Chapter One — Algebraic Fundamentals
The Humongous Book of Algebra Problems
2
Number Classification
Numbers fall into different groups
1.1 Describe the difference between the whole numbers and the natural numbers.
Number theory dictates that the set of whole numbers and the set of natural
numbers contain nearly all of the same members: {1, 2, 3, 4, 5, 6, …}. The
characteristic difference between the two is that the whole numbers also
include the number 0. Therefore, the set of natural numbers is equivalent to
the set of positive integers {1, 2, 3, 4, 5, …}, whereas the set of whole numbers is
equivalent to the set of nonnegative integers {0, 1, 2, 3, 4, 5, …}.
1.2 What set of numbers consists of integers that are not natural numbers? What
mathematical term best describes that set?
The integers are numbers that contain no explicit fraction or decimal.
Therefore, numbers such as 5, 0, and –6 are integers but 4.3 and are not.
Thus, all integers belong to the set {…, –3, –2, –1, 0, 1, 2, 3, …}. According to
Problem 1.1, the set of natural numbers is {1, 2, 3, 4, 5, …}. Remove the natural
numbers from the set of integers to create the set described in this problem:
{…, –4, –3, –2, –1, 0}. This set, which contains all of the negative integers and
the number 0, is described as the “nonpositive numbers.”
1.3 Is the number 0 even or odd? Positive or negative? Justify your answers.
By definition, a number is even if there is no remainder when you divide it by 2.
To determine whether 0 is an even number, divide it by 2: . (Note that 0
divided by any real number—except for 0—is equal to 0.) The result, 0, has no
remainder, so 0 is an even number.
However, 0 is neither positive nor negative. Positive numbers are defined as
the real numbers greater than (but not equal to) 0, and negative numbers are
defined as real numbers less than (but not equal to) 0, so 0 can be classified
only as “nonpositive” or “nonnegative.”
1.4 Identify the smallest positive prime number and justify your answer.
A number is described as “prime” if it cannot be evenly divided by any number
other than the number itself and 1. According to this definition, the number 8
is not prime, because the numbers 2 and 4 both divide evenly into 8. However,
the numbers 2, 3, 5, 7, and 11 are prime, because none of those numbers is
evenly divisible by a value other than the number itself and 1. Note that the
number 1 is conspicuously absent from this list and is not a prime number.
By definition, a prime number must be divisible by exactly two unique values,
the number itself and the number 1. In the case of 1, those two values are equal
and, therefore, not unique. Although this might seem a technicality, it excludes
1 from the set of prime numbers, so the smallest positive prime number is 2.
The natural
numbers are also
called the “counting
numbers,” because
when you read them,
it sounds like you’re
counting: 1, 2, 3, 4, 5,
and so on. Most people
don’t start counting
with 0.
So you
get the
whole numbers
by taking the
natural numbers
and sticking 0
in there.
Numbers, like
8, that aren’t prime
because they are
divisible by too many
things, are called
“composite numbers.”
Chapter One — Algebraic Fundamentals
The Humongous Book of Algebra Problems
3
1.5 List the two characteristics most commonly associated with a rational number.
The fundamental characteristic of a rational number is that it can be expressed
as a fraction, a quotient of two integers. Therefore, and are examples
of rational numbers. Rational numbers expressed in decimal form feature
either a terminating decimal (a finite, rather than infinite, number of values
after the decimal point) or a repeating decimal (a pattern of digits that repeats
infinitely). Consider the following decimal representations of rational numbers
to better understand the concepts of terminating and repeating decimals.
615384
615384615384
1.6 The irrational mathematical constant p is sometimes approximated with the
fraction
. Explain why that approximation cannot be the exact value of p.
When expanded to millions, billions, and even trillions of decimal places,
the digits in the decimal representation of p do not repeat in a discernable
pattern. Because p is equal to a nonterminating, nonrepeating decimal, p is an
irrational number, and irrational numbers cannot be expressed as fractions.
1.7 Which is larger, the set of real numbers or the set of complex numbers?
Explain your answer.
Combining the set of rational numbers together with the set of irrational
numbers produces the set of real numbers. In other words, every real number
must be either rational or irrational. The set of complex numbers is far larger
than the set of real numbers, and the reasoning is simple: All real numbers are
complex numbers as well. The set of complex numbers is larger than the set of
real numbers in the same way that the set of human beings on Earth is larger
than the set of men on Earth. All men are humans, but not all humans are
necessarily men. Similarly, all real numbers are complex, but not all complex
numbers are real.
Little bars
like this are
used to indicate
which digits of a
repeating decimal
actually repeat.
Sometimes, a few digits
in front won’t repeat,
but the number is still
rational. For example,
is a rational number.
Complex
numbers are
discussed in
more detail later
in the book, in
Problems 13.37-
13.44.
Chapter One — Algebraic Fundamentals
The Humongous Book of Algebra Problems
4
1.8 List the following sets of numbers in order from smallest to largest: complex
numbers, integers, irrational numbers, natural numbers, rational numbers,
real numbers, and whole numbers.
Although each of these sets is infinitely large, they are not the same size. The
smallest set is the natural numbers, followed by the whole numbers, which is
exactly one element larger than the natural numbers. Appending the negative
integers to the whole numbers results in the next largest set, the integers. The
set of rational numbers is significantly larger than the integers, and the set
of irrational numbers is significantly larger than the set of rational numbers.
The real numbers must be larger than the irrational numbers, because all
irrational numbers are real numbers. The complex numbers are larger than
the real numbers, as explained in Problem 1.7. Therefore, this is the correct
order: natural numbers, whole numbers, integers, rational numbers, irrational
numbers, real numbers, and complex numbers.
1.9 Describe the number 13 by identifying the number sets to which it belongs.
Because 13 has no explicit decimal or fraction, it is an integer. All positive
integers are also natural numbers and whole numbers. It is not evenly divisible
by 2, so 13 is an odd number. In fact, 13 is not evenly divisible by any number
other than 1 and 13, so it is a prime number. You can express 13 as a fraction
, so 13 is a rational number. It follows, therefore, that 13 is also a real
number and a complex number. In conclusion, 13 is odd, prime, a natural
number, a whole number, an integer, a rational number, a real number, and a
complex number.
1.10 Describe the number by identifying the number sets to which it belongs.
Because is less than 0 (i.e., to the left of 0 on a number line), it is a negative
number. It is a fraction, so by definition it is a rational number and, therefore, it
is a real number and a complex number as well.
According
to Problem 1.1,
the single element
that the whole
numbers contain and
the natural numbers
exclude is the
number 0.
Any innitely
long decimal
that has no
pattern of repeating
digits represents an
irrational number. On
the other hand, rational
decimals either have
to repeat or terminate.
Because there are a
lot more ways to write
irrational numbers as
decimals than there
are to write rational
numbers as decimals,
there are a lot more
irrational numbers
than rational
numbers.
Any number
divided by itself
is 1, so
.
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