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nature of mathematics

Questions and Answers of Nature Of Mathematics

Let Ο be an arbitrary operation in Problems 52–59. Describe the operation Ο for each problem. 109 11; 207 = 10; 900 = 10; 90 8 = 18;
Let Ο be an arbitrary operation in Problems 52–59. Describe the operation Ο for each problem. 800 = 1;504 = 21; 100 = 1; 5 0 6 = 31; -
Let Ο be an arbitrary operation in Problems 52–59. Describe the operation Ο for each problem. 406=20; 802= 20; 709 = 32; 608 = 28;
Let Ο be an arbitrary operation in Problems 52–59. Describe the operation Ο for each problem. 407 = 1; 405 = 3; 703 = 11; 1209 = 15;..
Let Ο be an arbitrary operation in Problems 52–59. Describe the operation Ο for each problem. 407 = 17;506 = 26; 604 37; 208 = 5; . =
Cut out a small square and label it as shown in Figure 5.19.Figure 5.19Be sure that 1 is in front of 1´, 2 is in front of 2´, 3 is in front of 3´, and 4 is in front of 4´. We will study certain
Problems 41– 46 involve the set {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10} and addition and multiplication modulo 11.Make a table for addition and multiplication.
Problems 41– 46 involve the set {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10} and addition and multiplication modulo 11.Is the set a group for addition?
Problems 41– 46 involve the set {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10} and addition and multiplication modulo 11.Is the set a group for multiplication?
Problems 41– 46 involve the set {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10} and addition and multiplication modulo 11.Is the set a commutative group for addition?
Problems 41– 46 involve the set {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10} and addition and multiplication modulo 11.Is the set a commutative group for multiplication?
Problems 41– 46 involve the set {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10} and addition and multiplication modulo 11.Is the set a field for the operations of addition and multiplication?
Create the modular designs in Problems 47–52.(19, 2)
Create the modular designs in Problems 47–52.(19, 18)
Create the modular designs in Problems 47–52.(21, 5)
Create the modular designs in Problems 47–52.(21, 10)
Create the modular designs in Problems 47–52.(65, 2)
Create the modular designs in Problems 47–52.(65, 3)
An International Standard Book Number (ISBN) is used to identify books. The ISBN number for the 9th edition of The Nature of Mathematics was 0-534-36890-5. The first digit, 0, indicates the book is
An International Standard Book Number (ISBN) is used to identify books. The ISBN number for the 9th edition of The Nature of Mathematics was 0-534-36890-5. The first digit, 0, indicates the book is
What are the possible check digits for ISBNs? What is the check digit for the 10th edition of The Nature of Mathematics with ISBN 0-534-40023-X? What do you think the X stands for in this ISBN?
If it is now 2 p.m., what time will it be 99,999,999,999 hours from now?
Write a schedule for 12 teams so that each team will play every other team once and no team will be idle.
One Hundred Fowl Problem (an old Chinese puzzle) A man buys 100 birds for $100. A rooster is worth $10, a hen is worth $3, and chicks are worth $1 a pair. How many roosters, hens, and chicks did he
Chinese Remainder Problem A band of 17 pirates decided to divide their doubloons into equal portions. When they found that they had 3 coins remaining, they agreed to give them to their Chinese cook,
What is the next smallest number of coins that would satisfy the conditions of Problem 59?Data from Problem 59Chinese Remainder Problem A band of 17 pirates decided to divide their doubloons into
You are serving food at a charity event and have a pizza cut into 12 slices, served with 48 chocolate kisses and 18 fruit drinks. What is the largest number of people that can be served with this
Shannon is planning a party and wants to give each person a bouquet of flowers. He has 32 carnations, 24 daisies, and 16 wild flowers. What is the greatest number of bouquets he can make if he wants
Boxes that are 12 inches tall are stacked next to boxes that are 18 inches tall. What is the shortest height at which the two stacks will be the same height?
Hot dogs come in packages of 10, while buns come in either 8 or 12. What is the smallest number of packages you have to buy to have the same number of hot dogs and buns?
Use the sieve in Problem 45a to make a conjecture about primes and multiples of 6.Data from problem 45We used a sieve of Eratosthenes in Table 5.2 by arranging the first 100 numbers into 10 rows and
Set up a sieve similar to the one illustrated here but using the first 100 counting numbers. 1 7 13 19 25 31 37 43 2 x 4 20 2 3 8 14 26 32 38 44 4 9 15 10 M 16 21 22 27 28 33 34 36 39 40 45 46 12
Pairs of consecutive odd numbers that are primes are called prime twins. For example, 3 and 5, 11 and 13, and 41 and 43 are prime twins. Can you find any others?
Three consecutive odd numbers that are primes are called prime triplets. It is easy to show that 3, 5, and 7 are the only prime triplets. Can you explain why this is true?
In 1742 the mathematician Christian Goldbach observed that every even number (except 2) seemed representable as the sum of two primes. Goldbach could not prove this result, known today as
Let S 5 {1, 2, 3, 5, 6, 10, 15, 30}, and define an operation M
Use an argument similar to the one in the text to show that 23 is not the largest prime.
Did you notice in Table 5.1 that no rule for 7 was given? Here is a rule for determining if counting number n is divisible by 7. Any number has a units digit (it may be 0), a tens digit, a hundreds
In the text, we showed that 19 is not the largest prime byconsidering M 2-3-5-7 11 13 17-19 + 1 Now, M is either prime or composite. If it is prime, then since it is larger than 19, we have a prime
There are three brothers. The product of their ages is 36. The sum of their ages is equal to the age of their only sister. The three brothers never admit their ages, but their sister always admits
What is the smallest natural number that is divisible by the first 20 counting numbers?
Some primes are 1 more than a square. For example, 5 = 22 + 1.Can you find any other primes p so that p = n2 + 1?
Some primes are 1 less than a square. For example, 3 = 22 - 1. Can you find any other primes p so that p = n2 - 1?
The Pythagoreans studied numbers to find certain mystical properties in them. Certain numbers they studied were called perfect numbers. A perfect number is a natural number that is equal to the sum
The Pythagoreans studied numbers that they called amicable or friendly. A pair of numbers is friendly if each number is the sum of the proper divisors of the other (a proper divisor includes the
Simplify the expressions in Problems 38–50. a. 6-(-2) b. -5-(-3)
Simplify the expressions in Problems 38–50. a. 15 (6) b. 4 +68]
Simplify the expressions in Problems 38–50. a. -32-5-(-7) b.--4-(-8)
Simplify the expressions in Problems 38–50. a. -3 [(-6) - 4] b. 5+ (-19) + |15|
Simplify the expressions in Problems 38–50. a. |-3-[-(-2)] b. 15-(-7)
Simplify the expressions in Problems 38–50. a. [-54 (-9)] 3 b. 54 [(-9) 3]
Simplify the expressions in Problems 38–50. a. [48 (-6)] = (-2) b. 48 [(-6)= (-2)] =
Simplify the expressions in Problems 38–50. a. 15 (3)-|4 11| - b. -12 + (-7) - 10 - 14 -
Simplify the expressions in Problems 38–50. a. (-2)(3)(-4)(5)(-6)(7)(-8)(9)(-10) b. 1 + (-2) + 3 + (4) + 5 + (6) + 7 +(-8) + 9 + (-10)
Perform the indicated operations. Let k be a natural number.a. 16b. 167c. 12007d. 12ke. 12k+1f. 12k-1
Perform the indicated operations. Let k be a natural number.a. 22b. 23c. 24d. 25e. Is 22k+1 positive or negative?
Perform the indicated operations. Let k be a natural number.a. (-1)6b. (-1)67c. (-1)2007d. (-1)2ke. (-1)2k+1f. (-1)2k-1
Perform the indicated operations. Let k be a natural number. a. (-2) c. (-2)4 e. Is (-2)2+ positive or negative? b. (-2) d. (-2)5
a. State the commutative property.b. Is Z commutative for addition? Give reasons.c. Is Z commutative for subtraction? Give reasons.d. Is Z commutative for multiplication? Give reasons.e. Is Z
a. State the associative property.b. Is Z associative for addition? Give reasons.c. Is Z associative for subtraction? Give reasons.d. Is Z associative for multiplication? Give reasons.e. Is Z
Show that the set Z is closed for subtraction.
Find a finite subset of Z that is closed for multiplication.
Multiply 1,234,567 × 9,999,999a. Using a calculatorb. Using patterns
B.C. apparently has a mental block against fours, as we can see from the cartoon.See if you can handle fours by writing the numbers from 1 to 10 using four 4s, operation symbols, or possibly grouping
Perform the indicated operations in Problems 39–48. a. + + b. -21 + 4 - 70
Perform the indicated operations in Problems 39–48. a. b. + 119 -3 81 200 200 -3/119 81 =(200+200) 4
Perform the indicated operations in Problems 39–48. a. 3+3 +3-2 b. 5-+52 +5
Perform the indicated operations in Problems 39–48. 14 90 7 + 50 60 11 50
Perform the indicated operations in Problems 39–48. 11 17 7 144 + 300 + 108
Perform the indicated operations in Problems 39–48. 11 108 7 23 + 144 300
Perform the indicated operations in Problems 39–48. 7 60 19 3909090 21 51
Perform the indicated operations in Problems 39–48. 143 15 + + 210 124 11 1,085
Show how the circles in Figure 5.4 can be related to the set of rational numbers.Figure 5.4
Show that the set Q of rationals is closed for subtraction.
Show that the set Q of rationals is closed for nonzero division.
Is the set Z of integers closed for addition, subtraction, multiplication, and nonzero division? Explain.
Is Q, the set of rationals, associative and/or commutative for addition? Explain.
Is Q, the set of rationals, associative and/or commutative for multiplication? Explain.
A unit fraction (a fraction with a numerator of 1) is sometimes called an Egyptian fraction. In Section 4.1, we said that the Egyptians expressed their fractions as a sum of distinct (different) unit
A unit fraction (a fraction with a numerator of 1) is sometimes called an Egyptian fraction. In Section 4.1, we said that the Egyptians expressed their fractions as a sum of distinct (different) unit
A unit fraction (a fraction with a numerator of 1) is sometimes called an Egyptian fraction. In Section 4.1, we said that the Egyptians expressed their fractions as a sum of distinct (different) unit
A unit fraction (a fraction with a numerator of 1) is sometimes called an Egyptian fraction. In Section 4.1, we said that the Egyptians expressed their fractions as a sum of distinct (different) unit
Recall that the Egyptians used only unit fractions to represent numbers. Is the answer given on the papyrus correct?A quantity and its two-thirds and its half and its one-seventh together make 33.
The sum and difference of the same two squares may be primes, as in this example: 945 and 9 + 4 = 13 Can the sum and difference of the same two primes be squares? Can you find more than one example?
An antenna is to be erected and held by guy wires. If the guy wires are 15 ft from the base of the antenna and the antenna is 10 ft high, what is the exact length of each guy wire?What is the length
What is the exact length of the hypotenuse if the legs of a right triangle are 2 in. each?
What is the exact length of the hypotenuse if the legs of a right triangle are 3 ft each?
An empty lot is 400 ft by 300 ft. How many feet would you save by walking diagonally across the lot instead of walking the length and width?
A diagonal brace is to be placed in the wall of a room. The height of the wall is 8 ft and the wall is 20 ft long. What is the exact length of the brace? What is the length of the brace to the
A balloon rises at a rate of 4 ft per second when the wind is blowing horizontally at a rate of 3 ft per second. After three seconds, how far away from the starting point, in a direct line, is the
Consider a square inch as shown in Figure 5.10.a. Find the total length of the segments making up the stairs in each part of Figure 5.10.b. Find the length of the diagonal of each square.c. It seems
Find an irrational number between 1 and 3.
Find an irrational number between 0.53 and 0.54.
Find an irrational number between 1 / 11 and 1 / 10.
Without using a radical symbol, write an irrational number using only 2s and 3s.
Suppose three squares of uniform thickness are made of gold plate and you are offered either the large square or the two smaller ones. Which choices would you make for each of the squares having
The Pythagorean theorem tells us that the sum of the squares of the lengths of the legs of a right triangle is equal to the square of the length of the hypotenuse. Verify that the theorem is true by
Repeat Problem 53 for the following squares.Data from Problem 53The Pythagorean theorem tells us that the sum of the squares of the lengths of the legs of a right triangle is equal to the square of
A man wanted to board a plane with a 5-ft long steel rod, but airline regulations say that the maximum length of any object or parcel checked on board is 4 ft. Without bending or cutting the rod, or
The Historical Note on page 218 introduces the great mathematician Karl Gauss. Gauss kept a scientific diary containing 146 entries, some of which were independently discovered and published by

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