01 Irrational Numbers
Section 1.1 gives a proof of the Fundamental Theorem of Arithmetic and uses
it to show that various real numbers are irrational. The proof of this
theorem is not important for this course, and the theorem itself
is only used to prove the existence of irrational numbers. The exercises in
Section 1.1 explore some other facts about irrational numbers.
(Note that the historical background information in this section, and elsewhere
in the textbook, is not required for this course. But it can be interesting to
read it anyway.)
Mathematicians work with various "number systems." The set of natural numbers,
is the most basic. This set contains the positive integers:
Adding the negative integers and zero, we get the set
of integers:
Two natural numbers can be added but not always subtracted.
In some sense, is invented to make subtraction possible; when you
subtract a natural number from a smaller natural number, the result is not a natural number, but it is
a negative integer. We say that the natural numbers are closed
under addition but not closed under subtraction. The set of integers is closed
under both addition and subtraction.
Simlarly, the set of integers is not closed under division.
The rational numbers are invented to make
division possible (except of course for division by zero).
The set of rational numbers contains all quotients where and
are integers and where the denominator, is not zero. Of course, the same rational
number can be represented by many different fractions; for example
If you want to
be fancy, a rational number can be defined as an equivalence class of fractions
where is equivalent to if and only if
Every non-zero rational number can be represented uniquely as a fraction
where is a positive integer and and have no common factors.
The set of real numbers, is different.
If the motivation for inventing and is arithmetic, the motivation
for inventing is more geometric. The set of real numbers is identified with
a geometric object, a "line." There is a one-to-one correspondence between
real numbers and points on a line. To get a correspondence, you have to decide
which two points correspond to the numbers 0 and 1, but once you do that,
the correspondence is determined. This gives the familiar picture of the
real number line:
It is not immediately obvious that you need anything more than
the rational numbers to account for all of the points on the line. However, the
geometry of the line leads us to numbers that are not rational. That is, if we define
the real numbers to be the points on a line, then there are real numbers that are not
rational. Real numbers that are not rational are called irrational.
The original geometric proof of this fact used a square whose sides have length 1.
According to the Pythagorean theorem, the diagonal of that square has length
or But cannot be a rational number.
However, it is certainly a real number. You can locate that number on the real line geometrically
simply by laying the diagonal on the line; if one endpoint of the diagonal is at zero, then the other
endpoint is at
The well-known proof that is irrational is given in the textbook. In fact, we can use the Fundamental
Theorem of Arithmetic to get a large supply of irrational numbers:
Theorem: Let be a positive integer
greater than 1, and suppose that has prime factorization
(where are distinct primes, and
are integers greater than zero). Then is rational if and only
if every is an even number.
Proof: Suppose that every is
an even number. Then where are
integers. So which is a
rational number. (Note that this square root is in fact an integer, not just a rational number.
The square root of an integer can only be an integer or an irrational number.)
Conversely, suppose that one of the exponents, say , is odd.
Suppose, for the sake of contradiction,
that is rational. Write where and are
integers. Then and
By the Fundamental Theorem of Arithmetic,
we can write and where and are non-negative integers
and and are integers that are not divisible by That is, occurs
times as a factor of and it occurs times as a factor of
(We have simply
factored all possible factors of out of and ) But then
occurs times as a factor of and it occurs times as a factor
of Since occurs an odd number of times as a factor of
and it occurs an even number of times as a factor in and in
it follows that occurs an odd number of times as a factor of
and an even number of times as a factor in But, by the Fundamental Theorem
of Arithmetic, this means that cannot equal since they have different
prime factorizations. This contradicts the fact that
and the contradiction shows that cannot be rational.
But taking square roots of integers does not produce all irrational numbers.
For example, and are irrational, but and are not integers.
The number can be defined
geometrically as the circumference of a circle that has diameter equal to 1.
The number is the base of the natural logarithm, and its definition requires
calculus.
For lots more irrational numbers, we can turn to non-repeating decimals.
(See Exercises 17 through 21 in Section 1.1.)
Real numbers are often represented in decimal form,
where is an integer and are digits, that is, integers in the range
0 to 9. (This really represents the number as the sum of an infinite
series, ) A number in decimal form is
rational if and only if it is repeating. In a repeating decimal, after some
point, there is a finite block of one or more digits that is just
repeated over and over forever. For example,
or
or The line over 3 or 51 or 0 indicates that
that digit or block of digits repeats forever.
But this means that any non-repeating decimal represents an irrational
number. For example:
0.101001000100001000001000000100000001000000001000000000...
In this decimal, the number of zeros between the ones increases by one
from each group of zeros to the next, making it impossible for the decimal to
terminate in a finite, repeating block. The ones could be replaced with other digits
to give other irrational numbers. For example, we could replace some of the
ones with twos:
0.102002000100002000001000000100000002000000001000000000...
We can choose any subset of the ones to replace with twos.
Since there is a countable number of ones, there is an uncountable number of
subsets of the ones, so we can get an uncountable number of different irrational
numbers from this one example. (This assumes that
you know about countable versus uncountable infinities, and the fact that the
set of subsets of an infinite set is uncountable. If so, you
probably know that the set of rational numbers is countable — that is, it can
be put into one-to-one correspondence with the set of natural numbers —
while the set of real numbers is uncountable. But it's nice to have a
concrete example of uncountably many irrational numbers.)
The question remains, is there some way that we can produce all of the
irrational numbers? That is the topic of the
next reading.
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