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nature of mathematics

Questions and Answers of Nature Of Mathematics

Describe Pólya’s problem-solving model.
Design a piece of art based on a magic square.
Test the validity of the following arguments.a. All dictionaries are books.This is a dictionary.Therefore, this is a book.b. If you like potato chips, then you will like Krinkles.You do not like
A chessboard consists of 64 squares, as shown in Figure 1.19. The rook can move one or more squares horizontally or vertically. Suppose a rook is in the upper-left-hand corner of a chessboard. Tell
Do you plan to use a calculator for working the problems in this text? If so, what type of logic does it use?
Assuming that light travels at 186,000 miles per second, what is the distance (in miles) that light travels in 1 year? This is the unit of length known as a light-year. Give your answer in scientific
Describe the differences in evaluating exponents and using scientific notation on a calculator.
A jokester tells you that he has a group of cows and chickens and that he counted 13 heads and 36 feet. How many cows and chickens does he have?
Compute 111,111,111 3 111,111,111. Do not use direct multiplication; show all your work.
Show the calculator steps and display for 4 + 3 × 5 - 7.
Answer the question posed in Problem 59, Section 1.3 for your own state. If you live in California, then use Florida.Data from Problem 59a. If the entire population of the world moved to California
What is the largest number whose name you know? Describe the size of this number.
If a family has 5 children, in how many different birth orders could the parents have a 3-boy, 2-girl family?
Repeat Example 3 using a calculator; that is, findData from Example 3Assuming that light travels at 186,000 miles per second, what is the distance (in miles) that light travels in 1 year? This is the
What is meant by “order of operations”?
What is a trillion? Do not simply define this number, but discuss its magnitude (size) in terms that are easy to understand.
Read the article, “Mathematics at the Turn of the Millennium,” by Philip A. Griffiths, The American Mathematical Monthly, January 2000, pp. 1–14. Briefly describe each of these famous
“I’m nine years old,” says Adam.“I’m ten years old,” says Eve.“My tenth birthday is tomorrow,” says Adam.“My tenth birthday was yesterday,” says Eve.“But I’m older than
What is scientific notation?
Write each given number in scientific notation and in calculator notation.a. 745b. 1,230,000,000c. 0.00573d. 0.00000 06239
Write each of the numbers in Problems 7–10 in scientific notation and in floating-point notation (as on a calculator).a. 3,200b. 0.0004c. 64,000,000,000
Does the story in the news clip illustrate inductive or deductive reasoning? Q: What has 18 legs and catches flies? A: I don't know, what? Q: A baseball team. What has 36 legs and catches flies? A: I
Nick and Marsha are driving from Santa Rosa, CA, to Los Angeles, a distance of 460 miles. They leave at 11:00 a.m. and average 50 mph. On the other hand, Mary and Dan leave at 1:00 p.m. in Dan’s
If your salary is $14.75 per hour, your annual salary is approximatelyA. $5,000B. $10,000C. $15,000D. $30,000E. $45,000
Write each of the numbers in Problems 7–10 in scientific notation and in floating-point notation (as on a calculator).a. 23.79b. 0.000001c. 35,000,000,000
What is wrong, if anything, with the following “Great Tapes” advertisement? Instead of reading the 100 greatest books of all time, buy these beautifully transcribed books on tape. If you listen
Use the map in Figure 1.16 to estimate the distance from Orlando International Airport to Disney World.Figure 1.16 Lake Apopka Florida's Turnpike 192 Walt Disney World Resort 17 92 441 4 5 miles
If you have been reading the historical notes in this chapter, you may have noticed that Blaise Pascal was born in 1623 and died in 1662. You may also have noticed that the first time Pascal’s
Write each of the numbers in Problems 7–10 in scientific notation and in floating-point notation (as on a calculator).a. 5,629b. 630,000c. 0.00000 0034
Show the calculator keys you would press as well as the calculator display for the result. Also state the brand and model of the calculator you are using.a. 263b. 9.22 108 6.34 X 106
Under 3/4 impulse power, the starship Enterprise will travel 1 million kilometers (km) in 3 minutes. Compare full impulse power with the speed of light, which is approximately 1.08 . 109 kilometers
Write each of the numbers in Problems 7–10 in scientific notation and in floating-point notation (as on a calculator).a. Googolb. 1,200,300c. 0.00000 123
Assume that there is a $281.9 billion budget “windfall.”Which of the following choices would come closest to liquidating this windfall?A. Buy the entire U.S. population a steak dinner.B. Burn
Estimate the magnitude of the grains of wheat on the last square of a chessboard.
Write each of the numbers in Problems 11–14 in fixed-point notation.a. 72b. 7.2 × 1010c. 4.56 + 3
In 1995, it was reported that an iceberg separated from Antarctica. The size of this iceberg was reported to be equal to 7 × 1016 ice cubes. Convert this size to meaningful units.
The national debt in 2016 soared to $18,300,000,000,000. Write this number in scientific notation. Suppose that in 2016 there were 322,000,000 people in the United States. If the debt is divided
Write each of the numbers in Problems 11–14 in fixed-point notation.a. 26b. 2.1 × 10-3c. 4.07 + 4
Rearrange the cards in the formation shown here so that each horizontal, vertical, and diagonal line of three adds up to 15. A A ovV+ 2 5 2 5 8 34 6 3 6
Write each of the numbers in Problems 11–14 in fixed-point notation.a. 63b. 4.1 × 10–7c. 4.8 –7
Write each of the numbers in Problems 11–14 in fixed-point notation.a. 6–2b. 3.217 × 107c. 8.89 –11
Write each of the numbers in Problems 15–18 in scientific notation. THE SUN IS 93,000,000 MILES AWAY. Trust hea Hartlehn Hari Alghe HOW DO YOU KNOW THAT? IF IT WERE FARTHER AWAY, THERE WOULD BE
Assume that your classroom is 20 ft × 30 ft × 10 ft. If you fill this room with dollar bills (assume that a stack of 233 dollar bills is 1 in. tall), how many classrooms would it take to contain
Use Euler circles to check the validity of each of the arguments given in Problems 15–18.All birds have wings.All flies have wings.Therefore, some flies are birds.
Write each of the numbers in Problems 15–18 in scientific notation.The velocity of light in a vacuum is about 30,000,000,000 cm/sec.
Use Euler circles to check the validity of each of the arguments given in Problems 15–18.No apples are bananas.All apples are fruit.Therefore, no bananas are fruit.
Use Euler circles to check the validity of each of the arguments given in Problems 15–18.All artists are creative.Some musicians are artists.Therefore, some musicians are creative.
Write each of the numbers in Problems 15–18 in scientific notation.The distance between Earth and Mars (220,000,000 miles) when drawn to scale is 0.00000 25 in.
Consider the following pattern:Find one unhappy number as well as one happy number. 1 is happy. 10 is happy because 1 + 0 = 1, which is happy. 13 is happy because 1 + 3 = 10, which is happy. 19 is
Use Euler circles to check the validity of each of the arguments given in Problems 15–18.All rectangles are polygons.All squares are rectangles.Therefore, all squares are polygons.
In 2010, the national debt was approximately $12 trillion. It has been proposed that this number be used to define a new monetary unit, a light buck. That is, one light buck is the amount necessary
A kilowatt-hour is about 3.6 × 106 joules.Write each of the numbers in Problems 19–22 in fixed-point notation.
Suppose you could write out 71000. What is the last digit?
A ton is about 9.06 × 102 kilograms.Write each of the numbers in Problems 19–22 in fixed-point notation.
The volume of a typical neuron is about 3 × 10-8 cm3.Write each of the numbers in Problems 19–22 in fixed-point notation.
Estimate the distance from Los Angeles International Airport to Disneyland. Los Angeles 2 405 Scale: 1 inch = 15 miles L.A. Int'l Airport 110 42) 710, (90) 605 5 (39 10 Disneyland 60 Anaheim
If the sun were a light bulb, it would be rated at 3.8 × 1025 watts.Write each of the numbers in Problems 19–22 in fixed-point notation.
Compute the results in Problems 31–36. Leave your answers in scientific notation. a. (5 10) (8 105) 4 X 106
Compute the results in Problems 31–36. Leave your answers in scientific notation. a. (6 10)(2 10)
Estimate the distance from Fish Camp to Yosemite Village in Yosemite National Park. (120) (140) (49) 0 5 10 15 U (41) miles Yosemite Village Fish Camp 120
Compute the results in Problems 31–36. Leave your answers in scientific notation. a. (6 x 107) (4.8 x 10-6) 2.4 x 105
Compute the results in Problems 31–36. Leave your answers in scientific notation. a. (2xy-)(2-x-y4) ,2 c-y X
Compute the results in Problems 31–36. Leave your answers in scientific notation. a. 0.00016 500 2,000,000
Compute the results in Problems 31–36. Leave your answers in scientific notation. a. 4,500,000,000,000 0.00001 50 X 0.0003
Estimate the number of items in each photograph in Problems 37–40.
Estimate the number of items in each photograph in Problems 37–40.
Estimate the number of items in each photograph in Problems 37–40.
Estimate the number of items in each photograph in Problems 37–40.
a. What is the largest number you can think of using only three digits?b. Use scientific notation to estimate this number. Your calculator may help, but will probably not give you the answer directly.
Zerah Colburn (1804–1840) toured America when he was 6 years old to display his calculating ability. He could instantaneously give the square and cube roots of large numbers. It is reported that it
Jedidiah Buxion (1707–1772) never learned to write, but given any distance he could tell you the number of inches, and given any length of time he could tell you the number of seconds. If he
George Bidder (1806–1878) not only possessed exceptional power at calculations but also went on to obtain a good education. He could give immediate answers to problems of compound interest and
The Library of Congress contains about 35 million books, and this is about 10 terabytes. A recent study by IBM estimates that humanity creates about 2.5 quintillion bytes of data every day. Compare
A sheet of notebook paper is approximately 0.003 in. thick. Tear the sheet in half so that there are 2 sheets. Repeat so that there are 4 sheets. If you repeat again, there will be a pile of 8
If it takes one second to write down each digit, how long will it take to write down all the numbers from 1 to 1,000?
If it takes one second to write down each digit, how long will it take to write down all the numbers from 1 to 1,000,000?
Imagine that you have written down the numbers from 1 to 1,000. What is the total number of zeros you have recorded?
Imagine that you have written down the numbers from 1 to 1,000,000. What is the total number of zeros you have recorded?
It is known that a person’s body has about one gallon of blood in it, and that a cubic foot will hold about 7.5 gallons of liquid. It is also known that Central Park in New York has an area of 840
Let C = cities in California, a = city of Anaheim, and b = city of Berlin. Use set membership notation to describe relations among a, b, and C.
What do we mean by the operations of union, intersection, and complement?
Why do you think mathematics accepts the word set as an undefined term?
Let U = {1, 2, 3, 4, 5, 6, 7, 8, 9}, A = {2, 4, 6, 8}, B = {1, 3, 5, 7}, and C = {5, 7}.A Venn diagram showing these sets is shown in Figure 2.10.Find:a. A ∪ Cb. B ∪ Cc. B ∪ Cd. A ∩
Draw Venn diagrams for intersection and union.a. B ∩ Y b. B ∪ Y
Specify the given sets by roster. If the set is not well defined, say so.a. {counting numbers between 10 and 20}b. {x | x is an integer between -20 and 20}c. {distinct letters in the word
This chapter began with an open-ended question from an Examination of the California Assessment Program. Answer this question.James knows that half the students from his school are accepted at the
Distinguish between equal and equivalent sets.
Suppose a survey indicates that 45 students are taking mathematics and 41 are taking English. How many students are taking math or English?
a. What English word is used to describe union?b. What English word is used to describe intersection?c. What English word is used to describe complement?
Specify the given sets by description.a. {1, 2, 3, 4, 5, c}b. {0, 1, 2, 3, 4, 5, c}c. {x | x ∉ Z}d. {12, 14, 16, c,.... 98}e. {4, 44, 444, 4444,... c}f. {m, a, t, h, e, i, c, s}
a. What is a universal set?b. What is the empty set?
State formulas for:a. Cardinality of an intersectionb. Cardinality of a union
Which of the following sets are equivalent? Are any equal? {O, A, D, 0},{5, 8, 11, 14}, {, }, {, 0, , E}, {1, 2, 3, 4}
a. Give three descriptions of the empty set.b. Give an example of a set with cardinality 0.
Under what conditions is the following formula true? |XU Y = |X| + |Y|
Find the cardinality of each of the following sets.b. S = { }c. T = {states of the United States} a. R= {5, A, Y, T}
Under what conditions is the following formula true? |X| = |U|-|X|
Tell whether each set in Problems 5–8 is well defined. If it is not well defined, change it so that it is well defined.a. The set of counting numbers between 3 and 4.b. The set of happy people in
Let the universal set be all of the cards in a deck of cards.* Draw a Venn diagram for the set of hearts.
Perform the given set operations in Problems 7–18.Let U = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}.a.b.c. {2, 6, 8) U {6, 8, 10}

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