Chapter 1
|Parentheses |Parentheses override all other operations. When nested " { [ ( ) ] } ", the inner parentheses have highest priority |
|& |and when sequential " [ ] [ ] [ ] ", the left most parentheses have highest priority. Brackets, extended fraction |
|Groupings |dividers, radicals and all other grouping symbols infer parentheses and thus have the same priority as parentheses. |
|Exponents |If exponents are stacked " 234 ", the upper-most pair has highest priority. If exponents are sequential " 23 + 32 – |
|& |43 ", the left-most exponent has highest priority. An individual root is equivalent to an exponent and the root of |
|Roots |an expression implies parentheses since it is a grouping symbol. See § |
|Multiplication |These share the same priority level. As before, with multiple operations the left-most operation has highest |
|& |priority. e. g. 3 · 8 ÷ 4 · 2. Work from left to right |
|Division | |
|Addition |These share the same priority level. As before, with multiple operations the left-most operation has highest |
|& |priority. e. g. 3 + 8 – 4 + 2. Work from left to right. |
|Subtraction | |
|Negation |Though equivalent to subtraction in priority there is the common problem of correctly interpreting -ab or |
| |-(expression). Note! -ab = -(ab) ≠ (-a)b. |
( Beware These Common Pitfalls!
|The calculation 25 − 15 × 43 tempts us to perform the subtraction first because that calculation is enticingly simple and it comes first. Such |
|pitfalls can be avoided by writing 25 − (15 × 43). Although the parentheses are redundant they help eliminate a potential error. |
| |
|The calculation 123 × 25 − 15 might also tempt us to perform the subtraction first because the calculation is enticingly simple compared to the |
|multiplication. As before, consider including redundant parentheses, e.g. (123 × 25) − 15, to help eliminate a potential error. |
| |
|Another common error seems to occur when the order of operations is out of sync with the left-to-right priority. Remember, the operational |
|priority take precedence. The left-to-right priority only comes into play with multiple operations all having the same priority. |
|e. g. 3 + 4 + 5 × 6 |
|It’s tempting to work left to right but the multiplication operation 5 × 6 has priority! |
| |
| |
|Watch out for other grouping symbols besides parentheses. Fraction dividers and radicals are a grouping which imply parentheses. Since we so |
|often use a calculator to key in such expressions it is imperative to understand the order of operations (PEMDAS). |
|→ (expression)/(expression) |
| |
|→ |
| |
| |
|When in doubt use extra parentheses to remove any chance of ambiguity and to help the reader avoid mistakes. This is especially important if you |
|plan to use a calculator. But be careful, parentheses can change the value of an expression if they change the order of operations. |
|= 4/(2 + 5) = 4/7 ( ≠ 4/2 + 5 = 7 ( |
|= 4/(5π) ( ≠ 4/5π ( |
|= = = 3 ( ≠ + 5= 2 + 5 ( |
| |
|What is the difference between (-a)n vs. -an ? Notice that parentheses make a difference. |
|(-3)2 = (-3)(-3) = 9 ( these are different! -32 = -(3 · 3) = -9 ( |
|-32 ≠ (-3)(-3) ( this is a very common mistake! |
Practice with Order of Operations (Try these without using a calculator)
|1) |a) |4 · 5 – 3 = |b) |52 – 32 = |c) |(3 + 2)·10 – 5 = |
|2) |a) | = |b) | = |c) | = |
|3) |a) |10 ÷ 12 ÷ ÷ = |b) | = |c) | = |
|4) |a) |-22 = |b) | = |c) |[28 − 4]2 = |
|5) |a) |16 – 3 · 5 + 5 = |b) | = |c) |8(9 – 6)10 – 5 = |
|6) |a) |0 ÷ 10 = |b) |3 · 10 – 10 · 2 = |c) |(3 · 5)2 + 3 · 52 = |
Answers-
|1) 17, 16, 45 |3) 12, 1/2, 32 |5) 6, 6, 235 |
|2) 5/2, Ø, 2/5 |4) -4, 5, 576 |6) 0, 10, 300 |
1.2 Exercises- Order of Operations
Simplify to an exact integer or a reduced fraction. Do not use a calculator.
|1) |2 · 8 − 4 |2) |3 · 10 − 10 |3) |12 · 8 ÷ 4 |
|4) |1 + 9 · 10 |5) |2 + 8 · 8 + 2 |6) |10 − 5 (10 − 5) |
|7) |-32 |8) |23 · 32 |9) |(4 − 2)2 |
|10) |4(2 − 2) |11) |4 · 3 − 2 · 6 |12) |2 · 12.1 − 2.1 |
|13) |9[7 − 2 + 6] |14) |3.2 · 6.5 + 3.5 |15) |12 ÷ 4 ÷ 3 |
|16) |24 ÷ 12 ÷ 6 · 3 |17) |21 − 4 · 5 − 5 |18) |1000 ÷ 10 ÷ 10 ÷ 10 |
|19) |10 ÷ 0 |20) |101 − 10 ÷ 10 |21) |0 ÷ 20 |
|22) | |23) | |24) | |
|25) |5 + 5[50 − 5(10 − 5)] |26) |2+2{2+2[2+2(2+2)]} |27) |1 + |
|28) |13 – 18 |29) |-18 – (-12) |30) |7 + (-11) |
|31) |-3 + 5 + (-2) – (-7) |32) |-(-4) – 3 + (-12) – (-27) |33) |-15 – 27 |
|34) |-2(6 – 13) |35) |-5 (7 + (-16)) |36) |-11 + 13 – 16 + (-5) + 8 |
|37) |-(5)(-3) |38) |(2 – 8)(-7) |39) |(8 – 11)(-5 – 2) |
|40) | |41) | |42) | |
|43) | |44) | |45) | |
|46) | |47) | |48) | |
|49) |(-4)3 |50) |[-3(5–7)+(-2)]6÷16–15 |51) |33 – 33 |
|52) |(-4)3 – 4·34 |53) |[-3(5 – 8)+4]4÷13−26 |54) |-24 · 42 |
|55) |-15 ÷ 3 |56) |[5(2–5)+3]2÷(-8)–(-6) |57) | |
|58) | |59) |200 − 100[27 − 32] |60) |[696 − 701]400 − 300 |
|61) |1 + 23 − 3 |62) |(2 – 8) ÷ (-3) |63) |(2 – 8)/2 |
|64) |8 − 2(6 – 13) |65) |3 − 2(4 + 6) |66) |20 − 10(4 − 6) |
|67) |(5)(-4) ÷ -10 |68) |10 − 5 (2) |69) |24 − 4 (3)/6 |
|70) | |71) | |72) | |
|73) | |74) | |75) | |
|76) |(32 + 4)2 |77) |(23 − 32)3 |78) |(-1)27 |
Practice Compute to a single fraction.
|1) |(a) + |(b) – |(c) + |
|2) | – | | |
|3) | 2 + 5 | 12 – 7 | 6 + 5 |
|4) | 9 – 2 | · | 4 · 6 |
|5) | ÷ | 6 ÷ 1 | |
Answers-
|1) 5/4, 2/3, 39/40 |3) 8 1/2, 4 5/9, 11 5/8 |5) 1 1/4, 4 1/2, 1 23/27 |
|2) 5/8, 121/60, 19/16 |4) 6 7/20, 5/7, 30 1/12 | |
(Beware!
|When entering calculations with mixed numbers it is often necessary to use parentheses to ensure correct order of operations. For example: |
| |
|1½ × 2¾ is entered as (1+1/2)(2+3/4). -1½ × -2¾ is entered as -(1+1/2)*-(2+3/4). Both "-" symbols are negative keys, not subtraction! -1½ |
|is entered as -(1+1/2). The negative must be outside the parentheses. Why? 1½ − 2¾ is entered as (1+1/2)−(2+3/4). What happens if the |
|parentheses are omitted? |
| |
|To avoid mistakes when using a calculator wrap all mixed fraction in a parentheses. |
| |
|-1⅔ CANNOT be entered into a calculator as -1+2/3. ( Why? |
|-1⅔ → -(1+2/3) ( |
| |
|8 − 5 ⅔ CANNOT be entered into a calculator as 8−5+2/3. ( Why? |
|8 − 5 ⅔ → 8−(5+2/3) ( |
| |
|7 ÷ 1½ CANNOT be entered into a calculator as 7/1+1/2. ( Why? |
|7 ÷ 1½ → 7/(1+1/2) ( |
| |
|4(-1⅔) CANNOT be entered into a calculator as 4*-1+2/3. ( Why? |
|4(-1⅔) → 4*-(1+2/3) ( |
|Watch out for other grouping symbols besides parentheses. Fraction dividers | → (expression)/(expression) |
|and radicals imply a grouping and imply parentheses. Since we so often use a| |
|calculator to key in such expressions to preserve order of operations |→ |
|(PEMDAS) get in the habit of including the necessary parentheses when using a| |
|calculator. | |
| |
|CANNOT be entered into a calculator as 6*15/25*9. ( Why? |
|→ (6 * 15) / (25 * 9) ( |
|CANNOT be entered into a calculator as 4+10/10−3. ( Why? |
|→ (4+10)/(10−3) ( |
|CANNOT be entered into a calculator as 3(32−4×5). ( Why? |
|→ (3(32−4×5)) ( |
| |
|When in doubt use extra parentheses to remove any chance of ambiguity and to help the reader avoid mistakes. But when using a calculator |
|only rounded parentheses ( ) may be used for arithmetic in a calculator. DO NOT use { } or [ ] for calculator arithmetic. |
| |
|Most (if not all) calculators have an exponentiation key. On the TI-83+ the exponentiation key is black: ^ . To compute (-2)(-2)(-2)(-2) |
|= (-2)4 use (-2)^4 = 16. If you mistakenly omit the parentheses and type -24 the TI-83+ computes the result as -16. Why? Notice that the |
|parentheses were crucial. |
| |
|What is the difference between (-a)n vs. -an ? Notice that parentheses make a difference. |
|(-3)2 = (-3)(-3) = 9 ( these are different! -32 = -(3 · 3) = -9 ( |
|-32 ≠ (-3)(-3) ( this is a very common mistake! |
| |
|Remember, extra parentheses will not cause an error. Leaving them out often does. |
Example 2 Translate into Calculator Format '-' negative key, '−' subtraction key
|Printed Version |→ |Entered into TI-83+ |Printed Version |→ |Entered into TI-83+ |
|4[3 + 5(2 + 1)] |→ |4(3 + 5(2 + 1)) |{2 + 3[1 + (2)(3)]} |→ |(2+3(1+(2)(3))) |
| |→ |4/(2 + 5) | |→ |(2+3)/(3+4) |
| |→ |4/5/π or 4/(5π) | |→ |1/(1+1/2) or 1/1.5 |
| |→ | |1⅔ × 3¾ |→ |(1+2/3)(3+3/4) |
|-1⅔ + -3¾ |→ |-(1+2/3)+-(3+3/4) | |→ |(1+2/3)/(3+3/4) |
|1⅔ − 3¾ |→ |(1+2/3)−(3+3/4) | |→ |/10 |
1.4 Exercises - Calculator Arithmetic
Use a calculator to simplify to a decimal number accurate to the hundredths place.
|1) | |2) | |3) | |
|4) | |5) | |6) | |
|7) |5 + 5[50 − 5(10 − 5)] |8) |2+2{2+2[2+2(2+2)]} |9) |1 + |
|10) | |11) | |12) | |
|13) | |14) | |15) | |
|16) |(-4)3 |17) |[-3(5–7)+(-2)]6÷16–15 |18) | |
|19) | |20) |[-3(5 – 8)+4]4÷13−26 |21) | |
|22) | |23) | |24) | |
|25) |(-10)5 |26) |(-0.1)6 |27) |(¼ − ¾)4 |
|28) | |29) | |30) | |
|31) | |32) | |33) | |
|34) |(32 + 4)2 |35) |(23 − 32)3 |36) |(-1)27 |
|37) | |38) | |39) | + 5.54 |
|40) |π |41) | |42) |-π4 + 2400 π – 3 |
|43) | |44) | |45) | |
|46) | |47) | |48) | |
|49) | |50) | |51) | |
|52) | |53) |2.5+6.5 |54) | |
|55) | |56) | |57) | |
|58) | |59) | |60) | |
|61) | |62) | |63) | |
|64) | |65) | |66) | |
|67) | |68) | |69) | |
|70) | |71) | |72) | |
|73) | + 6.39 |74) | – 7.39 |75) | + 5.54 |
|76) | |77) | |78) | |
|79) | |80) | |81) | |
|82) | − 10 |83) | + 8 |84) |2 + 9 |
|85) | |86) | |87) | ÷ 10 |
|88) | + |89) | + |90) |(½) |
Use your calculator to simplify to a single fraction. Write improper fractions as mixed fractions.
|91) | + |92) | – |93) | + |
|94) |4 – |95) |2 ⅝ + ¾ – 1 ⅜ |96) | – + |
|97) | |98) | |99) | |
|100) |3 ⅝ − 2 ¾ |101) |5 ⅞ + 9 ⅔ |102) |3·(1/3 – 3/2) + 4 |
|103) | |104) | |105) |-6 |
|106) | |107) |-6 ⅜ ÷ 4 ⅔ |108) |4 ⅞ + 8 ¾ ÷ 2 ½ |
|109) | |110) | |111) | |
|112) | |113) | |114) | |
|§ 1.2 |Arithmetic of Signed Numbers | | |
|1. |12 |2. |20 |
|3. |24 |4. |91 |
|5. |68 |6. |-15 |
|7. |-9 |8. |72 |
|9. |4 |10. |0 |
|11. |0 |12. |22.1 |
|13. |99 |14. |24.3 |
|15. |1 |16. |1 |
|17. |-4 |18. |1 |
|19. |undefined |20. |100 |
|21. |0 |22. |3 |
|23. |undefined |24. |0 |
|25. |130 |26. |46 |
|27. |1 3/5 = 8/5 |28. |-5 |
|29. |-6 |30. |-4 |
|31. |7 |32. |16 |
|33. |-42 |34. |14 |
|35. |45 |36. |-11 |
|37. |15 |38. |42 |
|39. |21 |40. |-5/8 |
|41. |3/4 |42. |-4/5 |
|43. |-1/3 |44. |1 |
|45. |1/2 |46. |undefined |
|47. |1/3 |48. |0 |
|49. |-64 |50. |-27/2 |
|51. |6 |52. |-388 |
|53. |-22 |54. |-256 |
|55. |-5 |56. |9 |
|57. |9/28 |58. |1/1500 |
|59. |700 |60. |-2300 |
|61. |6 |62. |2 |
|63. |-3 |64. |22 |
|65. |-17 |66. |40 |
|67. |2 |68. |0 |
|69. |22 |70. |10/3 = 3 ⅓ |
|71. |-19/14 |72. |-19/2 = -9 ½ |
|73. |0 |74. |-1/9 |
|75. |0 |76. |169 |
|77. |-1 |78. |-1 |
| | | | |
|§ 1.4 |Calculator Arithmetic | | |
|1. |0.75 |2. |-9 |
|3. |-2 |4. |3 |
|5. |undefined |6. |0 |
|7. |130 |8. |46 |
|9. |5/3 = 1 ⅔ ≈ 1.67 |10. |-⅓ ≈ -0.33 |
|§ 1.4 |continued | | |
|11. |1 |12. |½ = 0.5 |
|13. |undefined |14. |⅓ ≈ 0.33 |
|15. |0 |16. |-64 |
|17. |-27/2 = -13.5 |18. |1 5/16= 1.3125 |
|19. |0.35 |20. |-22 |
|21. |0.30 |22. |0.16 |
|23. |0.24 |24. |9/28 ≈ 0.32 |
|25. |-100,000 |26. |0.000001 |
|27. |1/16 = 0.0625 |28. |-8/9 |
|29. |-19/14 ≈ -1.36 |30. |-19/2 = -9 ½ |
|31. |0 |32. |-1/9 |
|33 |0 |34. |169 |
|35. |-1 |36 |-1 |
|37. |-0.41 |38. |0.04 |
|39. |9.44 |40. |9.93 |
|41. |-0.613 |42. |7439.41 |
|43. |8.16 |44. |1.78 |
|45. |5.30 |46. |13.90 |
|47. |3.28 |48 |35.72 |
|49. |-1.39 |50. |24.33 |
|51. |Non Real |52. |undefined |
|53. |24.46 |54. |41.15 |
|55. |216 |56. |30.03 |
|57. |5.63 |58. |1.32 |
|59. |2.51 |60. |0.88 |
|61. |≈ 3.16 |62. |1522 |
|63. |1.45 |64. |≈ 6.32 |
|65. |732.57 |66. |-2.33 |
|67. |-1.20 |68. |8.93 |
|69. |10.64 |70. |1.16 |
|71. |0.38 |72. |1.15 |
|73. |5.46 |74. |-7.36 |
|75. |6.54 |76. |1.83 |
|77. |2.62 |78. |2.29 |
|79. |0.45 |80. |8 |
|81. |4 |82. |-4.08 |
|83. |10.83 |84. |14.66 |
|85. |½ |86. |1 |
|87. |0.28 |88. |5.66 |
|89. |10.86 |90. |2.45 |
|91. |1 |92. |6/5 or 1 1/5 |
|93. |8/5 or 1 3/8 |94. |10/3 or 3 1/3 |
|95. |2 |96. |83/120 |
|97. |1/7 |98. |0 |
|99. |51/16 or 3 3/16 |100. |7/8 |
|101. |373/24 or 15 13/24 |102. |1/2 |
|§ 1.4 |continued | | |
|103. |32/25 or 1 7/25 |104. |3/2 or 1 ½ |
|105. |-119/4 or -29 ¾ |106. |¾ |
|107. |-153/112 or -1 41/112 |108. |8 ⅜ |
|109. |2/39 |110. |0 |
|111. |1.19 |112. |1.45 |
|113. |1.73 |114. |¾ |
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