First please: Spoken Reals Point and Pictures**.** Then**:**

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Preface: -1

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

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Introduction: 0

I believe Niels Bohr said of quantum physics that you don't

Session: 1

It's the numbers and addition and multiplication.

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Session: 2

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 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 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

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 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

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

If

then

and

a + b = b + a and a · b = b · a

pictures

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Session: 8

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

If

then

adding the sum of

equals

adding

and

multiplying the product of

equals

multiplying

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

pictures

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Session: 9

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

If

then

multiplying

equals

adding

a · ( b + c ) = a · b + a · c

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Session: 10

There is a real number called zero which is a neutral element for addition.

If

There is

If

a + 0 = a and a · 1 = a

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Session: 11

For every real number

there is a real number called minus

For every real number

there is a real number called

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Session: 12

Given any two statements

The material implication is true if

The material implication is false if

The material implication is true if

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Session: 13

Now let

If

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Session: 14

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

If

are elements of R plus.

For any real number

one and only one of the following statements is true:

minus

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Session: 15

We say that the set

A real number

the following statements are true:

every other upper bound on

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Session: 16

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

**(c) 2008 Barry Davies. All Rights Reserved.**