READING:    II    §4    EXISTENCE OF SQUARE ROOTS                Back to reading:    II    §3    DECIMALS
begin quote ita"
CHAPTER II [:] The Real Number System [...] §4. THE EXISTENCE OF SQUARE ROOTS. "end quote

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THE EXISTENCE OF SQUARE ROOTS (
UNDER CONSTRUCTION - beware of and spot the unlinked concepts:   This page will serve as both the last page and the first page for my 3rd reading of chapter two. The '1st page' status will require many links to be added which would not be needed if the page only served as last. The page will be permanently unfinished because the task of linking all the concepts is very far from done.
)

Please note: The "square" of a given number is the number we get when we multiply the given number by itself: , where the center dot indicates multiplication and where the x with superscript 2 is a symbol for the squared number.

begin quote ita-2-4-2-pg28 "
A square root of a given number is a number whose square is the given number. Since [see items (3) and (5) in the table] the square of any nonzero number is positive [see the order property], only non-negative numbers can have square roots. The number zero has one square root, which is zero itself.
"end quote

It is also convenient here to say that the number one has one square root, which is one itself.

begin quote ita-2-4-2-pg28 "
Proposition.   Every positive number has a unique positive square root.
"end quote

Rearrange O4, which says the following:
For any real numbers   a, b, c, d   such that    ab0,   and   cd0   we have   acbd

as follows:
if     0 b a    and    0 d c,     then    bd ac ,

and replace:

to get:
begin quote ita-2-4-3-pg28 "
If       then    . That is, bigger positive numbers have bigger squares. Thus any given real number can have at most one positive square root.
"end quote

The truth of the last part of the quote immediately above, the "at most" part, was not immediately evident to me - in spite of the fact that I have worked through this material before. But two candidate square roots for a given positive real number must have the property that one is larger than the other. Thus, the larger one has a larger square so that the two candidates can not be square roots of the given number.

begin quote ita-2-4-3-pg28 "
It remains to show that if     then   a   has at least one positive square root. For this purpose consider the set

.
"end quote

I found thinking about the set   S   to be quite difficult, and I found it helpful to look at a few examples:

 a is less than 1: a = 1/4 = a = 1/9 = a = 1/16 = ------------------ a is greater than 1: a = 16 = a = 9 = a = 4 =

begin quote ita-2-4-3-pg28 "
This set   []   is nonempty, since   ,   and bounded from above, since if     we have   .
"end quote

It took me quite a while to see this last point immediately above, even though I understood each of the steps from   "x squared"   to   "greater than a". It was these two ends read together,   ,   that made me understand.

begin quote ita-2-4-3-pg28 "
Hence   [by the least upper bound property]     exits. We proceed to show that   .
"end quote

The next parts of this argument will be presented in tables with the steps in the left column and the justifications in the right column. Recall, from the material above that,     and   . Also: In the table below, 'min' is an abbreviation for 'minimum' - meaning that the lesser of the two elements in the given set is to be chosen. This problem has been stated in such a way that the two elements will not be equal to one another.

begin quote ita-2-4-3-pg28 "
First     [because both 1 and a are greater than zero so that]   ,   since

[The 1st column here is the quote.]
reasons for step 1

O4:    ,
case I :     so that
 a 1 a a Yes 0 e f 0 g h eg fh
case II:     so that
The expression at the left reduces to     with answer   Yes
identidy element for multiplication
See the two cases above.   I: equality.   II: inequality.
steps
reasons
"end quote

begin quote ita-2-4-3-pg28 "
Next, for any     such that     we have     so   , since bigger positive numbers have bigger squares.
"end quote

We get the first part,   ,   from the definition of the less than symbol. We get the next two parts,     and   ,   from O3 as shown in the table immediately below.

O3:   For any real numbers   a, b, c, d:     ab,   and   cd   imply that   a+cb+d.

b
a dc
b
+
d a + c
O3 re-arranged
-0 y= y -
+
y 0 + y
application
b
a dc
b
+
d a + c
O3 re-arranged
0
y= y
0
+ y + y
application

begin quote ita-2-4-3-pg28 "
By the definition of   y   there are numbers greater than     in   S   [],   but   .   Again using the fact that bigger positive numbers have bigger squares,
 we get .
"end quote

We will need here to negate an inequality. Then look at the photo here.

begin quote ita-2-4-3-pg28 "
 Hence
"end quote

And:

begin quote ita-2-4-3-pg28 "
so    .
"end quote

begin quote ita-2-4-3-pg28 "
The inequality     holds for any     such that   ,   and by choosing     small enough we can make     less than any preassigned positive number. Thus     is less than any positive number. Since   ,   we must have   ,   proving   .
"end quote

(which is to say: CONSTRUCTION SITE - beware of unlinked concepts)

on to Chapter 3: Metric Spaces