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# Decimals

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### Decimals

1. 1. Index The set of reals Decimals Reading real numbers Powers of ten Approximating a quantity Exercises Decimals Matem´ticas 2o E.S.O. a Alberto Pardo Milan´s e -
2. 2. Index The set of reals Decimals Reading real numbers Powers of ten Approximating a quantity Exercises 1 The set of reals 2 Decimals 3 Reading real numbers 4 Multiplying and dividing by 10, 100, 1000, etc 5 Approximating a quantity 6 Exercises
3. 3. Index The set of reals Decimals Reading real numbers Powers of ten Approximating a quantity Exercises The set of realsAlberto Pardo Milan´s e Decimals
4. 4. Index The set of reals Decimals Reading real numbers Powers of ten Approximating a quantity Exercises The set of reals Sets of numbers IN is the set of natural numbers. The set Z of natural numbers, negative numbers, and zero are all called integers. A rational number is a number that can be expressed as a fraction p/q where p and q are integers and q = 0. The set of rational numbers is named Q. There are numbers that can’t be expressed as a fraction. An irrational number is a num- ber that can’t be expressed as a fraction p/q for any integers p and q. √ Example: π, φ, 2, . . . can’t be expressed as a fraction as they are irrational numbers. The set R of rational and irrational numbers is named the set of real numbers.Alberto Pardo Milan´s e Decimals
5. 5. Index The set of reals Decimals Reading real numbers Powers of ten Approximating a quantity Exercises DecimalsAlberto Pardo Milan´s e Decimals
6. 6. Index The set of reals Decimals Reading real numbers Powers of ten Approximating a quantity Exercises Decimals Decimal expansion The decimal expansion of a number is its representation in the decimal system. Example: the decimal expansion of 252 is 625, of π is 3.14159 . . . , and of 1/9 is 0.1111 . . . The decimal expansion of a number may terminate, become periodic or continue inﬁnitely without repeating. A ﬁnite decimal (or terminating decimal) is a number that has a ﬁnite decimal expansion. Example: 1/8 = 0.125.Alberto Pardo Milan´s e Decimals
7. 7. Index The set of reals Decimals Reading real numbers Powers of ten Approximating a quantity Exercises Decimals Decimal expansion A decimal number is a repeating decimal if at some point it becomes periodic: there is some ﬁnite sequence of digits that is repeated indeﬁnitely. The repeating portion of a decimal expansion is conventionally denoted with a vinculum (a horizontal line placed above multiple quantities). Example: 5/3 = 1,66666666 · · · = 1.6, read it as one point six recurring. Note the possibility of repeating decimals that begin with a non-repeating part. Example: 61/30 = 2,03333333 · · · = 2.03, read it as two point zero, three recurring.Alberto Pardo Milan´s e Decimals
8. 8. Index The set of reals Decimals Reading real numbers Powers of ten Approximating a quantity Exercises Decimals Decimal expansion Irrational numbers have decimal expansions that neither terminate nor become periodic. Example: π = 3. 14159265358979323846264338327950288419716 9399375105820974944592307816406286208998628034825342117 0679821480865132823066470938446095505822317253594812848 1117450284102701938521105559644622948954930381964428810 9756659334461284756482337867831652712019091456485669234 6034861045432664821339360726024914127372458700660631558 8174881520920962829254091715364367892590360011330530548 8204665213841469519415116094330572703657595919530921861 17381932611793105118548074462379962 . . . A fraction in lowest terms with a prime denominator other than 2 or 5 always produces a repeating decimal.Alberto Pardo Milan´s e Decimals
9. 9. Index The set of reals Decimals Reading real numbers Powers of ten Approximating a quantity Exercises Reading real numbersAlberto Pardo Milan´s e Decimals
10. 10. Index The set of reals Decimals Reading real numbers Powers of ten Approximating a quantity Exercises Reading real numbers Remember that the value of a digit depends on its place or position in the number and the decimal point shows where the fractional part of a number begins. Diﬀerent places of a ﬁgure gives diﬀerent names: Examples: hundred-thousandths hundred-thousands hundred-millionths hundred-millions ten-thousandths ten-thousands ten-millionths thousandths ten-millions hundredths thousands millionths hundreds billionths millions Billions tenths units tens 1 6 1 8 0 3 3 9 8 8 . 7 4 9 8 9 4 8 4 8Alberto Pardo Milan´s e Decimals
11. 11. Index The set of reals Decimals Reading real numbers Powers of ten Approximating a quantity Exercises Reading real numbers In 42.5 → ﬁve are the tenths and four are the tens. In 3,267.2558 → three are the thousands and eight are the ten thousandths. In 2,656,711.3 → two are the millions. Look at the following examples to learn how to read decimal numbers: Examples: 321.7 → Three hundred twenty-one and seven tenths. 5,062.57 → Five thousand sixty-two and ﬁfty-seven hundredths. 43.27 → Forty-three point two seven. \$4.76 → Four dollars and sixty-seven cents. 3.42 → Three point forty-two recurring. 12.37 → Twelve point three, seven recurring.Alberto Pardo Milan´s e Decimals
12. 12. Index The set of reals Decimals Reading real numbers Powers of ten Approximating a quantity Exercises Multiplying and dividing by 10, 100, 1000, etcAlberto Pardo Milan´s e Decimals
13. 13. Index The set of reals Decimals Reading real numbers Powers of ten Approximating a quantity Exercises Multiplying and dividing by 10, 100, 1000, etc When you multiply a number by 10 the digits move one place to the left, making the number bigger. Example: 1.414213 · 10 = 14.14213 When you multiply a number by 100 the digits move two places to the left, when you multiply a number by 1000 the digits move three places to the left,. . . When you divide a number by 10 the digits move one place to the right, making the number smaller, when you divide a number by 100 the digits move two places to the right, when you divide a number by 1000 the digits move three places to the right,. . . Examples: 63.256 · 100 = 6325.6 68.63 : 10 = 6.863 1234.5 : 100 = 12.345Alberto Pardo Milan´s e Decimals
14. 14. Index The set of reals Decimals Reading real numbers Powers of ten Approximating a quantity Exercises Approximating a quantityAlberto Pardo Milan´s e Decimals
15. 15. Index The set of reals Decimals Reading real numbers Powers of ten Approximating a quantity Exercises Approximating a quantity Rounding oﬀ and truncating a decimal are techniques used to estimate or approximate a quantity. Instead of having a long string of ﬁgures, we can approximate the value of the decimal to a speciﬁed decimal place. To truncate a decimal, we leave our last decimal place as it is given and discard all digits to its right. Examples: Truncate 123,237.23 to the tens place: 123,230. Truncate 35.77 euros to euros: 35 euros. Truncate 1.123 to the tenths: 1.1Alberto Pardo Milan´s e Decimals
16. 16. Index The set of reals Decimals Reading real numbers Powers of ten Approximating a quantity Exercises Approximating a quantity After rounding oﬀ, the digit in the place we are rounding will either stay the same (referred to as rounding down) or increase by 1 (referred to as rounding up). To round oﬀ a decimal ﬁrst ﬁnd the rounding place, then look at the digit to the right of the place being rounded and: • If the digit is 4 or less, the ﬁgure in the place we are rounding remains the same (rounding down). • If the digit is 5 or greater, add 1 to the ﬁgure in the place we are rounding (rounding up). After rounding, discard all digits to the right of the place we are rounding. Examples: Round 123,237.23 to the tens place:123,240 we are rounding up. Round 123,234.23 to the tens place:123,230 we are rounding down. Round 45.79 euros to the nearest euro: 46 we are rounding up.Alberto Pardo Milan´s e Decimals
17. 17. Index The set of reals Decimals Reading real numbers Powers of ten Approximating a quantity Exercises ExercisesAlberto Pardo Milan´s e Decimals
18. 18. Index The set of reals Decimals Reading real numbers Powers of ten Approximating a quantity Exercises Exercises Exercise 1 A mint costs .05 euros. How much would a roll of ten mints cost?Alberto Pardo Milan´s e Decimals
19. 19. Index The set of reals Decimals Reading real numbers Powers of ten Approximating a quantity Exercises Exercises Exercise 2 Lucy won a long-jump contest with a distance of 6.45 m. Samantha jumped 2.02 m less than Lucy, while Jenny jumped .73 m farther than Samantha, but 1.2 m less than Mary. How many meters did Mary jump?Alberto Pardo Milan´s e Decimals
20. 20. Index The set of reals Decimals Reading real numbers Powers of ten Approximating a quantity Exercises Exercises Exercise 3 A peach costs .62 euros. How much would two dozen peaches cost?Alberto Pardo Milan´s e Decimals
21. 21. Index The set of reals Decimals Reading real numbers Powers of ten Approximating a quantity Exercises Exercises Exercise 4 Round 7.601 to the nearest whole number: Round 68.94 to the nearest tenth: Round 1.25396 · 100 to the nearest hundredth: Truncate 1.787 to a whole number: Truncate 2.24 to a tenth: Truncate 2585.2 : 100 to a hundredth:Alberto Pardo Milan´s e Decimals
22. 22. Index The set of reals Decimals Reading real numbers Powers of ten Approximating a quantity Exercises Exercises Exercise 5 Mike swims 50 m every 32.54 seconds. Based on rounding, estimate the time he needs to swim 1000 m.Alberto Pardo Milan´s e Decimals
23. 23. Index The set of reals Decimals Reading real numbers Powers of ten Approximating a quantity Exercises Exercises Exercise 6 Anna bought 12 CDs with the same price each. The total cost was 255.00 euros. What was the price of each CD?Alberto Pardo Milan´s e Decimals
24. 24. Index The set of reals Decimals Reading real numbers Powers of ten Approximating a quantity Exercises Exercises Exercise 7 Kate earns 8.60 euros per hour working part-time. She worked a total of 15.7 hours one week. How much money did She earn?Alberto Pardo Milan´s e Decimals
25. 25. Index The set of reals Decimals Reading real numbers Powers of ten Approximating a quantity Exercises Exercises Exercise 8 Salmon cost 4.31 euros per kg at the supermarket. What is the price for a 800 g piece of salmon?Alberto Pardo Milan´s e Decimals
26. 26. Index The set of reals Decimals Reading real numbers Powers of ten Approximating a quantity Exercises Exercises Exercise 9 A can of mushrooms costs .0361 euros per ounce. To the nearest cent, how much does an ounce of mushrooms cost?Alberto Pardo Milan´s e Decimals
27. 27. Index The set of reals Decimals Reading real numbers Powers of ten Approximating a quantity Exercises Exercises Exercise 10 During the month of February, Alfred spent 14.78 euros on gas the ﬁrst week, 15.35 euros during the second week, 15.94 euros during the third week, and 14.07 euros during the fourth week. Which is closest to the total amount of money Alfred spent on gasoline during February? 35 euros 50 euros 60 euros 100 eurosAlberto Pardo Milan´s e Decimals
28. 28. Index The set of reals Decimals Reading real numbers Powers of ten Approximating a quantity Exercises Exercises Exercise 11 The length of a new swimming pool being built at the community recreation center is listed as 26 feet. What is the length of the new pool in yards? What is the length of the new pool in meters? (Note 1 yard=3 feet=0.9144 meters).Alberto Pardo Milan´s e Decimals
29. 29. Index The set of reals Decimals Reading real numbers Powers of ten Approximating a quantity Exercises Exercises Exercise 12 The speed of light is about 186,282 miles per second. Earth is about 92,976,000 miles from the sun. How long does it takes the sun´s light to reach the Earth to the nearest hundredth of a minute?Alberto Pardo Milan´s e Decimals
30. 30. Index The set of reals Decimals Reading real numbers Powers of ten Approximating a quantity Exercises Exercises Exercise 13 Renting a hall for a meeting will cost euros 5,000. Each person attending the meeting will be charged with euros 20.75. How many people will have to attend if you want all the expenses to be covered?Alberto Pardo Milan´s e Decimals
31. 31. Index The set of reals Decimals Reading real numbers Powers of ten Approximating a quantity Exercises Exercises Exercise 14 A circle is a set of points that are a given distance from a given point, the centre. Circumference is the distance around a circle. The diameter of a circle is the distance across a circle, through its center. If you divide the circumference of a circle by its diameter the result is always the number π. Find a circle at home (a window, a lamp, a round table,. . . ), measure its circumference and its diameter and write it down. Divide the circumference of your circle by its diameter and round the result to the nearest hundredth. π will appear!Alberto Pardo Milan´s e Decimals