MATH 135
Fall 2008
Algebraic and Transcendental Numbers
Number Systems
So far this term, we have looked at a number of different number systems:
Z
(the integers): the set of all whole numbers, including positive, negative and zero
Q
(the rational numbers): the set of all fractions of integers
R
(the real numbers): the set of all decimal numbers, including positive, negative and zero
C
(the complex numbers): the set of all complex numbers
We know that these form a chain:
Z
⊂
Q
⊂
R
⊂
C
. Each of these number systems includes numbers that
the previous one does not.
Examples
1
2
∈
Q
, but
1
2
∈
Z
.
2
i
∈
C
, but 2
i
∈
R
.
Can you remember a number
x
with
x
∈
R
but
x
∈
Q
?
x
=
√
2 works, and thus
√
2 is called
irrational
. We proved this in the first couple of weeks of term, but
here is another proof of this fact.
Theorem
√
2 is irrational.
Proof
Suppose that
r
=
√
2 is rational.
We know that
r
is a root of
x
2

2 = 0.
What are the possible rational roots of the polynomial
f
(
x
) =
x
2

2?
Suppose that
x
=
a
b
is a rational root with
a, b
∈
Z
and gcd(
a, b
) = 1.
By the Rational Roots Theorem,
b

1 and
a

2, so
b
=
±
1 and
a
=
±
1 or
±
2.
This means that
a
b
=
±
1 or
±
2.
But none of these square to give 2, so none equals
√
2.
Thus,
f
(
x
) =
x
2

2 has no rational roots, so
√
2 is irrational.
Note
We could use a similar method to prove that
√
5 and
3
√
2 are irrational.
Numbers as Solutions of Polynomial Equations
We can justify the need for each new system by looking at different polynomial equations with integer
coefficients that have or do not have solutions in that system:
x

2 = 0 has integer solutions
3
x

2 = 0 does not have integer solutions but does have rational solutions
x
2

This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
This is the end of the preview.
Sign up
to
access the rest of the document.
 Fall '08
 ANDREWCHILDS
 Algebra, Integers, real solutions

Click to edit the document details