is irrational since it results in a non-terminating and non-repeating number.. a), b). 8 6 0.107... 00 6.699... The cubes roots of the numbers in parts c and d will be irrational. 6. 98, 0, 00, 00, 7, 98 7. a) i) True Natural numbers are a subset of Integers ii) True Integers are a subset of Rational numbers. iii) False The set of Whole numbers includes 0. 0 is not a Natural number. iv) False Other irrational numbers are numbers like π and ζ and e. v) True Natural numbers are a subset of rational numbers b) iii) 0 iv) π Dr. Ron Licht www.structuredindependentlearning.com
8. Answers will vary. For example: a) any fraction or decimal 0.7 b) 0 c) any non-repeating, non-terminating number like 7 9. Additional numbers may vary. For example: 10. a) 0 6... Irrational number b) 81 9 Rational number 11. a) Yes. Any number that is not a perfect square like 0. b) No. If the original number is irrational then the square root will be doubly irrational. For example 7 1.667... Dr. Ron Licht 6 www.structuredindependentlearning.com