This is not a test! It’s a review of important general concepts you learned in the previous nine chapters. Read it though slowly and let it “sink in.” If you’re confused about anything here, or about anything in the section you’ve just finished, go back and study that material some more.
Chapter 1 Question 1-1
What’s the difference between a number and a numeral?
Answer 1-1
A number is an abstraction. Numerals are tangible or physical symbols that represent num-bers, such as 30 or 7 as they appear on this page. A numeral is not a number, just as your name is not you.
Question 1-2
In the Roman numeration system, how would we write the equivalents of the following decimal numerals?
(a) 10 (b) 30 (c) 40 (d) 50
(e) 100 (f ) 300 (g) 400 (h) 600
Answer 1-2
The Roman equivalents are:
(a) X (b) XXX (c) XL (d) L
(e) C (f ) CCC (g) CD (h) DC
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Question 1-3
The Hindu-Arabic system uses the numeral 0 to represent a quantity of nothing. Why is this significant?
Answer 1-3
The numeral 0 serves as a placeholder when writing numerals to represent large numbers. It makes arithmetic easier than it was with the Roman system, or with systems that used simple counting. Eventually, it made higher mathematics, such as algebra, possible.
Question 1-4
What’s the difference between an “English billion” and a “U.S. billion”?
Answer 1-4
In England, the term “billion” usually refers to the quantity represented by the numeral 1,000,000,000,000. In the United States, it means the quantity represented by 1,000,000,000.
The “English billion” is called a “trillion” in the United States.
Question 1-5
What’s the difference between the decimal, binary, octal, and hexadecimal numeration systems?
Answer 1-5
The decimal system works in base ten, where the numeral 10 represents the number of dots in the following group:
• • • • • • • • • •
If we use decimal numerals to represent quantities, then the binary system works in base 2, the octal system works in base 8, and the hexadecimal system works in base 16.
Question 1-6
The hexadecimal system needs some extra digits besides the usual 0 through 9 to denote num-bers. What are these digits, and what are their decimal equivalents?
Answer 1-6
Here are the extra digits and their decimal equivalents:
• Hexadecimal A stands for decimal 10.
• Hexadecimal B stands for decimal 11.
• Hexadecimal C stands for decimal 12.
• Hexadecimal D stands for decimal 13.
• Hexadecimal E stands for decimal 14.
• Hexadecimal F stands for decimal 15.
144 Review Questions and Answers
Question 1-7
In the octal numeration system, what follows 7? What follows 47? What follows 77? What follows 577?
Answer 1-7
In the octal system, 7 is followed by 10, 47 is followed by 50, 77 is followed by 100, and 577 is followed by 600.
Question 1-8
What is the main advantage of the binary numeration system?
Answer 1-8
It has only two states, which can be represented by the on/off states of simple switches. This makes the binary system ideal for use in electronic systems such as computers.
Question 1-9
In the binary numeration system, what follows 10? What follows 100? What follows 111?
What follows 1101?
Answer 1-9
In the binary system, 10 is followed by 11, 100 is followed by 101, 111 is followed by 1000, and 1101 is followed by 1110. Note that there are no commas in large binary numerals, as there usually are in large decimal numerals.
Question 1-10
In the hexadecimal numeration system, what follows 10? What follows FF? What follows 999? What follows FFF?
Answer 1-10
In the hexadecimal system, 10 is followed by 11, FF is followed by 100, 999 is followed by 99A, and FFF is followed by 1,000.
Chapter 2 Question 2-1
What’s the difference between a set and an element?
Answer 2-1
A set is a collection of elements. The elements of a set are also called its members. If we want to call a number or variable x an element of a set X, then we write x∈X. We can also say that x belongs to X, or that x is in X. If some other number or variable y is not an element of set X, then we write y∉ X. A set can be an element of another set.
Question 2-2
What’s the difference, if any, between the following sets?
A= {x,y,z} B= {z,y,x} C= {y,z,x} D= {x,y,z,y,x}
E= {y,x,z,z,z,z,z, ...}
Answer 2-2
These sets are all the same. When the elements of a set are listed, the order of the list isn’t important. Once we’ve listed something as an element of a set, we can list it again, or 10 times more, or even infinitely many times, and it doesn’t matter.
Question 2-3
Imagine two sets, one called R and the other called S. They share three elements, called x, y, andz. However, they both contain elements other than x, y, or z. What name can we give to the set {x, y, z}?
Answer 2-3
The set {x, y, z} contains all the elements that belong to both R and S. By definition, the set {x, y, z} is the intersection of R and S, written R∩S.
Question 2-4
What does the entire gray shaded region in Fig. 10-1 represent?
Answer 2-4
This region represents the set of all elements in set P, set Q, or both. That’s the union of sets P and Q, written P∪Q.
P
Q Figure 10-1 Illustration for
Questions and Answers 2-4 and 2-5.
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Question 2-5
What does the hatched region in Fig. 10-1 represent?
Answer 2-5
It portrays the set of elements belonging to both P and Q. That’s the intersection of the two sets, written P∩Q.
Question 2-6
What’s the union of two disjoint sets X and Y? What’s the intersection of two disjoint setsX andY?
Answer 2-6
The union of two sets X and Y is the set of all elements in either X or Y. Their intersection is the set of elements in both X and Y. If X and Y are disjoint, then there are no elements belong-ing to them both, so the intersection is the null set.
Question 2-7
What’s the union of two congruent sets? The intersection of two congruent sets?
Answer 2-7
By definition, if two sets X and Y are congruent, then they have exactly the same elements.
Every element in X is also in Y, and every element in Y is also in X. The two sets overlap totally. Their union is the set of all elements in either X or Y. That’s the same as either X or Y by itself. Their intersection is the set of elements in both X and Y. That’s also the same as either X or Y by itself. In this case, all the following sets are identical:
X Y X∪Y X∩Y Question 2-8
Can a set be a subset of itself ? A proper subset of itself ? Answer 2-8
Any set is a subset of itself. But no set is a proper subset of itself. By definition, a proper subset of any set S can’t contain all of the elements in S.
Question 2-9
Is there any set that is a subset of every possible set?
Answer 2-9
Yes. The null set is a subset of any set we can imagine, even itself.
Question 2-10
How can we describe the relationship between the following sets? How would we write it in symbolic form?
A= {4, 5, 6, 7, 8, 9, 10, ...}
B= {7, 8, 9, 10, 11, 12, ...}
Answer 2-10
In this case, B is a proper subset of A. That’s because every element in B is also in A, but there are some elements in A that are not in B. We write this fact as B⊂A.
Chapter 3 Question 3-1
The natural numbers are sometimes defined in terms of sets. How can we do this?
Answer 3-1
We can define the number 0 as the set containing nothing. That’s the null set:
0= { } = ∅
Once we’ve defined the number 0, then we can define the number 1 as the set containing the number 0, like this:
1= {{ }} = {∅}= {0}
After that, we can build the rest of the natural numbers upon each other:
2= {0, 1}
3= {0, 1, 2}
4= {0, 1, 2, 3}
↓
n+ 1 = {0, 1, 2, ..., n}
↓ and so on, forever Question 3-2
How can we write the natural numbers 0 through 4 purely in terms of set braces and the null set symbol?
Answer 3-2
We start with 0, which is equal to ∅ by definition. The numbers are built upon each other as sets within sets, like this:
0= ∅ 1= {0} = {∅}
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2= {0, 1} = {∅, {∅}}
3= {0, 1, 2} = {∅, {∅}, {∅, {∅}}}
4= {0, 1, 2, 3} = {∅, {∅}, {∅, {∅}}, {∅, {∅}, {∅, {∅}}}}
Question 3-3
How is the set of natural numbers symbolized? What elements does it contain?
Answer 3-3
In this book, we symbolize the set of natural numbers as N, and define it as N= {0, 1, 2, 3, 4, ...}
The three dots, called an ellipsis, tell us that the sequence goes on without end. The set N is also known as the set of whole numbers. In some texts, N is defined without including 0:
N= {1, 2, 3, 4, 5, ...}
This is also called the set of counting numbers.
Question 3-4
According to the set-based definition of the natural numbers 0, 1, 2, 3, and so on, what num-ber is represented by the entire set N?
Answer 3-4
Mathematicians call this an infinite ordinal or transfinite ordinal, and denote it using the lowercase Greek letter omega (ω). We can imagine it as a form of “infinity.”
Question 3-5
How can we generate the set of even natural numbers (call it Neven) from the set N of natural numbers? How can we generate the set of odd natural numbers (call it Nodd) from Neven? Answer 3-5
We can generate Neven by taking each element of N and multiplying it by 2. We can generate Nodd from Neven by taking every element of Neven and adding 1.
Question 3-6
In the set N, what is a prime number? What’s a composite number? Are there any natural numbers that are neither prime nor composite? Are there any natural numbers that are both prime and composite?
Answer 3-6
A prime number is a natural number larger than 1 (in other words, 2 or larger) that can only be factored into a product of itself and 1. A composite number is a natural number that’s a product of two or more primes. All the nonprime numbers larger than 1 are composite.
According to these definitions, the numbers 0 and 1 are neither prime nor composite. No natural number can be both prime and composite.
Question 3-7
In the set N, what is a perfect square? Can any perfect square be prime?
Answer 3-7
A perfect square is the result of multiplying a natural number by itself. The first few perfect squares are 0, 1, 4, 9, 16, 25, 36, 49, and 64. No perfect square can be prime. By definition, 0 and 1 are not prime. Any perfect square larger than 1 can be broken down into a product of two or more primes, so it’s composite.
Question 3-8
How can we write down the set Z of integers as an “implied, two-ended list”? As an “implied, one-ended list”?
Answer 3-8
Here’s an “implied, two-ended list” that can give any reader the basic idea concerning the elements of Z:
Z= {..., −4,−3,−2,−1, 0, 1, 2, 3, 4, ...}
To create the “implied, one-ended list,” we start with 0 and then go through the positive and negative integers alternately, like this:
Z= {0, 1, −1, 2, −2, 3, −3, 4, −4, ...}
Question 3-9
How can we quickly and easily tell if a large natural number is divisible by 2, 3, 5, 9, or 10 without a remainder?
Answer 3-9
A natural number is divisible by 2 if its last digit is 0, 2, 4, 6, or 8. A natural number is divis-ible by 3 if the sum of its digits is a natural-number multiple of 3, by 5 if its numeral ends in 0 or 5, by 9 if the sum of its numeric digits is a natural-number multiple of 9, and by 10 if its numeral ends in 0.
Question 3-10
How can we find the prime factors of a large natural number n?
Answer 3-10
We start by finding the square root of n. We ignore the digits after the decimal point, so we have a whole number. We add 1 to that whole number and call the result s. Then we divide the original number n by all the primes less than or equal to s, one by one, starting with the largest prime and working our way down. If we ever get a whole-number quotient, then we know that the divisor and the quotient are both factors of n. Sometimes the quotient is prime, and sometimes it is not. If it isn’t prime, then it can be factored down further. We keep dividing n by smaller and smaller primes until we get down to 2. Once we’ve found all Part One 149
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the primes smaller than s that divide n without leaving a remainder, we have found the prime factors of n. Some of these prime factors may occur more than once.
Chapter 4 Question 4-1
What is the meaning of the term absolute value with respect to integers?
Answer 4-1
The absolute value is the extent to which a number differs from 0. If we imagine the integers as points on a straight line with a “number reflector” passing through the point for 0, then the absolute value of any integer is its distance from the “number reflector.” Figure 10-2 shows a couple of examples. The absolute value of an integer is always positive or 0, because it is an expression of distance without taking direction into account. The absolute value of any natural number is equal to that natural number. The absolute value of any negative integer can be obtained by removing the minus sign.
Question 4-2
Suppose we have two integers a and b. How can we define a+b on a vertical number line where values increase as we move upward?
Answer 4-2
We start by finding the point on the line that corresponds to a. Then we move upward along the line for a distance of b units. We end up at the point for a+b.
Figure 10-2 Illustration for Answer 4-1.
Question 4-3
Suppose we have two integers c and d. How can we define c−d on a vertical number line where values increase as we move upward?
Answer 4-3
We start by finding the point on the line that corresponds to c. Then we move downward along the line for a distance of d units. We end up at the point for c−d.
Question 4-4
How do signs work when adding and subtracting positive and negative integers?
Answer 4-4
When we add a positive or subtract a negative, the result grows larger. When we subtract a positive or add a negative, the result grows smaller. For any two integers p and q,
p+ (−q)=p−q and
p− (−q)=p+q Question 4-5
Suppose that we start with −6, add −8 to it, then subtract 12 from that, then subtract −5 from that, then add −2 to that, and finally subtract −23 from that. What’s the result?
Answer 4-5
Let’s work this through in steps, paying careful attention to signs and using parentheses when we need them:
−6+ (−8)= −6− 8 = −14
−14− 12 = −26
−26− (−5)= −26+ 5 = −21
−21+ (−2)= −21− 2 = −23
−23− (−23)= −23+ 23 = 0 Question 4-6
What does the commutative law tell us about the sum of two integers? What does the associa-tive law tell us about the sum of three integers? What do these two laws, taken together, allow us to do?
Answer 4-6
The commutative law tells us that when we add two integers, we can do it in either order and the sum will be the same. If a and b are integers, then
a+b=b+a
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The associative law says that we can group a sum of three integers in a certain order by twos either way, and the result will always be the same. If a, b, and c are integers, then
(a+b)+c=a+ (b+c)
In combination, the commutative and associative laws allow us to arrange and group a sum of integers in any possible way, and the result will always be the same.
Question 4-7
Does the commutative law for addition apply to sums of more than two integers? Does the associative law apply for sums of more than three integers?
Answer 4-7
Both of these laws will work for sums having as many addends as we want, as long as the number of addends is finite.
Question 4-8
Does the commutative law work for subtraction?
Answer 4-8
We cannot apply the commutative law to subtraction problems and expect valid results. Let’s look at these two subtractions:
5− 10 = −5 but
10− 5 = 5
Question 4-9
Does the associative law work for subtraction done twice?
Answer 4-9
We can’t apply the associative law to subtraction done twice and expect valid results. Here’s an example showing its failure:
(5− 10) − 15 = −5− 15 = −20 but
5 − (10 − 15) = 5 − (−5)
= 5 + 5
= 10
Question 4-10
Suppose a,b, and c are integers and we see the expression a−b+c without any parentheses in it. Can we use parentheses for grouping, apply the associative law, and expect valid results in this situation?
Answer 4-10
No! Here’s an example:
(5− 10) + 15 = −5+ 15 = 10 but
5− (10 + 15) = 5 − 25 = −20 Chapter 5
Question 5-1
When we multiply a positive integer c by another positive integer d, it’s the equivalent of start-ing with c and then addstart-ing c repeatedly a certain number of times. How many times? Give an example.
Answer 5-1
The product cd is equivalent to starting with c and then adding c a total of (d− 1) times. For example, we get 5 × 12 when we start with 5 and then add 5 over and over, a total of 11 times.
Another way to look at this is to imagine that 5 × 12 is what we get when we start with 0 and then add 5 repeatedly, a total of 12 times.
Question 5-2
What happens if we multiply a positive integer p by a negative integer n? How can we describe that in terms of repeated addition?
Answer 5-2
It works the same way as it does when adding a positive integer. The only difference is that, as we keep adding the negative integer repeatedly, the sum gets smaller instead of larger.
Question 5-3
In which of the following quotients is there a remainder?
(a) 20/10 (b) 33/11 (c) 51/17 (d) 95/19 (e) 105/21 (f ) 116/29 (g) 218/31 (h) 301/43 Answer 5-3
There is a remainder only in case (g). It’s easy to see this by using a calculator to divide the numerator by the denominator in each of the expressions.
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Question 5-4
Suppose a mathematician says, “The operations of addition, subtraction, and multiplication are closed over the set of integers, but the operation of division is not.” What does he/she mean by this?
Answer 5-4
He/She means that we always get an integer if we add, subtract, or multiply one integer by another. But he/she also warns us that we don’t always get an integer if we divide one integer by another.
Question 5-5
Which of the following “rules” is actually false? How can its wording be changed to make it right?
(a) When we multiply a positive integer by 2 or more, the result stays positive and the absolute value increases.
(b) When we divide a positive integer by 2 or more, the result stays positive and the absolute value decreases.
(c) When we multiply a negative integer by −2 or less, the result stays negative and the absolute value increases.
(d) When we divide a negative integer by −2 or less, the result becomes positive and the absolute value decreases.
Answer 5-5
All of the above “rules” are true except (c). Remember that if we multiply a negative by a negative, we get a positive! The correct way to state this rule would be, “When we multiply a negative integer by −2 or less, the result becomes positive and the absolute value increases.”
Question 5-6
Sometimes we’ll come across an expression that doesn’t contain parentheses, brackets, or braces. This can be confusing if we don’t know the order in which the operations should be done. Suppose we see this:
6× 8 − 14/2 + 3 What number does this represent?
Answer 5-6
Remember the rules for precedence of operations when we see expressions without parenthe-ses. The steps go in this order:
• Do all the multiplications.
• Do all the divisions.
• Convert all the subtractions to negative additions.
• Do all the additions.
The product should be found first, and then the ratio. Then the subtraction should be changed to negative addition, and finally the additions should be done. This gives us
6 × 8 − 14/2 + 3 = 48 − 14/2 + 3
= 48 − 7 + 3
= 48 + (−7)+ 3
= 41 + 3
= 44 Question 5-7
How can we change the expression in Question 5-6 to indicate that the subtraction should be done first, then the multiplication, then the division, and finally the last addition? What will the result be then?
Answer 5-7
We should place an opening parenthesis to the left of the 8 and a closing parenthesis to the right of the 14, like this:
6× (8 − 14)/2 + 3
Now we’ve isolated the subtraction problem so it must be done first. We don’t need to change it to negative addition in this case, because there’s no risk of ambiguity with the subtraction
Now we’ve isolated the subtraction problem so it must be done first. We don’t need to change it to negative addition in this case, because there’s no risk of ambiguity with the subtraction