Math 100
Square Roots
Martin H. Weissman
1.
The real square root of two
Since we have not thoroughly discussed it, we begin by mentioning the
intermediate value theorem
for
polynomials:
Suppose that
P
(
x
) is a polynomial with real coeﬃcients, with one variable
x
. Suppose that
a,b
∈
R
, and let
u
=
P
(
a
), and
v
=
P
(
b
). Suppose that
a < b
, and
u < v
. If
u < w < v
,
then there exists
c
∈
R
, such that
a < c < b
, and
P
(
c
) =
w
.
Rather than proving such a theorem, we take it as an
axiom
: a fact about numbers which we assume. This
axiom guarantees that every positive real number has a real square root:
Theorem 1.
If
y
is a real number, and
y >
0
, then there exists a real number
x
such that
x
2
=
y
.
Proof.
Suppose that
y
is a positive real number. Let
P
be the polynomial
P
(
x
) =
x
2
. Then, 0
< y
, and
P
(0)
< P
(
y
+ 1), since
P
(0) = 0 and
P
(
y
+ 1) = (
y
+ 1)(
y
+ 1) and the product of positive numbers is
positive.
We claim that 0
< y < P
(
y
+ 1). Note that 0
< y
since
y
is positive. Moreover, we can compare
P
(
y
+ 1)
and
y
as follows:
P
(
y
+1) =
y
2
+2
y
+1, and so
P
(
y
+1)

y
=
y
2
+
y
+1. Since 1,
y
, and
y
2
are all positive,
it follows that
P
(
y
+ 1)

y
is also positive. Hence
P
(
y
+ 1)

y >
0, and so
P
(
y
+ 1)
> y
. We have proven
that 0
< y < P
(
y
+ 1).
Now, we may apply the intermediate value theorem: since
P
(0)
< y < P
(
y
+ 1), and 0
< y
+ 1, we can
choose
x
∈
R
such that 0
< x < y
+ 1 and
P
(
x
) =
y
. Therefore
x
2
=
y
.
/
Note that the above proof used many of the basic axioms for real numbers. Moreover, it was not a completely