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<div align="center">
<h3><a name="1: Numbers"></a><a href="#MATH">1: Numbers</a></h3>
</div>
<a href="#1A.">1A. The Counting (Natural) Numbers</a>
<p>
<a href="#1B.">1B. Lists and Names</a>
<p>
<a href="#1C.">1C. Iteration (Repetition)</a>
<p>
<a href="#1D.">1D. Inverse</a>
<p>
<a href="#1E.">1E. Addition and Subtraction (+ and -)</a>
<p>
<a href="#1F.">1F. Bonding<p>
<a href="#1G.">1G. Multiplication or Times (*)<p>
<a href="#1H.">1H. Power and Exponent (^)<p>
<a href="#1I.">1I. Monomials and Polynomials (p.)<p>
<a href="#1J.">1J. Division (%)<p>
<a href="#1K.">1K. Review and Supplement<p>
<a href="#1L.">1L. Notes<p>
<a name="1A."></a>
<h3><a href="#1: Numbers">1A. The Counting (Natural)
Numbers</a>
<a href="#S1A.">
<font size="-1">S1A.</font>
</a>
</h3>
The <i>Counting numbers</i> begin with <code>1 2 3 4</code>
and go on forever, because every
counting number has a <i>successor</i> that comes next after
it. Thus:
<pre>
>: 1
2
>: 2
3
>: 3
4
>: 4
5
>: >: 1
3
>: >: >: >: 2
6
>: 1 2 3 4 5 6
2 3 4 5 6 7</pre>
<p><code>>:</code> "performs an action" upon the number
to which
it is applied, and is therefore analogous to an "action
word" or
<i>verb</i> in
English. In math, such a verb is also called a
<i>function</i> (from
Latin <i>fungi</i>, to perform or execute).
<p>The number to which a verb applies may be called a
<i>noun</i>. In
math it is also called an <i>argument</i>, in the sense
of a
<i>topic</i> or <i>subject</i> ("You and love are still
my
argument" -- Shakespeare).
<a name="1B."></a>
<h3><a href="#1: Numbers">1B. Lists and Names</a>
<a href="#S1B.">
<font size="-1">S1B.</font>
</a>
</h3>
A collection of numbers may be written as a <i>list</i>, in
the form <code>2 3 5 7 11</code>, and verbs may be applied
to
such lists. For example:
<pre> >: 2 3 5 7 11
3 4 6 8 12
>: >: >: 2 3 5 7 11
5 6 8 10 14</pre>
A <i>name</i> may be assigned to a number or list by using
the <i>copula</i> <code>=:</code>, and the name may then be
used
to refer to its <i>referent</i>. For example:
<pre> primes=: 2 3 5 7 11
>: primes
3 4 6 8 12
b=: >: primes
>: >: b
5 6 8 10 14</pre>
The main copulas used in English are "is" and "are". They
may
be used in reading mathematical expressions aloud, as in
"primes are
<code>2 3 5 7 11</code>"
for <code>primes=:2 3 5 7 11</code>, and "first is one" for
<code>first=:1</code>.
<p>Names may also be assigned to verbs. There appears to be
no
English word that corresponds to <code>>:</code>
except, perhaps,
the command "next" used to summon a successor from a
queue.
Thus:
<pre> next=: >:
next 5
6
next next 5
7</pre>
<a name="1C."></a>
<h3><a href="#1: Numbers">1C. Iteration (Repetition)</a>
<a href="#S1C.">
<font size="-1">S1C.</font>
</a>
</h3>
The expression <code>next ^: 3</code> produces a verb that
is
equivalent to applying the verb <CODE>next</code> <b>for</b>
three
times.
Thus:
<pre> primes
2 3 5 7 11
next ^: 3 primes
5 6 8 10 14
next next next primes
5 6 8 10 14
for=: ^:
next for 3 primes
5 6 8 10 14
list=: 1 2 3 4
next for list primes
3 4 6 8 12
4 5 7 9 13
5 6 8 10 14
6 7 9 11 15</pre>
<h3><a name="1D."></a>
<a href="#1: Numbers">1D. Inverse</a>
<a href="#S1D.">
<font size="-1">S1D.</font>
</a>
</h3>
The verb <code>previous=: <:</code> "undoes" the work of
the verb
<code>next=: >:</code>,
and is said to be its <i>inverse</i>. Thus:
<pre> previous=: <:
primes
2 3 5 7 11
previous primes
1 2 4 6 10
next previous primes
2 3 5 7 11
previous next primes
2 3 5 7 11
previous 3
2
previous 2
1
previous 1
0
previous 0
_1
previous _1
_2
</pre>
<p>Because <code>1</code> is the <b>first</b>
counting number, the zero (<code>0</code>)
and the negative numbers (<code>_1</code> and
<code>_2</code>)
shown above are not counting numbers. But they are
<i>integers</i>,
so called because they are unbroken or intact (in=not,
tag=touch), as
distinguished from the <i>fractions</i> (broken or
fractured)
referred
to in the preface.
<p>Adopting the seemingly-innocent notion of a verb that
undoes the
effect of <code>next</code> has, rather surprisingly,
led us out of the original domain
of counting numbers, and forced the adoption of a
broader class,
the <i>integers</i>, that includes the zero and negative
numbers.
<p>In adopting further inverses we will again experience the
same need to broaden our domain, to include
<i>rational</i>
numbers, and <i>imaginary</i> numbers. Moreover, the
surprising
effect of inverse processes is not confined to
mathematics.
<p>Consider the effect of reversing a movie projector to run
a
film backward. If the film shows a locomotive moving
forward
along a track, the result of reversal is unremarkable.
But if
it concerns the dropping of an egg on the pavement, or a
dive
from a diving-board, the result is a startling
illustration
of the important distinction between <i>reversible</i>
and
<i>irreversible</i> processes.
<p>Finally, the inverse of a function can be obtained by
using
iteration (<code>for</code>) with the right argument
<code>_1</code>.
Thus:
<pre> next for _1 primes
1 2 4 6 10
>: ^: _1 primes
1 2 4 6 10
>: ^: _1
<:
next for _3 _2 _1 0 1 2 3 primes
_1 0 2 4 8
0 1 3 5 9
1 2 4 6 10
2 3 5 7 11
3 4 6 8 12
4 5 7 9 13
5 6 8 10 14</pre>
<a name="1E."></a>
<h3><a href="#1: Numbers">1E. Addition and Subtraction (+
and -)</a>
<a href="#S1E.">
<font size="-1">S1E.</font>
</a>
</h3>
The effect of the verb <code>next for 3</code> is to
<i>add</i>
<code>3</code> to its argument, and
<code>next for 3 primes</code>
is equivalent to the addition <code>primes + 3</code> (using
the familiar Saint George's cross to denote the verb). Thus:
<pre> next for 3 primes
5 6 8 10 14
primes + 3
5 6 8 10 14
0 1 2 3 4 5 + 0 1 2 3 4 5
0 2 4 6 8 10
+ table 0 1 2 3 4 5
+-+------------+
| |0 1 2 3 4 5|
+-+------------+
|0|0 1 2 3 4 5|
|1|1 2 3 4 5 6|
|2|2 3 4 5 6 7|
|3|3 4 5 6 7 8|
|4|4 5 6 7 8 9|
|5|5 6 7 8 9 10|
+-+------------+</pre>
The last result is an <i>addition table</i>, which may be
"read"
as follows: <blockquote>
To find the result of <code>3 + 4</code>, choose the
result
(<code>7</code>)
found in the row headed by <code>3</code> and the column
headed by <code>4</code>.</blockquote>
The verb <code>+ table</code> is only one example of a
<i>function</i>
table, and other functions may be used. For example:
<pre> previous for 3 primes
_1 0 2 4 8
primes - 3
_1 0 2 4 8
- table 0 1 2 3 4 5
+-+----------------+
| |0 1 2 3 4 5|
+-+----------------+
|0|0 _1 _2 _3 _4 _5|
|1|1 0 _1 _2 _3 _4|
|2|2 1 0 _1 _2 _3|
|3|3 2 1 0 _1 _2|
|4|4 3 2 1 0 _1|
|5|5 4 3 2 1 0|
+-+----------------+</pre>
<b>
<div align="center">Exercises</div>
</b>
<ol>
<li>Read
from the addition table the sums <code>2 + 5</code>
and
<code>5 + 2</code>
and verify that they agree. Make similar comparisons
of additions
of numbers that are similarly interchanged or
<i>commuted.</i>
<p>
<li>Because
of the agreements noted in Exercise 1, addition is
said to
be <i>commutative</i>. Use the subtraction table to
find
whether subtraction is commutative.
</ol>
<a name="1F."></a>
<h3><a href="#1: Numbers">1F. Bonding (&)</a>
<a href="#S1F.">
<font size="-1">S1F.</font>
</a>
</h3>
The verb <code>+ & 3</code> is equivalent to "add
<code>3</code>",
that is, to <code>next for 3</code>. Thus:
<pre> + & 3 primes
5 6 8 10 14
primes + 3
5 6 8 10 14
with=: &
+ with 3 primes
5 6 8 10 14
next for 3 primes
5 6 8 10 14
- with 3 primes
_1 0 2 4 8
primes - 3
_1 0 2 4 8
- with 3 + with 3 primes
2 3 5 7 11
+ with 2 primes
4 5 7 9 13
+ with 2 for 0 1 2 3 4 primes
2 3 5 7 11
4 5 7 9 13
6 7 9 11 15
8 9 11 13 17
10 11 13 15 19</pre>
Although the referent of <code>primes</code> is the list
<code>2 3 5 7 11</code>, it would not be correct to
substitute the
referent for the name in the foregoing
expression, because the resulting
<code>0 1 2 3 4 2 3 5 7 11</code>
would be treated as a single list argument to
<code>for</code>.
Thus:
<pre> + with 2 for 0 1 2 3 4 2 3 5 7 11
+&2^:0 1 2 3 4 2 3 5 7 11</pre>
The lists may, however, be separated by parentheses:
<pre> + with 2 for 0 1 2 3 4 (2 3 5 7 11)
2 3 5 7 11
4 5 7 9 13
6 7 9 11 15
8 9 11 13 17
10 11 13 15 19</pre>
<a name="1G."></a>
<h3><a href="#1: Numbers">1G. Multiplication or Times
(*)</a>
<a href="#S1G.">
<font size="-1">S1G.</font>
</a>
</h3>
Three times four <code>(3 * 4)</code>
is said to be the addition of four copies ("plies") of
three. Thus:
<pre> 3 + 3 + 3 + 3
12
3 * 4
12
* table 0 1 2 3 4 5 6 7 8 9 10
+--+--------------------------------+
| |0 1 2 3 4 5 6 7 8 9 10|
+--+--------------------------------+
| 0|0 0 0 0 0 0 0 0 0 0 0|
| 1|0 1 2 3 4 5 6 7 8 9 10|
| 2|0 2 4 6 8 10 12 14 16 18 20|
| 3|0 3 6 9 12 15 18 21 24 27 30|
| 4|0 4 8 12 16 20 24 28 32 36 40|
| 5|0 5 10 15 20 25 30 35 40 45 50|
| 6|0 6 12 18 24 30 36 42 48 54 60|
| 7|0 7 14 21 28 35 42 49 56 63 70|
| 8|0 8 16 24 32 40 48 56 64 72 80|
| 9|0 9 18 27 36 45 54 63 72 81 90|
|10|0 10 20 30 40 50 60 70 80 90 100|
+--+--------------------------------+
</pre>
<b>
<div align="center">Exercises</div>
</b>
<ol>
<li>Study
the multiplication table, and comment on its
properties (such
as commutativity).<p>
<li>The numbers in column <code>5</code> are
<i>multiples</i> of <code>5</code>, that is, they
result
from multiplying by <code>5</code>. Verify that they
progress
by "counting by fives", and check for similar
properties in other
columns and rows.<p>
<li>Comment on any patterns you find in the rows,
columns, or diagonals of other function tables.<p>
<li>Make
the table <code>* table 2 3 4 5 6 6 7 8 9 10</code>
and make
a list of the counting numbers beginning with
<code>2</code> (and
up to perhaps <code>19</code>)
that <i>do not</i> occur in it. These numbers are
called
<i>primes</i>.
</ol>
<h3><a name="1H."></a>
<a href="#1: Numbers">1H. Power and Exponent (^)</a>
<a href="#S1H.">
<font size="-1">S1H.</font>
</a>
</h3>
Just as repeated application of addition is equivalent to
another important function
(multplication, or <code>*</code>),
so repeated multiplication is equivalent to <i>power</i>, or
<code>^ </code>. Thus,
<code>3 ^ 4</code> is equivalent to
<code>3 * 3 * 3 * 3</code>.
The right argument (in this case <code>4</code>)
is often called the <i>exponent</i>, and the
expression <code>3 ^ 4</code> is read as "three power
four" or "three to the power four". For example:
<pre>
3 ^ 4
81
3*3*3*3
81
0 1 2 3 4 5 ^2
0 1 4 9 16 25
^ table 0 1 2 3 4 5
+-+-------------------+
| |0 1 2 3 4 5|
+-+-------------------+
|0|1 0 0 0 0 0|
|1|1 1 1 1 1 1|
|2|1 2 4 8 16 32|
|3|1 3 9 27 81 243|
|4|1 4 16 64 256 1024|
|5|1 5 25 125 625 3125|
+-+-------------------+</pre>
<h3><a name="1I."></a>
<a href="#1: Numbers">1I. Monomials and Polynomials
(p.)</a>
<a href="#S1I.">
<font size="-1">S1I.</font>
</a>
</h3>
An expression such as <code>5*4^3</code> is called a
<i>monomial</i>
(one name) with the
<i>coefficient</i> <code>5</code>, the argument
<code>4</code>,
and the exponent
<code>3</code>; a sum of monomials is called
a <i>polynomial</i> (many names). For example:
<pre> 5 * 4 ^ 3
320
(5*4^3) + (_2*4^4) + (1*4^1)
_288</pre>
<p>
A polynomial in which the exponents in the
successive monomials are successive
integers beginning with zero, is said to be
in <i>standard form</i>, and may be expressed
using the polynomial function <code>p.</code>, with the
list of coefficients as the left argument.
For example:
<pre>
x=: 4
(2*x^0) + (3*x^1) + (4*x^2)
78
2 3 4 p. x
78</pre>
<b>
<div align="center">Exercises</div>
</b>
<ol>
<p>
<lI>
Evaluate the following expressions, and comment
on the results:
<pre> a=: 0 1 2 3 4 5
1 2 1 p. a
(a+1) ^ 2
1 3 3 1 p. a
(a+1) ^ 3
c4=: 0 1 3 3 1 + 1 3 3 1 0
c4 p. a
(a+1) ^ 4</pre>
</ol>
<h3><a name="1J."></a>
<a href="#1: Numbers">1J. Division (%)</a>
<a href="#S1J.">
<font size="-1">S1J.</font>
</a>
</h3>
Division (to be denoted by <code>%</code>) "undoes" the work
of
multiplication. For example:
<pre> a=: 0 1 2 3 4 5 6
b=: a * 2
b
0 2 4 6 8 10 12
b % 2
0 1 2 3 4 5 6
b % 3
0 0.666667 1.33333 2 2.66667 3.33333 4</pre>
Just as the inverse of addition introduced new numbers
outside the domain of the counting numbers, so
some of the results of this inverse function
lie outside of the domain of integers. These non-integral
results (such as <code>0.666667</code>) are "decimal
approximations to"
a new class of numbers, called <i>fractions</i> or
<i>rationals</i>.
<p>Just as we introduced a way to represent negative
numbers, so
we introduce a representation for rationals:
<code>2r3</code> for
the fraction two-thirds, <code>4r3</code> for the
fraction
four-thirds,
etc. Thus:
<pre> 1r3+1r3
2r3
a=: 0 1r2 1r3 1r4 1r5 1r6
a+a
0 1 2r3 1r2 2r5 1r3
a-a
0 0 0 0 0 0
a * a
0 1r4 1r9 1r16 1r25 1r36
+ table a
+---+------------------------------+
| | 0 1r2 1r3 1r4 1r5 1r6|
+---+------------------------------+
| 0| 0 1r2 1r3 1r4 1r5 1r6|
|1r2|1r2 1 5r6 3r4 7r10 2r3|
|1r3|1r3 5r6 2r3 7r12 8r15 1r2|
|1r4|1r4 3r4 7r12 1r2 9r20 5r12|
|1r5|1r5 7r10 8r15 9r20 2r5 11r30|
|1r6|1r6 2r3 1r2 5r12 11r30 1r3|
+---+------------------------------+
- table a
+---+--------------------------------+
| | 0 1r2 1r3 1r4 1r5 1r6|
+---+--------------------------------+
| 0| 0 _1r2 _1r3 _1r4 _1r5 _1r6|
|1r2|1r2 0 1r6 1r4 3r10 1r3|
|1r3|1r3 _1r6 0 1r12 2r15 1r6|
|1r4|1r4 _1r4 _1r12 0 1r20 1r12|
|1r5|1r5 _3r10 _2r15 _1r20 0 1r30|
|1r6|1r6 _1r3 _1r6 _1r12 _1r30 0|
+---+--------------------------------+
* table a
+---+--------------------------+
| |0 1r2 1r3 1r4 1r5 1r6|
+---+--------------------------+
| 0|0 0 0 0 0 0|
|1r2|0 1r4 1r6 1r8 1r10 1r12|
|1r3|0 1r6 1r9 1r12 1r15 1r18|
|1r4|0 1r8 1r12 1r16 1r20 1r24|
|1r5|0 1r10 1r15 1r20 1r25 1r30|
|1r6|0 1r12 1r18 1r24 1r30 1r36|
+---+--------------------------+
% table a
+---+---------------------+
| |0 1r2 1r3 1r4 1r5 1r6|
+---+---------------------+
| 0|0 0 0 0 0 0|
|1r2|_ 1 3r2 2 5r2 3|
|1r3|_ 2r3 1 4r3 5r3 2|
|1r4|_ 1r2 3r4 1 5r4 3r2|
|1r5|_ 2r5 3r5 4r5 1 6r5|
|1r6|_ 1r3 1r2 2r3 5r6 1|
+---+---------------------+
</pre>
<b>
<div align="center">Exercises</div>
</b>
<ol>
<li>Study the foregoing tables,
and comment on their properties. (See the Exercises
in Section
1G.)<p>
<li>
Comment on the <code>_</code> that occurs in the
first column
of the division table (it denotes <i>infinity</i>).
<p>
<li>
Study the multiplication table, and try to formulate
rules for
the multiplication of rationals.
</ol>
<h3><a name="1K."></a>
<a href="#1: Numbers">1K. Review and Supplement</a>
<a href="#S1K.">
<font size="-1">S1K.</font>
</a>
</h3>
Using notation provided by a programming language (that
will allow us to experiment with our math on a computer),
we have introduced:<ol>
<li>
The <i>counting numbers</i> <code>1 2 3 4</code>
etc.
<p>
<li>The use of <i>lists</i> and of <i>names</i>
that are assigned <i>referents</i> by a
<i>copula</i>.<p>
<li>
The <i>iteration</i> operator <code>for=: ^:</code>
that applies a function for a specified number of
times.<p>
<li>
The function <i>addition</i>.<p>
<li>Its <i>inverse</i> (<i>subtraction</i>).
<p>
<li>The class of <i>integers</i> that includes
<i>zero</i>, and
the <i>negative numbers</i> <code>_1 _2 _3 _4</code>
etc.
introduced
by subtraction.
<p>
<li>The <i>multiplication</i> that is given by iterated
addition.
<p>
<li>The <i>division</i> that is inverse to
multiplication.
<p>
<li>The <i>fractions</i> or <i>rationals</i> that are
introduced by
division.<p>
<li>The <i>power</i> function and the <i>polynomials</i>
based upon it.
</ol>
Some of the assigned names (such as the <code>for</code>
assigned by
the expression <code>for=: ^:</code>) are "utilities" that
will be utilized throughout the text,
and are collected for easy reference in Appendix 1.<p>
The pace of this text is brisk, moving on quickly to
further topics
as soon
as the essential foundations for them are established.
For example,
although the Polynomials of Chapter 4 could provide a
rich subject
in itself, we pass on after a brief three pages to the
Power Series
of Chaper 5, and the Slope and Derivative of Chapter 6.
<p>This gives a quick exposure to significant, and sometimes
surprising,
consequences of otherwise dull foundations. On the other
hand, the
reader
may eventually (or immediately) want further treatments
of the
successive foundations: these are provided in a separate
part of the
book
called Supplement.
<p>In a printed text, the need to move between a given
section and the
corresponding supplemental section in a different part
of the book
might prove onerous. However, in reading the text from a
computer
(through
a Browser), this switching back and forth is made by a
click of a
mouse.
<p>For example, click on the S1K that appears to the right
of the
heading
for this section to read the further discussion in the
supplemental
section S1K, and click on its heading to return.
<p>The <i>Find</i> facility can be used as an index to find
the
occurrences of
words in the text; invoke it by pressing F while
holding down the control key, or select it from the
Search menu.
<h3><a name="1L."></a>
<a href="#1: Numbers">1L. Notes</a>
<a href="#S1L.">
<font size="-1">S1L.</font>
</a>
</h3>
On page 75 Hogben says:<blockquote>
<font size="-1">
<p>It may therefore be soothing for many to whom
mathematical expressions evoke a
malaise comparable to being seasick, if they
can learn to think of mathematics less as an
exploit in reasoning than as an exercise in
translating an unfamiliar script like Braille or
the Morse code.
<p>In this chapter we shall
therefore abandon the historical approach and
deal mainly with two topics: for what sort
of communication do we use this highly
space-saving " now international " written
language, and on what sort of signs do we
rely. To emphasize that the aim of this
chapter is to accustom the reader to approach
mathematical rules as Exercises in economical
translation, every rule in the sign
language of mathematics will have an
arithmetical
illustration "
</font>
</blockquote>
<p>
He continues with:
<blockquote>
<font size="-1">
<p>In contradistinction to common speech which
deals largely with the quality of things,
mathematics deals only with matters of size,
order, and shape. " First let us consider
what different sorts of signs go to the
making of a mathematical statement. We may
classify these as:
<ol>
<LI> punctuation;
<LI> models;
<LI> labels (e.g. 5 or x) for enumeration,
measurement, and
position in a sequence;
<LI> signs for relations;
<LI> signs for operations.
</ol>
</font>
</blockquote>
<p>
Conventional mathematical notation uses three
pairs of symbols for punctuation: <code>( )</code>
and <code>[ ]</code> and <code>{ }</code>. We will use
only the
pair <code>( )</code> for this purpose, and will use the
others for operations: <code>{</code> for
indexing (selection), <code>{.</code> and
<code>}.</code> for
take and drop of a
first item, and <code>{:</code> and <code>}:</code> for
take and drop of a last item.
<p>Hogben offers the following suggestions for
study:
<blockquote>
<font size="-1">
<p>Although care has been taken to see that all the
logical, or,
as we ought to say, the
grammatical rules are put in a continuous
sequence, you must
not expect that you will
necessarily follow every step in the argument
the first time
you read it. An eminent
Scottish mathematician gave a very sound piece
of advice for
lack of which many people
have been discouraged unnecessarily. "Every
mathematical book
that is worth
anything", said Chrystal, "must be read
backwards and forwards
""
<p>---------------------
<p>"Always have a pen and paper, preferably squared
paper, in hand
". when you read
the text for serious study, and work out all the
numerical
examples as you read ". What
you get out of the book depends on your
co-operation in the business of learning.
</font>
</blockquote>
<p>To this we may add the advice to use the computer in the
manner
introduced in the
next chapter, and illustrated throughout the entire
text. In
particular, do not
hesitate to do any computer experiments that may occur
to you " the
worst that
can happen is the appearance of an error message of some
kind, after
which you
may continue without any special action.
<p>