Sets of Numbers
Problem with Zero and Division
begin with the various types of numbers called Real Numbers. Together, these numbers can be ordered and
create a solid line, without gaps.
Numbers: These are counting numbers, the
smallest of which is 1. There is not a
largest Natural Number.
Numbers: All of the natural numbers and zero.
Zero is the only number that is a Whole Number but not a Natural Number.
Integers: The integers are all of the Whole Numbers and
their opposites. For example, the
opposite of 11 is -11.
Numbers: A Rational Number is a ratio of
two integers. All of the integers, whole and natural numbers are rational.
that terminate or repeat (have patterns) are rational as they can be written as
a ratio of integers.
Numbers: A number that cannot be written as a ratio of two integers is
irrational. Famous examples are π, and
the square root of a prime number (which will be discussed next).
these make up the Real Numbers. The
name, Real, is a misnomer, leading people to conclude that the word real in this context has the same
definition as used in daily language.
That misconception is only strengthened when the Imaginary numbers are
introduced, as the word imaginary here
harkens back to a day when the nature of these numbers, and their practical
use, was unknown.
Is zero rational?
rational number is a number that is the ratio of two integers. Before we tackle the issues that arise from
zero, let’s reframe how we think about rational numbers (fractions) and develop
a different language for these to promote greater proficiency in Algebra and
allow for greater ease in understanding how zero causes real problems with
rational numbers. (If you understand the
nature of what follows you do not have to memorize or remember the tricks, you
the fraction . You
were likely taught to think of this fraction as division and would also likely
be taught to ask the question, “How many times does two go into eight?” That is sufficient for this level of
mathematics, but the Algebra ahead is seemingly more complicated, but by simply
rephrasing the language we use to talk about fractions, we can expose the
seemingly more complex as being the same level of difficulty.
of asking, “How many times does two go into eight,” the better question is,
“Two times what is eight?”
true that because
two times four is eight. Simply answer
the question “Two times what is eight,” and you’ve found the answer.
will come into play with Algebra when we begin reducing Algebraic Fractions
(also called Rational Expressions) like:
ask the question, “How many times does three x going into nine x squared,” you’ll likely be stuck,
especially when the expressions become more complicated.
asking, “three x times what is nine x squared,” is a little easier to
will be much more on reducing Algebraic Expressions later in this chapter. Let’s turn our attention to zero and how it
“behaves” in with rational numbers.
an integer, and again, a rational number is a ratio of two integers. Consider the
first expression asks, “Zero times what is five?”
second expressions asks, “Five times what is zero?” (Again, phrase the question in this fashion
to provide easier insight into the math.)
product of zero and any number is zero.
So, the answer to, “zero times what is five,” is … well, there is no
answer. There is no number times zero
that is five. There is not a number
times zero that equals anything except zero.
We say this is undefined, meaning, there is no definition for such a
second expression, “five times what is zero,” is zero. Five times zero is zero.
these two expressions is rational, the other is not a number at all. It does not just fail to fit within the Real Numbers,
it fails to fit in with any number.
Written as Fractions
the fraction This is a rational number because it is the
ratio of two integers, 1 and 3. Yet, the
decimal approximation of one-third is (the bar above the three means it is repeating
how to express a repeating decimal as a fraction. Let us begin with the number .
We don’t know
what number, as a fraction is , so we will write the unknown x.
repeating after the hundredths place, we will multiply both sides of the
equation by 100.
0.333333… we would multiply by 10, since the decimal repeats after the 10ths
place, but we would multiply 0.457457457457…by 1,000 since it repeats after
the thousandths place.)
The following step is done by a
procedure learned with solving systems of equations, which will be covered
later. (In fact, this procedure would be a great topic to review when systems
of equations is learned.)
first equation from the second.
Divide both sides
by 99 to solve for x.
Recall that x was originally defined as the
fractional equivalent of the repeating decimal.
Change the following into rational numbers:
is a the following called undefined: ?
all of the sets of numbers to which the following numbers belong:
0 b. 9
c. -5 d. e.
a rational number also be a whole number?
number is whole but not natural?