First please:   Spoken Reals Point and Pictures.     Then:

The Spoken Reals

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Preface: -1

This page can be translated into the spoken word. It is a seed for many oral (micro) traditions.

It is one possible foundation for numeracy, and mathematicians might call it an introduction to the real number system. My source is a Dover republication of the book Introduction to Analysis by Maxwell Rosenlicht. My first reading is presented in what I call 'peak to ground' form at the next link. After you click on the next link, just have a quick look and then keep on clicking on the first link on each page until there are no more links. This will show what I mean by peak to ground. Also, if you would like to buy the book, you can find it here.


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Introduction: 0

I believe Niels Bohr said of quantum physics that you don't understand it - you just get used to it.

I think it would be a good idea to treat this page in the same way because it took the greatest minds thousands of years to formulate these ideas. Also, I think that the mind only mathematics which I am advocating here is to the study of mathematics what thinking ahead is to playing chess. And: Demanding (of yourself) instant understanding in mathematics is like expecting to win every game.



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Session: 1

The Real Number System
(the first micro tradition)

It's the numbers and addition and multiplication.

One of my original intentions in this page was to put in one place all seven of Rosenlicht's properties of the real number system - before working out any of the consequences of the properties. This is done below along with the presentation of a few needed ideas from set theory and from mathematical logic. Another intention was deductively to create the real number system from scratch - on a clean slate in our mind's eye. This means we start with nothing - as if we have forgotten everything we ever knew about numbers, and the statement that begins this session is a preview. The emphasis on the spoken word has more benefit than the analogy to chess mentioned above. I find that too much focus on the written word introduces material that is not essential. This extra stuff usually has to do with the names we give to concepts rather than the concepts themselves. The translation of the page into the spoken word does not mean reading the page aloud. It means telling someone the material from memory by voice alone, or voice and scratch pad. Thinking of those who have nothing, my favorite scratch pad would be the earth itself - with a pointed stick as pencil.




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Session: 2

Ordered Pairs & Sets

A set is a collection of things where order does not matter, and we call the things elements.

An ordered pair is a set of two elements where order matters:
There is a first element and a second element, and
if you switch them you have a different ordered pair.




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Session: 3

The Null Set

The set without any elements is very important, and it is called the null set.

At this point, we don't have any specific numbers, and this means that
the 'set of objects' in the real number system is the null set.




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Session: 4

The Cartesian Product of Two Sets ...

... is the set of all ordered pairs that can be formed in which
the first element of each pair is taken from the first set, and in which
the second element of each pair is taken from the second set.




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Session: 5

Functions from One Set to Another

Given two sets,
a function from one to the other does the following:
Given ANY element of the first set, ONE AND ONLY ONE element of the second set is determined.




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Session: 6

The Real Number System
(a more detailed description)

The real number system is a set of objects we call the real numbers together with two functions
from
the Cartesian product of the real numbers with the real numbers
to
the real numbers.

The two functions are called addition and multiplication.




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Session: 7

Field Property - Commutativity

Addition and Multiplication are commutative, and this means the following:

If   a   and   b   are any real numbers,
then
a plus b     equals     b plus a,
and
a times b     equals     b times a.

Visualize from earth, chalkboard, computer, or mobile phone:

a + b    =    b + a             and             a · b    =    b · a

pictures




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Session: 8

Field Property - Associativity

Addition and Multiplication are associative, and this means the following:

If   a,   b,   and   c   are any real numbers,

then

adding     the sum of a and b     plus     c
equals
adding     a     plus     the sum of b and c

and

multiplying     the product of a and b     times     c
equals
multiplying     a     times     the product of b and c.

Visualize from earth, chalkboard, computer, or mobile phone:

( a + b ) + c    =    a + ( b + c )             and             ( a · b ) · c    =    a · ( b · c )

pictures




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Session: 9

Field Property - Distributivity

Multiplication is Distributive over Addition, and this means the following:

If   a,   b,   and   c   are any real numbers,
then
multiplying     a     times     the sum of b and c
equals
adding     a times b     plus     a times c.

Visualize from earth, chalkboard, computer, or mobile phone:

a · ( b + c )    =    a · b + a · c




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Session: 10

Field Property - Existence of Neutral Elements

There is a real number called zero which is a neutral element for addition.
If   a   is any real number, then   a plus zero   equals   a.

There is another real number called one which is a neutral element for multiplication.
If   a   is any real number, then   a times one   equals   a.

Visualize from earth, chalkboard, computer, or mobile phone:

a + 0   =   a             and             a · 1   =   a




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Session: 11

Field Property - Existence of Inverses

For every real number   a,
there is a real number called   minus a   such that
a   plus   minus a      equals      zero.

For every real number   a   that is not zero,
there is a real number called   a to the minus 1   such that
a   times   a to the minus 1      equals      1.

Visualize from earth, chalkboard, computer, or mobile phone:

a + (-a)   =   0             and             a · (a to the minus 1)   =   1




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Session: 12

Material Implication

In previous sessions we let   a   and   b   stand for real numbers.   Now let   a   and   b   stand for statements that must be either true or false.   Material implication is a restriction of meaning that says the statement   'If a, then b'   means the following, and it means NOTHING MORE THAN THE FOLLOWING:   'If a, then b'   means that we can't have the situation where   a   is true and   b   is false.   Another way of saying a material implication is to say that   'a implies b'.

Given any two statements   a   and   b   that must be either true or false, we have four possibilities for the truth of the material implication   'a implies b'.   The truth of the implication is determined by whether or not   a   is true and   b   is false.   This is the case when the material implication is false, and in all other cases it is true.   This is because we restricted the meaning of   'If a, then b'.   Consequently we have the following truth listing for material implication.

The material implication is true if   a   and   b   are both true.

The material implication is true if   a   is false and   b   is true.

The material implication is false if   a   is true and   b   is false.

The material implication is true if   a   is false and   b   is false.




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Session: 13

Subsets

Now let   a   and   b   be sets.
a   is a subset of   b   if the following material implication is true:
If   x   is an element of   a,   then   x   is also an element of   b.




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Session: 14

The Order Property

There is a subset of the real numbers called   R plus   such that the following statements are true:

If   a   and   b   are elements of   R plus,   then both
a plus b    and    a times b
are elements of   R plus.

For any real number   a,
one and only one of the following statements is true:
a   is in   R plus,
a   is equal to   zero,   or
minus a   is in   R plus.




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Session: 15

Remember One Thing About Numbers

In order to be mathematically precise, we would need to work out the consequences of the field and order properties before we state the least upper bound property.   However, it seems very desirable to state all seven properties before getting into the details of working out consequences.   Therefore, let's remember one thing we knew about numbers before we started.   Some numbers are bigger than others.   With this fact remembered, we can informally define an upper bound on a set of real numbers as follows:

A real number   a   is an upper bound of a set   s   of real numbers ... if
a   is greater than or equal to every element of   s.

We say that the set   s   is bounded from above.

A real number   a   is a least upper bound of a set   s   of real numbers ... if
the following statements are true:
a   is an upper bound of   s,   and
every other upper bound on   s   is greater than or equal to   a.




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Session: 16

The Least Upper Bound Property

Any nonempty set of real numbers that is bounded from above has a least upper bound.




Consequences

(c) 2008 Barry Davies. All Rights Reserved.