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# Integers

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Integers

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### Integers

1. 1. Index Sets Number line Add and subtract Multiply and divide Exercises Integers Matem´ticas 2o E.S.O. a Alberto Pardo Milan´s e -
2. 2. Index Sets 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 ExercisesAlberto Pardo Milan´s e Integers
3. 3. Index Sets Number line Add and subtract Multiply and divide Exercises Sets of NumbersAlberto Pardo Milan´s e Integers
4. 4. Index Sets Number line Add and subtract Multiply and divide Exercises Sets of Numbers Counting numbers IN is the set of natural numbers, also called counting numbers or positive numbers. Positive numbers, negative numbers, and zero are called integers. Zero is neither positive nor negative and is also called the origin. Positive numbers represent data that are greater than 0. They are written with a + sign or no sign at all. Example: read +7, positive seven or plus seven.Alberto Pardo Milan´s e Integers
5. 5. Index Sets Number line Add and subtract Multiply and divide Exercises Sets of Numbers Integers Negative numbers represent data that are less than 0. They are written with a − sign. Example: Read −2, negative two or minus two. For temperatures you can also use two below zero. Z is the set of integers: Z = {· · · − 5, −4, −3, −2, −1, 0, 1, 2, 3, 4, 5 . . . }Alberto Pardo Milan´s e Integers
6. 6. Index Sets Number line Add and subtract Multiply and divide Exercises Sets of Numbers Absolute value and opposite The absolute value of an integer is the value of the number regardless of its sign. Example: read | − 4| = 4 the absolute value of negative four is four, read | + 6| = 6 the absolute value of positive six is six. Opposites are numbers that have the same absolute value. Example: the opposite of 6 is Op(+6) = −6, the opposite of −10 is Op(−10) = 10.Alberto Pardo Milan´s e Integers
7. 7. Index Sets Number line Add and subtract Multiply and divide Exercises The number lineAlberto Pardo Milan´s e Integers
8. 8. Index Sets 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. The absolute value of a number is its distance from zero on the number line. Opposites are numbers that are the same distance from zero on the number line, but in opposite directions. 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.Alberto Pardo Milan´s e Integers
9. 9. Index Sets Number line Add and subtract Multiply and divide Exercises The number line Examples: • 3 < 4 because 3 lies to the left of 4 in the number line. • −2 > −4 because −2 lies to the right of −4 in the number line. • −2 < 1 because −2 lies to the left of 1 in the number line. • −3 < 0 because 0 lies to the right of −3 in the number line.Alberto Pardo Milan´s e Integers
10. 10. Index Sets Number line Add and subtract Multiply and divide Exercises Adding and subtracting integersAlberto Pardo Milan´s e Integers
11. 11. Index Sets Number line Add and subtract Multiply and divide Exercises Adding and subtracting integers Add 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
12. 12. Index Sets Number line Add and subtract Multiply and divide Exercises Adding and subtracting integers Add using absolute values: 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+2 = 5, so (−3) + (−2) = −5. Find (+2)+(+1): the signs are the same (positive) and 2 + 1 = 3, so (+2) + (+1) = +3.Alberto Pardo Milan´s e Integers
13. 13. Index Sets Number line Add and subtract Multiply and divide Exercises Adding and subtracting integers Add using absolute values: 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 − 2 = 1, so (−3) + (+2) = −1, because 3 > 2. Find (−1) + (+5): the signs are diﬀerent and 5 − 1 = 4, so (−1) + (+5) = +4, because 1 < 5.Alberto Pardo Milan´s e Integers
14. 14. Index Sets Number line Add and subtract Multiply and divide Exercises Adding and subtracting integers Subtraction 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
15. 15. Index Sets Number line Add and subtract Multiply and divide Exercises Multiplying and dividing integersAlberto Pardo Milan´s e Integers
16. 16. Index Sets Number line Add and subtract Multiply and divide Exercises Multiplying and dividing integers To multiply integers, multiply the absolute values and then use the rule of signs to ﬁnd the corresponding sign. To divide integers, divide the absolute values and then use the rule of signs to ﬁnd the corresponding sign. The Rule of Signs for multiplying and dividing says unlike signs produce negative numbers and like signs produce positive numbers. +·−=− − ·+ = − +:−=− −:+=− +·+=+ − ·− = + +:+=+ −:−=+ The product of two integers with diﬀerent signs is negative, the product of two integers with the same sign is positive. The division of two integers with diﬀerent signs is negative, the division of two integers with the same sign is positive.Alberto Pardo Milan´s e Integers
17. 17. Index Sets Number line Add and subtract Multiply and divide Exercises Multiplying and dividing integers Examples: Find 2 · (−1): the integers have diﬀerent signs, the product is negative 2 · (−1) = −2. Find −2 · (−4): the integers have the same signs, the product is positive −2 · (−4) = +8. Find 5 : (−1): the integers have diﬀerent signs, the quotient is negative 5 : (−1) = −5. Find −4 : (−2): the integers have the same signs, the quotient is positive −4 : (−2) = +2.Alberto Pardo Milan´s e Integers
18. 18. Index Sets Number line Add and subtract Multiply and divide Exercises ExercisesAlberto Pardo Milan´s e Integers
19. 19. Index Sets Number line Add and subtract Multiply and divide Exercises Exercises Exercise 1 How can we read these numbers? −53 624 −136 −2, 753 1, 384 −25, 242 −189, 777 2, 195, 020Alberto Pardo Milan´s e Integers
20. 20. Index Sets Number line Add and subtract Multiply and divide Exercises Exercises Exercise 2 Write the correct number: Positive forty-two: Negative one hundred and twenty: Negative two hundred and ﬁfty-three: Four thousand, seven hundred and eleven: Negative one hundred and three thousand, nine hundred and thirty: Positive ﬁve million, ﬁfty thousand, ﬁve hundred and forty:Alberto Pardo Milan´s e Integers
21. 21. Index Sets Number line Add and subtract Multiply and divide Exercises Exercises Exercise 3 Complete the gaps with the correct word (not a symbol): • +4 is than 5, because +4 lies to the of 5. • −13 is than −3, because −13 to the of −3. • −75 is than 0, because 0 lies the of −75. • 4 is than −12, because −12 to the of 4.Alberto Pardo Milan´s e Integers
22. 22. Index Sets Number line Add and subtract Multiply and divide Exercises Exercises Exercise 4 Find out the opposite of each number. Write down a sentence and the expression: −12 45 −15 0 25 −48 Sentences:Alberto Pardo Milan´s e Integers
23. 23. Index Sets Number line Add and subtract Multiply and divide Exercises Exercises Exercise 5 Find out the absolute value of each number. Write down a sentence and the mathematic expression: 13 −65 −90 77 −69 0Alberto Pardo Milan´s e Integers
24. 24. Index Sets Number line Add and subtract Multiply and divide Exercises Exercises Exercise 6 Calculate the following and write down how we read the answer: (−12) · 3 − 7 · 13 = 15 − 7 · 10 = 64 : (−4) + 63 : 7 = −8 · 3 − 66 : 3 = (−20) · (−11) − 15 · 4 − 81 : 9 =Alberto Pardo Milan´s e Integers
25. 25. Index Sets Number line Add and subtract Multiply and divide Exercises Exercises Exercise 7 Put these numbers in order from least to greatest: −235, 12, −330, −42, 131, −25Alberto Pardo Milan´s e Integers
26. 26. Index Sets Number line Add and subtract Multiply and divide Exercises Exercises Exercise 8 Write an integer beside each sentence: Temperature increases 12 degrees Farenheit: Peter withdraws 40e from his savings account: A submarine can operate at 1,300 meters: Fred hikes at a height of three thousand one hundred metres above sea level: Mary earns 1,870e a month: A jet ﬂies ﬁve thousand metres high: The temperature outside is ﬁfteen degrees Celsius bellow zero:Alberto Pardo Milan´s e Integers
27. 27. Index Sets Number line Add and subtract Multiply and divide Exercises Exercises Exercise 9 Last week, high temperatures in Berlin were 8, 5, 1, −3, 4, −1, and −6. Complete these sentences: The highest temperature is . The lowest temperature is . The range of the data is .Alberto Pardo Milan´s e Integers
28. 28. Index Sets Number line Add and subtract Multiply and divide Exercises Exercises Exercise 10 Thales of Miletus was the ﬁrst known Greek philosopher, scientist and mathematician. He was born in 624 B.C, and he lived 78 years. Find out the year of his death.Alberto Pardo Milan´s e Integers
29. 29. Index Sets Number line Add and subtract Multiply and divide Exercises Exercises Exercise 11 Parts of Netherlands are below sea level. A hiker starts at an elevation of 22 feet below sea level. Then she hikes to an elevation that is 2 feet above sea level. How many feets does she hike?Alberto Pardo Milan´s e Integers
30. 30. Index Sets Number line Add and subtract Multiply and divide Exercises Exercises Exercise 12 A grandfather has e1,235 and three grandsons. He wants to give the same amount of money to each boy. How much does each boy get? Is there any money left?Alberto Pardo Milan´s e Integers