Unit5

1. Indice Sets of Numbers The number line Add and subtract Multiply and divide Exercises Integers Matem´ticas 1o E.S.O. a Alberto Pardo Milan´s e -

2. Indice Sets of Numbers The number line Add and subtract Multiply and divide Exercises 1 Sets of Numbers 2 The number line 3 Adding and subtracting integers 4 Multiplying and dividing integers 5 Exercises Alberto Pardo Milan´s e Integers

3. Indice Sets of Numbers The number line Add and subtract Multiply and divide Exercises Sets of Numbers Alberto Pardo Milan´s e Integers

4. Indice Sets of Numbers The number line Add and subtract Multiply and divide Exercises Sets of Numbers Counting numbers and Integers IN is the set of natural numbers, also called counting numbers or positive numbers. Positive numbers, negative numbers, and zero are called integers. Positive numbers represent data that are greater than 0, they are written with a + sign or no sign at all. Example: read +3 positive three or plus three. Negative numbers represent data that are less than 0, they are written with a − sign. Example: read −5 negative five or minus five, for temperatures you can also use five below zero. Alberto Pardo Milan´s e Integers

7. Indice Sets of Numbers The number line Add and subtract Multiply and divide Exercises Sets of Numbers The set Z, absolute value and opposite Z is the set of integers: Z = {· · · − 5, −4, −3, −2, −1, 0, 1, 2, 3, 4, 5 . . . } The absolute value of an integer is the value of the number regardless of its sign. Example: read | − 2| = 2 the absolute value of negative two is two. The absolute value of a number is its distance from zero on a number line. Opposites are numbers that are the same distance from zero on a number line, but in opposite directions (so they both have the same absolute value). Example: the opposite of −5 is Op(−5) = 5. Alberto Pardo Milan´s e Integers

10. Indice Sets of Numbers The number line Add and subtract Multiply and divide Exercises The number line Alberto Pardo Milan´s e Integers

11. Indice Sets of Numbers The number line Add and subtract Multiply and divide Exercises The number line What´s the number line? The number line is a straight line in which the integers are shown. The line continues left and right forever. If a number is to the left of a number on the number line, it is less than the other number. If it is to the right then it is greater than that number. Example: 2 < 4 because 2 lies to the left of 4 in the number line. −1 > −3 because −1 lies to the right of −3 in the number line. −4 < 1 because −4 lies to the left of 1 in the number line. −2 < 0 because 0 lies to the right of −2 in the number line. Alberto Pardo Milan´s e Integers

12. Indice Sets of Numbers The number line Add and subtract Multiply and divide Exercises Adding and subtracting integers Alberto Pardo Milan´s e Integers

13. Indice Sets of Numbers The number line Add and subtract Multiply and divide Exercises Adding and subtracting integers Adding Add using the number line: Add a positive integer by moving to the right on the number line. Add a negative integer by moving to the left on the number line. Alberto Pardo Milan´s e Integers

14. Indice Sets of Numbers The number line Add and subtract Multiply and divide Exercises Adding and subtracting integers Adding Add using absolute values: The signs are the same: If the signs are the same, add the numbers’ absolute values and retain the same sign. The sum of two positive integers is always positive, the sum of two negative integers is always negative. Examples: Find −3 + (−2): the signs are the same (negative) and | − 3| = 3, | − 2| = 2, 3 + 2 = 5, so −3 + (−2) = −5. Find 2 + (+1): the signs are the same (positive) and | + 2| = 2, | + 1| = 1, 2 + 1 = 3, so 2 + (+1) = +3. Alberto Pardo Milan´s e Integers

15. Indice Sets of Numbers The number line Add and subtract Multiply and divide Exercises Adding and subtracting integers Adding Add using absolute values: The signs are different: If the signs are different, subtract the numbers’ absolute values and retain the sign of the number with the greater absolute value. Examples: Find −3 + (+2): the signs are different and | − 3| = 3, | + 2| = 2, 3 − 2 = 1, so −3 + (+2) = −1, because 3 > 2. Find −1 + (+5): the signs are different and | − 1| = 1, | + 5| = 5, 5 − 1 = 4, so −1 + (+5) = +4, because 1 < 5. Alberto Pardo Milan´s e Integers

16. Indice Sets of Numbers The number line Add and subtract Multiply and divide Exercises Adding and subtracting integers Subtracting Subtract an integer by adding its opposite. Examples: Find −4 − (−3): adding its opposite −4 − (−3) = −4 + (+3) = −1. Find −3 − 1: adding its opposite −3 − 1 = −3 − (+1) = −3 + (−1) = −4. Find 6 − (−10): adding its opposite 6 − (−10) = 6 + (+10) = 16. Alberto Pardo Milan´s e Integers

17. Indice Sets of Numbers The number line Add and subtract Multiply and divide Exercises Multiplying and dividing integers Alberto Pardo Milan´s e Integers

18. Indice Sets of Numbers The number line Add and subtract Multiply and divide Exercises Multiplying and dividing integers Multiplying To multiply integers, multiply the absolute values and then use these rules to find the corresponding sign: The product of two integers with different signs is negative, the product of two integers with the same sign is positive. Examples: Find 2 · (−1): the integers have different signs, the product is negative 2 · (−1) = −2. Find −4 · 3: the integers have different signs, the product is negative −4 · 3 = −12. Find 3 · 5: the integers have the same signs, the product is positive 3 · 5 = +15. Find −2 · (−4): the integers have the same signs, the product is positive −2 · (−4) = +8. Alberto Pardo Milan´s e Integers

22. Indice Sets of Numbers The number line Add and subtract Multiply and divide Exercises Multiplying and dividing integers Dividing To divide integers, divide the absolute values and then use these rules to find the corresponding sign: The division of two integers with different signs is negative, the division of two integers with the same sign is positive. Examples: Find 5 : (−1): the integers have different signs, the quotient is negative 5 : (−1) = −5. Find −12 : 3: the integers have different signs, the quotient is negative −12 : 3 = −4. Find 15 : 5: the integers have the same signs, the quotient is positive 15 : 5 = +3. Find −4 : (−2): the integers have the same signs, the quotient is positive −4 : (−2) = +2. Alberto Pardo Milan´s e Integers

26. Indice Sets of Numbers The number line Add and subtract Multiply and divide Exercises Multiplying and dividing integers Rule of Signs for multiplying and dividing: Unlike signs produce negative numbers: +·−=− − ·+ = − +:−=− −:+=− Like signs produce positive numbers: +·+=+ − ·− = + +:+=+ −:−=+ Alberto Pardo Milan´s e Integers

27. Indice Sets of Numbers The number line Add and subtract Multiply and divide Exercises Exercises Alberto Pardo Milan´s e Integers

28. Indice Sets of Numbers The number line Add and subtract Multiply and divide Exercises Exercises Exercise 1 Write the correct number: Positive forty-eight: Negative one hundred and two: Negative two hundred and seventy-three: Three thousand, nine hundred and eleven: Negative one hundred and two thousand, eight hundred and ten: Positive two million, twenty thousand, ﬁve hundred and ﬁfty-four: Alberto Pardo Milan´s e Integers

29. Indice Sets of Numbers The number line Add and subtract Multiply and divide Exercises Exercises Exercise 1 Write the correct number: Positive forty-eight: +48 Negative one hundred and two: −102 Negative two hundred and seventy-three: −273 Three thousand, nine hundred and eleven: +3911 Negative one hundred and two thousand, eight hundred and ten: −102810 Positive two million, twenty thousand, ﬁve hundred and ﬁfty-four: +2020554 Alberto Pardo Milan´s e Integers

30. Indice Sets of Numbers The number line Add and subtract Multiply and divide Exercises Exercises Exercise 2 How can we read these numbers? -34: 435: -543: -1,207: 5,673: -12,856: -174,730: 3,323,045: 1,100,305: Alberto Pardo Milan´s e Integers

31. Indice Sets of Numbers The number line Add and subtract Multiply and divide Exercises Exercises Exercise 2 How can we read these numbers? -34: Negative thirty-four 435:Four hundred thirty-five -543:Negative five hundred forty-three -1,207:Negative one thousand two hundred and seven 5,673:Positive five thousand six hundred seventy-three -12,856:Negative twelve thousand eight hundred fifty-six -174,730:Negative onehundred seventy-four thousand seven hundred thirty 3,323,045:Positive three million three hundred twenty-three thousand and forty-five 1,100,305:Positive one million one hundred thousand three hundred and five Alberto Pardo Milan´s e Integers

41. Indice Sets of Numbers The number line Add and subtract Multiply and divide Exercises Exercises Exercise 3 Complete: • +2 is than 5, because +2 lies to the left of 5 in the number line. • −5 is than −3, because −5 to the of −3. • −8 is than 0, because 0 lies the of −8. • 3 is than −2, because −2 to the of 3. Alberto Pardo Milan´s e Integers

42. Indice Sets of Numbers The number line Add and subtract Multiply and divide Exercises Exercises Exercise 3 Complete: • +2 is less than 5, because +2 lies to the left of 5 in the number line. • −5 is less than −3, because −5 lies to the left of −3. • −8 is less than 0, because 0 lies to the right of −8. • 3 is greater than −2, because −2 lies to the left of 3. Alberto Pardo Milan´s e Integers

46. Indice Sets of Numbers The number line Add and subtract Multiply and divide Exercises Exercises Exercise 4 Put these numbers in order from least to greatest: −25, 22, −3, 42, 31, 2 Alberto Pardo Milan´s e Integers

47. Indice Sets of Numbers The number line Add and subtract Multiply and divide Exercises Exercises Exercise 4 Put these numbers in order from least to greatest: −25, 22, −3, 42, 31, 2 −25 < − 3 < 2 < 22 < 31 < 42 Alberto Pardo Milan´s e Integers

53. Indice Sets of Numbers The number line Add and subtract Multiply and divide Exercises Exercises Exercise 5 Find out the opposite of each number. Write down a sentence and the expression: -3: 4: 2: -12: Alberto Pardo Milan´s e Integers

54. Indice Sets of Numbers The number line Add and subtract Multiply and divide Exercises Exercises Exercise 5 Find out the opposite of each number. Write down a sentence and the expression: -3: Op(−3) = +3, the opposite of negative three is positive three. 4: Op(4) = −4, the opposite of four is negative four. 2: Op(2) = −2, the opposite of two is negative two. -12:Op(−12) = +12, the opposite of negative twelve is positive twelve. Alberto Pardo Milan´s e Integers

58. Indice Sets of Numbers The number line Add and subtract Multiply and divide Exercises Exercises Exercise 6 Find out the absolute value of each number. Write down a sentence and the expression: 5: -7: -9: 11: -23: 0: Alberto Pardo Milan´s e Integers

59. Indice Sets of Numbers The number line Add and subtract Multiply and divide Exercises Exercises Exercise 6 Find out the absolute value of each number. Write down a sentence and the expression: 5: |5| = 5, the absolute value of ﬁve is ﬁve. -7: | − 7| = 7, the absolute value of negative seven is seven. -9: | − 9| = 9, the absolute value of negative nine is nine. 11: |11| = 11, the absolute value of eleven is eleven. -23: | − 23|=23,the absolute value of negative twenty-three is twenty-three 0: |0| = 0, the absolute value of zero is zero. Alberto Pardo Milan´s e Integers

65. Indice Sets of Numbers The number line Add and subtract Multiply and divide Exercises Exercises Exercise 7 Find: 4 + (−1) = 16 − 11 = 12 − 15 = 7 − (−4) = −8 + 2 = (−20) + (−9) = Alberto Pardo Milan´s e Integers

66. Indice Sets of Numbers The number line Add and subtract Multiply and divide Exercises Exercises Exercise 7 Find: 4 + (−1) =+3 16 − 11 =+5 12 − 15 =−3 7 − (−4) =+11 −8 + 2 =−6 (−20) + (−9) =−29 Alberto Pardo Milan´s e Integers

72. Indice Sets of Numbers The number line Add and subtract Multiply and divide Exercises Exercises Exercise 8 Find: 13 + (−1) − 14 − (−2) = 8 + (−15) − 13 + 24 = 2−8+2−5+2= 1 + (−2) + 6 − 12 + 6 + (−6) = −17 + 7 − 4 − 11 = Alberto Pardo Milan´s e Integers

73. Indice Sets of Numbers The number line Add and subtract Multiply and divide Exercises Exercises Exercise 8 Find: 13 + (−1) − 14 − (−2) =0 8 + (−15) − 13 + 24 =+4 2 − 8 + 2 − 5 + 2 =−7 1 + (−2) + 6 − 12 + 6 + (−6) =−7 −17 + 7 − 4 − 11 =−25 Alberto Pardo Milan´s e Integers

78. Indice Sets of Numbers The number line Add and subtract Multiply and divide Exercises Exercises Exercise 9 Find: (−3) · 4 − 5 · 2 = 13 − 5 · 10 = 8 : (−4) + 14 : 7 = 2 · (−3) − 25 : (−5) = −9 · 2 − 54 : 6 = (−20) · (−9) − 11 · 7 − 81 : 9 = Alberto Pardo Milan´s e Integers

79. Indice Sets of Numbers The number line Add and subtract Multiply and divide Exercises Exercises Exercise 9 Find: (−3) · 4 − 5 · 2 =−12 − 10 = −22 13 − 5 · 10 =13 − 50 = −37 8 : (−4) + 14 : 7 =−2 + 2 = 0 2 · (−3) − 25 : (−5) =−6 − (−5) = −1 −9 · 2 − 54 : 6 =−18 − 9 = −27 (−20) · (−9) − 11 · 7 − 81 : 9 =180 − 77 − 2 = 101 Alberto Pardo Milan´s e Integers

85. Indice Sets of Numbers The number line Add and subtract Multiply and divide Exercises Exercises Exercise 10 Write an integer beside each sentence: Mary hikes at a height of two thousand and twenty metres above sea level: Luis earns e1070 a month: An Alfa class submarine can operate at 1300 meters: The plane ﬂies three thousand metres high: The temperature outside is ﬁve degrees Celsius bellow zero: Alberto Pardo Milan´s e Integers

86. Indice Sets of Numbers The number line Add and subtract Multiply and divide Exercises Exercises Exercise 10 Write an integer beside each sentence: Mary hikes at a height of two thousand and twenty metres above sea level: +2020 Luis earns e1070 a month:+1070 An Alfa class submarine can operate at 1300 meters: −1300 The plane ﬂies three thousand metres high:+3000 The temperature outside is ﬁve degrees Celsius bellow zero:−5 Alberto Pardo Milan´s e Integers

91. Indice Sets of Numbers The number line Add and subtract Multiply and divide Exercises Exercises Exercise 11 Thales of Miletus was born in 624 b.C, and he lived 78 years. Find out the year of his death. Alberto Pardo Milan´s e Integers

92. Indice Sets of Numbers The number line Add and subtract Multiply and divide Exercises Exercises Exercise 11 Thales of Miletus was born in 624 b.C, and he lived 78 years. Find out the year of his death. Data: He was born in −624. He lived 78 years. −624 + 78 = −546 Answer: He died in 546 b.C. Alberto Pardo Milan´s e Integers

95. Indice Sets of Numbers The number line Add and subtract Multiply and divide Exercises Exercises Exercise 12 Parts of Death Valley in California are below sea level. A hiker starts at an elevation of 12 feet below sea level. Then she hikes to an elevation that is 8 feet above sea level. How many feets does she hike? Alberto Pardo Milan´s e Integers

96. Indice Sets of Numbers The number line Add and subtract Multiply and divide Exercises Exercises Exercise 12 Parts of Death Valley in California are below sea level. A hiker starts at an elevation of 12 feet below sea level. Then she hikes to an elevation that is 8 feet above sea level. How many feets does she hike? Data: She starts at −12m. She hikes to +8m. −12+ =8 8 + 12 = 20 Answer: She hikes 20m. Alberto Pardo Milan´s e Integers

99. Indice Sets of Numbers The number line Add and subtract Multiply and divide Exercises Exercises Exercise 13 Jackie buys three identical shirts in diﬀerent colors. She has to pay e3.24 in taxes. The total amount she pays is e57.24. What is the cost of each shirt without the taxes? Alberto Pardo Milan´s e Integers

100. Indice Sets of Numbers The number line Add and subtract Multiply and divide Exercises Exercises Exercise 13 Jackie buys three identical shirts in diﬀerent colors. She has to pay e3.24 in taxes. The total amount she pays is e57.24. What is the cost of each shirt without the taxes? Data: She buys three shirts. She pays e3.24 in taxes. She pays e57.24 in total. 57.24 − 3.24 = 54 54 : 3 = 18 Answer: Each shirt costs e18. Alberto Pardo Milan´s e Integers

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