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May 18th

May 18th

# College Algebra and Trigonometry

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.

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