Skills Practice Skills Practice for Lesson.1 Name Date Thinking About Numbers Counting Numbers, Whole Numbers, Integers, Rational and Irrational Numbers Vocabulary Define each term in your own words. 1. closure 2. rational numbers 3. irrational numbers. real numbers Problem Set Classify each number by its most restrictive category: rational, irrational, integer, whole, or counting. 1. 10 2. 701 counting 3. 0. 95 Chapter l Skills Practice 39
5. 2 5 6. 2 5 7. 20.5 8. 0. 16 9. 3 5 10. Answer each question. 11. When you add two integers, is your answer an integer? Explain. Yes. When you add two integers, your answer is always an integer. 12. When you multiply two integers, is your answer an integer? Explain. 13. When you subtract any integer from any other integer, is your answer an integer? Explain. 1. When you divide any integer by any other integer, is your answer an integer? Explain. 15. When you add two rational numbers, is your answer a rational number? Explain. 16. When you multiply two rational numbers, is your answer a rational number? Explain. 0 Chapter l Skills Practice
Name Date 17. When you subtract any rational number from any other rational number, is your answer a rational number? Explain. 18. When you divide any rational number by any other rational number, is your answer a rational number? Explain. Consider the four basic operations of addition, subtraction, multiplication, and division to answer each question. 19. Under which operations is the set of counting numbers closed? Explain. The set of counting numbers is closed under addition and multiplication because if you add or multiply two counting numbers, the sum or product is always another counting number. 20. Under which operations is the set of counting numbers not closed? Explain. 21. Under which operations is the set of integers not closed? Explain. Chapter l Skills Practice 1
22. Under which operations is the set of integers closed? Explain. 23. Under which operations is the set of rational numbers closed? Explain. 2. Under which operations is the set of rational numbers not closed? Explain. 25. Under which operations is the set of real numbers not closed? Explain. 26. Under which operations is the set of real numbers closed? Explain. 2 Chapter l Skills Practice
Name Date Write each repeating decimal as a fraction. 27. 0.3333 28. 0.6666 10x 3.3333 x 0.3333 9x 3 x 3 1 9 3 29. 0. 16 30. 0. 09 31. 0. 63 32. 0. 5 33. 2.1313 3..2626 35. 5. 35 36. 0. 12857 Chapter l Skills Practice 3
Chapter l Skills Practice
Skills Practice Skills Practice for Lesson.2 Name Date Real Numbers Properties of the Real Number System Vocabulary Provide an example of each property of the real number system. 1. commutative 2. associative 3. distributive. additive identity 5. multiplicative identity 6. additive inverse 7. multiplicative inverse Chapter l Skills Practice 5
Problem Set Each expression has been simplified one step at a time. Next to each step, identify the property, transformation, or simplification used in the step. 1. 8x (3x 7) 8x (12x 28) Distributive Property of Multiplication over Addition (8x 12x) 28 Associative Property of Addition 20x 28 Combine like terms 2. 1(2x 2 x) 1(2x x 2) 1(3x 2) 2x 28 3. 11(13 13 x 9) 11(0 x 9) 11(x 9) 11x 99. 7(x ) 28 7x 28 28 7x 0 7x 5. 3(5 7x 5) 3(7x 5 5) 3(7x 0) 3(7x) 21x 6. (10x 2) 0x 0x 8 0x 8 0x 0x 8 0 8 6 Chapter l Skills Practice
Name Date Each equation has been solved one step at a time. Next to each step, identify the property, transformation, or simplification used in the step. 7. x 19 23 x 19 ( 19) 23 ( 19) x 0 23 ( 19) x 23 ( 19) x Addition Property of Equality Combine like terms Additive Identity Combine like terms 8. x 7 3 x 7 7 3 7 x 0 3 7 x 3 7 x 1 9. 13x 52 10. 13x 1 52 1 13 13 x(13) 1 52 1 13 13 x(1) 52 1 13 x 52 1 13 x 1 7 x 9 x ( 1 7 ) 9 x ( 1 7 ) 7 9 7 x 1 9 7 x 9 7 x 63 Chapter l Skills Practice 7
11. 3(3x 8) 2 32 9x 2 2 32 9x 22 32 9x 22 22 32 22 9x 0 32 22 9x 32 22 9x 5 9x 1 5 1 9 9 x(9) 1 5 1 9 9 x(1) 5 1 9 x 5 1 9 x 6 12. 5(3 6x) 25 20 15 30x 25 20 30x 15 25 20 30x 10 20 30x 10 10 20 10 30x 0 20 10 30x 20 10 30x 30 30x 1 30 1 30 30 x(30) 1 30 1 30 30 x(1) 1 x 1 8 Chapter l Skills Practice
Name Date 13. 7x 1 12x 6 2 7x 1 12x 6 2 2 7x 1 6x 3 7x 1 1 6x 3 1 7x 6x 2 7x 6x 6x 2 6x 7x 6x 2 6x 6x x 2 1. x 8 11 3x 2 x 8 11 3x 2 2 2x 11 3x 2x 11 3x 2x 15 3x 2x 3x 15 3x 3x 5x 15 1 5x 1 15 5 5 x 3 15. 2x 5 2x 17 3 3 ( 2x 5 3 ) 3( 2x 17) 2x 5 3( 2x 17) 2x 5 6x 51 2x 5 5 6x 51 5 2x 6x 56 2x 6x 6x 56 6x 8x 56 1 8x 1 56 8 8 x 7 Chapter l Skills Practice 9
(x 9) 16. 2x 3 5 5(2x 3) 5 ( x 9 5 ) 5(2x 3) x 9 10x 15 x 9 10x 15 15 x 9 15 10x x 2 10x x x 2 x 6x 2 1 6x 1 2 6 6 x 50 Chapter l Skills Practice
Skills Practice Skills Practice for Lesson.3 Name Date Man-Made Numbers Imaginary Numbers and Complex Numbers Vocabulary Write the term that best completes each sentence. 1. The number 1 is a(n) represented by i., and it is usually 2. The set of real numbers is not closed under, because you cannot calculate the square roots of negative numbers; for instance, ( ) 1 2 is undefined in the real numbers. 3. An example of a(n) is 3 2i.. A(n) is an exponent that is a rational number. 5. The term a of the number a bi is called the. 6. The term bi of the number a bi is called the. Problem Set Simplify each power. 1. 1000 2 3 1000 2 3 3 1000 2 3 (10 3 ) 2 3 10 6 10 2 100 2. 10,000 5 3. ( 125) 3. ( 32) 2 5 5. 8 2 3 Chapter l Skills Practice 51
6. 9 3 2 7. ( 16) 3 8. ( 9) 3 2 Calculate each power of i. 9. i 16 i 16 ( i ) ( 1 ) 1 10. i 12 11. i 17 12. i 13 13. i 10 1. i 18 15. i 11 16. i 19 52 Chapter l Skills Practice
Name Date Using i, calculate each square root. 17. 9 9 9 1 7i 19. 169 18. 36 20. 256 21. 1 22. 10 Solve each quadratic equation. Check your work. 23. x 2 121 0 x 2 121 x 121 x 11i Check: ( 11i ) 2 121 121i 2 121 121 121 0 2. x 2 81 0 25. x 2 196 0 26. x 2 1 0 Chapter l Skills Practice 53
Identify the real term and the imaginary term of each complex number. 27. 2 6i Real term: 2 Imaginary term: 6i 28. 5 20i 29. 101i 30. 82i 31. 32. 10 33. 86 7 i 3. 6 51 i Solve each quadratic equation using the quadratic formula. Simplify your answer using imaginary numbers. 35. 2x 2 6x 5 0 x b b 2 ac 2a (6) (6) x 2 (2)(5) 2(2) x 6 36 0 x 6 6 2i 3 1 2 2 i 5 Chapter l Skills Practice
Name Date 36. 2x 2 7x 7 0 37. 11x 2 8x 2 0 38. 6x 2 x 3 0 Chapter l Skills Practice 55
56 Chapter l Skills Practice
Skills Practice Skills Practice for Lesson. Name Date The Complete Number System Operations with Complex Numbers Vocabulary Provide an example of each term. 1. conjugate of a complex number 2. power of a complex number 3. root of a complex number Problem Set For each pair of complex numbers, calculate the sum and the difference. 1. 8 5i, 16 i sum: 2 i difference: 8 6i 2. 3 10i, 13 2i 3. 6 5i, 20 3i. 9 2i, 11 i 5. 2.3 6.1i, 1.5.6i 6. 1. 2i, 16.1 0.3i Chapter l Skills Practice 57
For each pair of complex numbers, calculate the product. 7. i, 3 2i ( i)(3 2i) 12 8i 3i 2i 2 12 11i 2 10 11i 8. 1 2i, 3 i 9. 1 i, 6 7i 10. 10 i, 3 5i 11. 3 5i, 1 3 1 10 i 12. 1 2 5 i, 18i 6 5 For each complex number, write its conjugate. 13. 7 2i 1. 3 5i 7 2i 15. 8i 16. 7i 17. 2 11i 18. 9 i 19. 13 6i 20. 21 i 58 Chapter l Skills Practice
Name Date Calculate the product of each complex number and its conjugate. 21. 3 13i (3 13i)(3 13i) 9 39i 39i 169i 2 9 169 178 22. 2 11i 23. 6 5i 2. 7 2i Calculate each quotient. 25. 26. 3 i 5 6i 3 i 5 6i 8 7i 2 i 3 i 5 6i 5 6i 5 6i 15 2i 2 25 36 2 15 18i 20i 2i 25 30i 30i 36i 2 39 2i 61 39 61 2 61 i 27. 6 2i 2 3i 28. 1 5i 1 i Chapter l Skills Practice 59
Calculate the indicated power of each complex number. 29. (5 2i) 2 (5 2i) 2 25 10i 10i i 2 25 20i 21 20i 30. ( 6i) 2 31. (2 11i) 2 32. (7 3i) 2 33. (1 2i) 3 3. ( 3 i) 3 60 Chapter l Skills Practice
Name Date Determine the square root of each complex number. 35. 3 i 3 i a bi ( 3 i ) 2 ( a bi ) 2 3 i a 2 2abi b 2 i 2 3 i ( a 2 b 2 ) ( 2abi ) 3 a 2 b 2, i 2abi Solve for a on the second equation: a 2 b Substitute in the first equation: ( 2 b ) 2 b 2 3 b 2 b 2 3 b 3b 2 0 b 3b 2 0 ( b 2 )( b 2 1) b 2, b 2 1 b must be a real number, so b 2 1, or b 1 Replace to calculate a: a 2 b 2 2 1 3 i 2 i, 2 i Chapter l Skills Practice 61
36. 7 2i 62 Chapter l Skills Practice
Name Date 37. 51 10i Chapter l Skills Practice 63
38. 60 32i 6 Chapter l Skills Practice