Complexe cotangent et deformations I by L. Illusie

By L. Illusie

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34 31. 3 2 5 2 −3 −2 −1 0 1 2 32. 33. 4 −3 −2 −1 0 1 2 3 −3 −2 −1 0 1 2 3 0 1 2 3 34. 10. 6 ᭿ Ϫ2 − 43 −3 −2 −1 14. Ϫ 73 ᭿ Ϫ 52 In Exercises 35–48, evaluate the expression. See Example 6. 18. 75 20. 235 35. 10 37. Ϫ3 39. 4 16. Ϫ 38 ᭿ Ϫ 54 In Exercises 21–24, find the distance between a and zero on the real number line. 21. a ϭ 2 In Exercises 25–30, find the opposite of the number. Plot the number and its opposite on the real number line. What is the distance of each from 0? In Exercises 31–34, find the absolute value of the real number and its distance from 0.

7 ϩ 3 ϭ Ϫ ͑ Ϫ7 Ϫ 3 ͒ ϭ Ϫ4 e. Ϫ9 ϫ 2 ϭ Ϫ18 CHECKPOINT Now try Exercise 97. EXAMPLE 8 Finding a Pattern Complete each pattern. Decide which rules the patterns demonstrate. a. 3 и ͑3͒ 3 и ͑2͒ 3 и ͑1͒ 3 и ͑0͒ ϭ9 и ͑3͒ ϭ Ϫ9 Ϫ3 и ͑2͒ ϭ Ϫ6 Ϫ3 и ͑1͒ ϭ Ϫ3 Ϫ3 и ͑0͒ ϭ 0 Ϫ3 и ͑Ϫ1͒ ϭ ᭿ Ϫ3 и ͑Ϫ2͒ ϭ ᭿ Ϫ3 и ͑Ϫ3͒ ϭ ᭿ b. Ϫ3 ϭ6 ϭ3 ϭ0 и ͑Ϫ1͒ ϭ ᭿ 3 и ͑Ϫ2͒ ϭ ᭿ 3 и ͑Ϫ3͒ ϭ ᭿ 3 Solution a. 3 и ͑Ϫ1͒ ϭ Ϫ3 3 и ͑Ϫ2͒ ϭ Ϫ6 3 и ͑Ϫ3͒ ϭ Ϫ9 и ͑Ϫ1͒ ϭ 3 и ͑Ϫ2͒ ϭ 6 Ϫ3 и ͑Ϫ3͒ ϭ 9 b. Ϫ3 Ϫ3 The product of integers with unlike signs is negative, and the product of integers with like signs is positive.

2 −1 0 1 2 3 4 5 110. −3 −2 −1 0 1 2 3 4 113. Is it possible that the sum of two positive integers is a negative integer? Explain. 114. Is it possible that the difference of two negative integers is a positive integer? Explain. 3 Multiplying and Dividing Integers What You Should Learn NASA 1 ᭤ Multiply integers with like signs and with unlike signs. 2 ᭤ Divide integers with like signs and with unlike signs. 3 ᭤ Find factors and prime factors of an integer. 4 ᭤ Represent the definitions and rules of arithmetic symbolically.

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