Rational and irrational numbers
A rational number is a number that can be written as a
fraction, with both the numerator and
the denominator of the fraction being integers. (Negative integers are okay, and
0 is okay in the
numerator.) For example, 
are all rational
numbers. So are 0.6391 and -14.92,
because these can be written as the fractions 
All
integers are rational numbers,
because they can be written as fractions with denominator 1.
It is perhaps surprising to learn that there are some
numbers that cannot be written as fractions
with integer numerators and denominators. Such numbers are called irrational.
Examples of irrational
numbers include the numbers π, e, and 
. Every
real number is either rational or irrational
(but not both).
The ancient Greeks, and in particular the students of the
Greek mathematician Pythagoras (for
whom the Pythagorean theorem is named), believed that the whole universe could
be understood
through integers. They believed that every number was either itself an integer
or could be written as
a fraction of two integers-in other words, they believed that every number was
rational. However,
some time around the fifth century BC one of the Pythagoreans , a man named
Hippasus, proved
the startling fact that cannot be written as
a fraction of two integers. According to legend,
when he revealed his discovery to the other Pythagoreans, it disturbed their
philosophy so much
that Hippasus was drowned for heresy!
Properties of rational and irrational numbers
Rational numbers have decimal expansions that either
terminate or repeat some sequence of digits
over and over again forever. For example, the decimal expansion of the rational
number 
is 0.3125:
it terminates after four decimal places. The decimal expansion of
the sequence "142857" repeats over and over again, endlessly. Sometimes there
are some non-
repeating digits before the "block" of repeating digits begins. For example, the
decimal expansion
of 
the 3's repeat forever, but the 5 and the
8 only appear once. (In fact you
can think of terminating decimals as simply a special case of repeating
decimals, if you imagine a
bunch of 0's on the end. 
Irrational numbers, on the other hand, have decimal
expansions that never terminate and never
settle into a pattern of some repeating sequence of digits. For example, the
first 100 digits of π are
The decimal expansion π of continues forever, and it never
begins repeating the same sequence of
digits over and over. This is because π is irrational. For two other examples,
here are the first
100 digits of e and
We can use the fact that the decimal expansions of
irrational numbers do not repeat in order
to construct irrational numbers. For example, the number
is irrational, because there is no fixed sequence of
digits which repeats over and over. There is a
pattern to the digits of this number, but because there is one more 0 inserted
each time a 5 occurs,
it is not a repeating decimal, so it is irrational.
There are no "gaps" in the set of rational numbers.
between any two rational numbers there
is another rational number. For example, 2/3 is a rational number between 1/2
and 3/4 . Likewise, there
are no "gaps" in the set of irrational numbers. between any two irrational
numbers there is another
irrational number. In fact, between any two real numbers whatsoever, there is
some rational number
and some irrational number (actually infinitely many of them). Mathematicians
say that the set of
rational numbers is dense in the set of real numbers, and so is the set of
irrational numbers.
Square roots of positive integers are often irrational
numbers. The very first irrational number
discovered was 
, a square root of a positive
integer . In fact, whenever a square root of a positive
integer is not itself an integer, it is irrational. For example,
are integers
so these square roots are rational numbers;
but
and
are not integers, so they are irrational
numbers. (This is a special property of square
roots-it's not true in general. For instance, 22/7 is not an integer, but it is
a rational number.)
Irrational numbers may seem to be rather strange
creatures. Most of the numbers we use on a
daily basis are rational numbers, with a few notable exceptions like π, e, and
. It may seem that
there are "more" rational numbers than irrational numbers-that the irrational
numbers are quirky
"exceptions" hidden among the rational numbers. But the previous para graph shows
that there are
infinitely many of both of them. In fact, there is a meaningful sense in which
it can be said that
there are infinitely many times as many irrational numbers as rational numbers!
So actually it is
the rational numbers which are unusual-when you consider the set of all real
numbers, "almost all"
of them are irrational.
The Rational Zeros Theorem
Rational numbers are usually easier to work with than
irrational numbers. When we are trying
to solve a polynomial equation, it is nice if we can find some solutions that
are rational numbers.
Finding solutions that are irrational numbers is often more of a challenge. But
once we find one
solution to a polynomial equation , we can " divide out " that solution to get a
simpler equation to
solve. The textbook calls this simpler equation a depressed equation; see
Section 5.5 for details.
The Rational Zeros Theorem, covered in Section 5.5 of the
textbook, is a way to determine all
of the possible rational zeros of a polynomial. It gives us an explicit list of
the only possible zeros of
the polynomial which are rational numbers. Any zeros of the polynomial that are
rational numbers
must be in this list; any zeros of the polynomial that are not in this list must
be irrational numbers.
Not all of the rational numbers in the list produced by
the Rational Zeros Theorem will be zeros
of the polynomial; in fact, it's possible that none of them are. Conversely, not
all of the zeros of
the polynomial have to be one of the numbers in the list produced by the
Rational Zeros Theorem;
if the polynomial has some zeros which are irrational, then those zeros will not
appear in the list,
because the list contains only rational numbers.
Open questions
There are still many things which are not known about
rational and irrational numbers.
For example, Johann Heinrich Lambert proved in 1761 that
the numbers π and e are irrational.
In 1794 Adrien-Marie Legendre proved that 
is
irrational. Alexander Gelfond and Theodor Schneider
independently showed in 1934 that the number 
(that is, e raised to the power of π) is irrational.
However, it is still unknown whether any of the numbers
are rational. It seems
un likely that any of these can be written as a fraction of two integers, but so
far no one has been able
to prove that it's impossible. If any of these numbers is rational, then that
indicates a surprising
(and so far unexplained) relationship between e and π.
There is a number called the Euler-Mascheroni constant,
often written γ, which is important
in higher mathematics. It was first described in 1735 by the Swiss mathematician
Leonhard Euler.
Its value is approximately
The number γ is useful because of the fol lowing
approximation , which becomes a better and better
approximation as the value of n gets larger and larger.
No one knows whether γ is rational or irrational. However,
it has been shown that if it is rational,
its denominator (in lowest terms) must be greater than
. (This is an unbelievably huge
number!)
Another interesting question is how "random" the digits of
π are (or the digits of any other
irrational number, for that matter). Since the digits of never repeat and appear
to have no
particular pattern, it would be natural to suppose that every digit from 0
through 9 appears in
the decimal expansion of π just as often as every other digit. 1/10 of the
digits of π should be 0's,
another 1/10 should be 1's, and so on. Similarly, it would be natural to suppose
that every two-digit
sequence 00, 01, 02, 03, . . . , 99 occurs equally often, and every three-digit
sequence 000, 001, 002,
003, . . . , 999, and so on. It would be surprising if there were some bias to
the digits of π (if, for
example, the digit 6 occurs slightly more often than the other digits).
Mathematicians expect that
there is no such bias in the digits of π , but again no one has yet been able to
prove that this is
actually the case. (A number in which the digits show no bias in this sense is
called a normal
number. Very few numbers have been proven to be normal.)
Open questions like these are part of what makes
mathematics interesting. These are questions
to which no one in the world knows the answers.
Problems
1. Show that -74.816 is a rational number by writing it as
a fraction of two integers.
2. Find a rational number between 4/7 and 3/5 . Find an irrational number
between 6.4 and 6.5.
3. Let 
(a) Use the Rational Zeros Theorem to list all possible
rational zeros of f.
(b) Use the Intermediate Value Theorem to prove that f must have a zero
somewhere between
4 and 5.
(c) Explain why the zero described in part (b) must be an irrational number.
(d) Find one rational zero of f.
(e) "Divide out" the rational zero you found in part (d). Find the zeros of the
resulting
polynomial; these are the remaining zeros of f.
(f) Check that the zeros of f you have found really are zeros of f. Explain how
you know that
you have found all of the zeros of f.
(g) Which of the zeros that you found is between 4 and 5? In part (c) you
explained that this
zero must be irrational. Now that you have an exact expression for it, why does
it make
sense that it should be irrational? In other words, by looking at what the zero
actually is,
give a different justification for the claim that it's irrational.
