CONTENTS | 105.5 ARITHMETIC | NEXT |
ARITHMETIC: the symbolic extrapolations of real or hypothetical countings |
The most often-quoted statement purporting to be an example of self-evident and obvious 'truth' is the cryptic assertion that
'one and one equals two'.
A scrupulous analysis of the meanings of each of the words used however suggest the use of concepts that are locked into
unexamined assumptions and unexplained relationships that are skating on quite thin ice.
What is 'one' and what is 'two'?
Just exactly how does one 'and' one?
Does 'equals' mean 'identical with' or 'the same as' or 'measures the same as' or perhaps ' replaceable by'?
Self-evidence is a rather tenuous and unconvincing basis for certainty.
A CONCEPTUAL EVOLUTION PERSPECTIVE
The concepts and symbols of arithmetic evolved over many hundreds of years, during which time its methods both complexified and simplified
in ways that were responses to the demands of the computational environment.
Vocabulary, symbols, concepts and procedures were created to provide solutions, resolve dilemmas and explore previously unconsidered possibilities.
By describing arithmetic in a conceptually evolving manner, there will be an environment of continuous confrontation with intellectual uncertainty.
The selection of a resolution strategy will frequently both determine future evolution and generate latent ambiguities.
The intense microscopic introspections of mathematical analysis and the endless searchings for logical consequences of internalized definitions (theorems)
quickly results in complex abstract symbolism that can be a barrier to interest.
As well as this complexification impediment to comprehension, some humans find the process of attributing an independent reality to concepts irresistible,
so that 'real', 'complex', 'algebraic', and 'transcendental' numbers somehow start to acquire the existential reality of faeries.
Simplified reflections as to the consequences of number representation strategies, and comparing arithmetic processes with the patterns
apprehended in the observable universe, could assist in countering some of the more idealistic suppositions about numbers.
ARITHMETIC IS THE SYMBOLIC MANIPULATION OF NUMERICAL COUNTINGS
Arithmetic is the system of symbolic equivalences and processes that evolves in the language of a society... specifically for the purpose
of manipulating the logical consequences of numerical countings.
Many species exhibit behaviour that reflects a capacity to count, but there is little or no evidence... as yet... to support the suggestion that any of them
other than humans can 'do arithmetic'.
Human societies have devised many systems... each with different symbols, rules, and bases... that reflect the preoccupations of the
members with their trading inclinations, educational initiatives and existential perspective.
Contemporary arithmetic has evolved from the most useful and powerful ideas of former societies, and whilst evolution is inevitable, any profound and
revolutionary changes are likely to be impeded by a not inconsiderable inertia.
CONCEPTS OF COUNTABLE ENTITY SIMPLIFICATIONS
Arithmetic, and the mathematics that evolved from it, has its conceptual origins in the capacity of an awareness to simplify
some aspect of perception and experience in such a manner that it can be contemplated as an entity.
Some entities so conceived are not normally disputed... apples and axes and birds and bees... but many are less easily identified.
When is a building not a house, or a river just a stream... and who can decide whether a species or a political party is extinct?
It is not possible to absolutely identify things in the real world as primordial unitary entities, but arithmetic proceeds quite unperturbedly
on the assumption that it is... and therein is the source of many a metaphysical quandary.
Arithmetic thus assumes that there are things, each of which can be counted and conceptualized as a 'one', and that these 'ones'
can be aggregated and manipulated and recorded using an abstract symbolism.
In order to facilitate increasingly complex numerical manipulations... too involved to be reliably carried out in one average biological memory...
symbols are devised and procedures developed that enabled elaborate computations to be successfully performed.
THE SYMBOL FOR 'ONE'
Using common and widely accepted symbols... and a conscious attempt to identify any concepts and procedures involved...
an axiomatic symbolism can be developed to describe the various arithmetic procedures.
The symbol '1' is the most common symbol in use for 'one'... ie the abstract simplification of being a countable entity.
Shift an apple from here to there and it usually retains enough integrated coherence to be considered as an existential entity and thus be
abstracted in the memory as a countable 'one'.
Cut an apple into pieces with a knife and it will cease to be a single entity and become several different entities.
Normally, a 'one' is capable of being disintegrated into separate (smaller) pieces and it is only when entities of the dimensions of
'quarks' are being considered that this assumption of divisibility may need to be revisited.
ACCUMULATION, REPLACEMENT AND NUMBER SYMBOLS
If the symbol '+' is used to mean 'count and accumulate', then '1 + 1' symbolically represents the counting process of
accumulating a 'one' entity with another identical 'one' entity... which is located in another time and/or place.
This symbol gets used for many concepts which are not necessarily all equivalent... 'and', 'add', 'plus' and 'positive' for example...
so that establishing consistent and unambiguous meanings for their usage demands more than casual assumptions.
By next introducing the symbol '=' to mean 'may be replaced by', and the symbol '2' (called 'two') to stand for this most elementary of accumulations,
we are able to write the most profound initial statement of all arithmetic as
" 'one' 'counted and accumulated' with another 'one' may be replaced by 'two' "...
1 + 1 = 2 (in symbolic shorthand)
In contemporary usage the '=' symbol is used to mean 'equals'.
It will eventually be necessary to address the issue as to exactly what 'equals' might mean, but for the present, using the symbol '=' to simply
imply a direct conceptual substitution defers the philosophical problem until it becomes unavoidable.
Procrastination is not necessarily disadvantageous.
Further unique symbolic replacements are commonly used for accumulations of more 'ones', so that by subsequently defining
2 + 1 = 3 (three), 3 + 1 = 4 (four), 4 + 1 = 5 (five), 5 + 1 = 6 (six), 6 + 1 = 7 (seven), 7 + 1 = 8 (eight) and 8 + 1 = 9 (nine),
and with the inclusion of the symbol '0' (to indicate absence of an entity),
we arrive at the core symbolic set of numerals for the decimal arithmetic system.
As long as an accumulation does not exceed the maximum of nine, all sorts of symbolic counting representations can now be symbolized.
1 + 1 + 1 + 1 + 1 = 5, 0 + 5 = 5, 2 + 3 = 5, 4 + 1 = 5, 4 + 5 = 9, and so on.
At this early stage it can be noted that as long as we confine our introspections to the behaviour of the abstract symbolism, we can both
existentially assume that it does not matter in what order entities are counted and accumulated,
and with the help of 'grouping' symbols '(' ')' called 'brackets',
'prove' this 'commutative' property to ourselves using the symbolism itself.
Thus 3 + 4 = (1 + 1 + 1) + (1 + 1 + 1 + 1) = 1 + 1 + 1 + 1 + 1 + 1 + 1 = (1 + 1 + 1 + 1) + ( 1 + 1 + 1 ) = 4 + 3
Whilst this can be reassuring as to the integrity of the intellectual structure being created, it should be pointed out that the
'counting and accumulation' process is temporal... a fact that is often ignored... so that it takes time to perform the operation.
In the real world it can often matter when and where one starts counting.
When political poll counts are made, or biological species samples are taken, the choice of time and place has profound significance.
NUMBER REPRESENTATION EXTENSIONS
One of the first extensions to the symbolism described so far, is to consider how numbers other than the ones defined above could be represented
without without having to create more and more symbols.
Can the ten symbols defined so far be used and reused so that any number whatsoever can be symbolized?
There are and have been many possibilities, but the most widely accepted current system represents any number as a horizontal row of adjacent digits.
The core development step in this representation is to define 9 + 1 = 10 (ten), where the digits for '1' and '0' are reused in combination to represent the next
accumulation after nine.
The symbols '10'... where the '1' symbol is on the left of the '0' symbol...
are thus interpreted to mean there is one 'ten' and 'nothing else' ( zero units ).
The basic idea intrinsic to this notation is that counting is going to be recorded in packages of 'ten'... (called the 'base').
Other bases... binary, octal, hexadecimal for example... are used because they have very useful features, but base 10 is the one that is the most
widely understood.
The next nine accumulations in base 10 can thus be represented by using all the remaining defined numerals to the right of a '1' symbol...
thus 11, 12, 13, 14, 15, 16, 17, 18, 19, represent 'one ten accumulated with one', 'one ten accumulated with two' and so on.
Since the next accumulation after 19 is equal to two tens, the symbols '20' can be used to represent it.
Using the numeral set again, 21,22,23,24,25,26,27,28,29, will represent the next nine accumulations after which the symbols '30'
will be used to stand for 'three tens'.
Cycling thru all the digits in this manner, the 'decades' of 30-39, 40-49,... 90-99 allow all of the accumulations up to 'nine tens and nine' (ninety-nine)
to be represented unambiguously by this simple positional rule.
Since the next number after 99 is ten accumulations of ten units, the next step in this representational system is to define 99 + 1 = 100 (one hundred),
where the left hand digit '1' represents one hundred and the two zeros represent zero tens and zero units respectively.
Thus by using all the ten digits in the 3 positions, all the numbers from zero (000) up to nine hundred and ninety-nine (999) can be represented.
For many purposes... but not all... leading zeros are deemed unimportant and are not symbolized.
Thus zero is usually written as '0' and not '000', one is written as '1' and not '001'...and so on.
Continuing in this fashion, each additional position to the left is used to represent accumulations that are ten times larger than the position immediately
on its right.
Some of the positions have evolved to be considered significant... and have been named accordingly.
As well as the '10'(ten) and '100'(hundred) positions, '1000' is called 'one thousand', '1000000' is called 'one million' and
'1000000000 is called 'one billion'. (because counting the zeros is error prone, spaces are often left between groups of 3 digits,
so one billion might be written as 1 000 000 000 )
The positional system described so far has the ability to represent undivided numbers of unlimited magnitude... called 'whole numbers'... ,
but although the symbolism is quite efficient, verbal 'elaborations' of what the symbols stand for become increasingly incomprehensible.
Thus 'explaining' what '3479' (three thousand four hundred and seventy nine) means, would require an assertion that it was 'three accumulations
of ten accumulations of ten accumulations of ten accumulated with four accumulations of ten accumulations of ten
accumulated with seven accumulations of ten accumulated with nine'.
Time to introduce a couple of shorthand ideas and their symbolism.
IDEA AND SYMBOLISM FOR MULTIPLICATION
Certain repetitions occur so often in arithmetic that common shorthand symbols are used to simplify the representation.
Repetitions of accumulations are replaced by 'multiplication'.
Using 'x' as the symbol for 'multiplication'
3 repetitions of 4 accumulations would be
4 + 4 + 4 = 3 x 4
In general then 'm + m + m + ...(n times) = n x m' and 'n + n + n +...(m times) = m x n', where the intention is
that the alphabetic characters 'm' and 'n' stand for any counting number.
Just as '3 x 4 = 4 x 3' because they are both equivalent to '1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1' so it can be shown that in general that
the order does not matter.
m x n = n x m (multiplication is 'commutative' )
Defining multiplication thus in terms of accumulation repetitions is perfectly adequate for counting numbers, and serves as the engineering basis
for calculating machines, but it is too limiting for many number applications situations.
The meaning of 'multiplication' becomes more sophisticated as increasingly complex circumstances eventuate, but this fundamental 'repeated addition'
process will remain as a go-to method for machine-language computer implementations.
Using this definition of multiplication,
100 = 10+10+10+10+10+10+10+10+10+10 = 10x10
(ten repetitions of ten accumulations)
1000 = 100+100+100+100+100+100+100+100+100+100= 10x100
(ten repetitions of hundred accumulations)
70 = 10+10+10+10+10+10+10 = 7x10
400 = 100+100+100+100 = 4x100 = 4x10x10
3000 = 1000+1000+1000 = 3x1000 = 3x10x10x10
so that the number '3479' can now be symbolically 'explained' as 3 x 10 x 10 x 10 + 4 x 10 x 10 + 7 x 10 + 9... which is a distinct improvement over the verbal version.
The proliferation of multiplication repetitions suggested further shorthand.
The traditional symbolism chosen to indicate how many entities there were,
was to use a superscript number (called the 'index' or 'power') immediately after the repeatedly multiplied numeral.
Thus 5 x 5 x 5 = 5^{3} and 10 x 10 = 10^{2} and so on.
Using these shorthand symbols for repeated accumulations and multiplications we can now 'expand/explain' the example number by writing
3479 = 3 x 10^{3} + 4 x 10^{2} + 7 x 10 + 9
EXPANDING THE IDEA OF 'MULTIPLICATION'
The idea of 'multiplication' is often modified to suit different interpretation circumstances.
One approach is to suggest that it is a geometric operation that produces a 'scaling' in linear situations.
3 x 2 = 6 could be interpreted as 3 x (2 linear units) → 6 linear units.
DISTRIBUTION
The 'distribution' property of 'multiplication' over 'accumulation' (addition) is one of the most influential symbolic tools in the
arithmeticians toolbox, and the 'area' interpretation can be used to visualize its validity.
If the intention is to multiply an accumulation of several different accumulations, then the multiplication must be performed on each
and every one of the different accumulations.
eg it is easy to check a numerical example
2 x (3 + 5 + 4 + ... ) = 2x3 + 2x5 + 2x4 + ...
This property of counting numbers turns out to be true of most, if not all, of the different sorts of numbers that are now in common usage.
Once again, the symbolic device for expressing this fact is to use alphabetic characters to represent numbers.
Thus if the letters 'F, a, b, c, etc' are used to represent any numbers whatsoever, the geometric diagram below tries to illustrate the
fact that the 'F' number must be multiplied by each and every one of the added numbers, because the area of the whole rectangle must be
equal to the accumulated area of the individual separate areas.
SUBTRACTION AS THE OPPOSITE OF ACCUMULATION
Situations frequently occur when entities have to be removed from an established accumulation.
From the total of 37 sacks of grain stored after harvest, how many sacks are left in after the tax collector takes 9 for himself and the king?
The symbol '-' is used to mean 'count and remove' so that '37 - 9 = 28' can now symbolise the result of performing this reverse counting process
called 'subtraction'.
Initially, pragmatic 'common sense' prevailed and the idea of trying to subtract more than was there in the first place was judged as impossible.
It is not possible to subtract 43 sacks of grain from the store if there are only 37 there in the first place.
'37 - 43' was a nonsense and could not be replaced by anything sensible.
Sooner or later however, tax collectors came up with the idea that just because there were only 37 sacks in the store, that was not their problem...
the tax was 43 sacks so the farmer must therefore still owe the remaining 6 sacks.
'37 - 43 = 0 - 6' could be used to symbolise the fact that the store was empty and 6 sacks were still due from the farmer to the king.
At this stage a philosophical dilemma rapidly made itself evident.
Once '37 - 43 = 0 - 6' was shortened to '37 - 43 = -6', it started to look as if the 'count and remove' operation '-' had somehow
mutated into some sort of property of the 6.
'6' sacks of grain in the hand is not the same as '-6' sacks of grain on the tab.
'6' is kind of 'positive' whereas '-6' is kind of 'negative'.
THE OPERATION/PROPERTY DILEMMA RESOLVED WITH A RULE OF SIGNS
Although it might initially seem to be prudent to allocate unique property symbols to different sorts of numbers, it turns out that
using the '+' and '-' symbols for both operations and properties can be resolved by inventing a rule that determines what action
must result whenever combinations of them are encountered.
Using a 'number-line' as a geometric model, ordinary counting numbers are called 'positive' numbers and have a '+' symbol prefixed to them, numbers
with a '-' prefixed to them are called 'negative' numbers, and both types are envisaged as being positioned on a straight line.
Zero is the central reference number and 'positive' numbers increase in magnitude in equal increments to the right, and 'negative' numbers increase
in magnitude in equal increments to the left.
ACCUMULATING AND SUBTRACTING INTEGERS
All the 'positive' and 'negative' countable unitary entities along with 'zero' are collectively called 'integers'.
Accumulating (also called 'adding') and/or subtracting these numbers needs to be done according to strict rules.
One approach is to use the geometric model of the number line.
eg What single integer can symbolically replace +(-4) - (-5) - (+3) +(+6) -1 ?
ⓐ Resolve any juxtaposed signs into a single sign,
-4 +5 -3 +6 -1
ⓑ Start at '0'
ⓒ Interpret residual additions as a shift to the right
ⓓ Interpret residual subtractions as a shift to the left
+(-4) - (-5) - (+3) +(+6) -1 | can thus become |
-4 +5 -3 +6 -1 | after using rule of signs |
(-4 -3 -1) + ( +5 +6) | grouping -ves and +ves |
-8 + 11 | accumulating -ves and +ves separately |
(-8 +8) + 3 | pair -ve and +ve equivalence to zero |
= +3 | single equivalent number |
+(-4) - (-5) - (+3) +(+6) -1 | can thus become |
-4 +5 -3 +6 -1 | after using rule of signs |
(-4+5)-3+6-1 = (+1)-3+6-1 | combining 1st term with 2nd |
(+1-3)+6-1 = (-2)+6-1 | combining result with 3rd term |
(-2+6)-1 = +4-1 | combining result with 4th term |
(+4-1) = +3 | combining result with last term |
SUBDIVISIONS (fractions) OF THE CHOSEN UNIT ENTITY
The problem of symbolising the subdivision of an entity is unavoidable.
Food needs to be shared... taxes need to be computed... resources need to be rationed... and so on.
Convenience and/or perversity often dictate which symbol is used to represent 'division', but the most commonly useful notation
is a horizontal line drawn between the number of entities being divided and the number of divisions being computed.
symbols for 'division' of A entities into M partitions |
A | A/M A ÷ M A:M |
M |
2 | two accumulated units partitioned into three |
||
3 | |||
1 + 1 | equivalent because 1 + 1 = 2 |
||
3 | |||
1 | + | 1 | equivalent because each unit is partitioned separately then added |
3 | 3 | ||
2 | x | 1 | repeated accumulation replaced by multiplication shorthand |
3 |
A | = | 1 | + | 1 | + | ... | (A repetitions) | = | A | x | 1 |
M | M | M | M |
ADDITIONS ( and SUBTRACTIONS) OF SUBDIVISIONS
Accumulations (additions) and subtractions of subdivisions is straightforward when they all have a common partition N.
The symbolism developed so far can be used to determine further equivalences.
Thus 3/5 + 4/5 = ( 3 + 4 )/5 = 7/5
In a similar fashion it can be shown that subtractions involve a similar symbolism so that if the symbol '±' means 'plus or minus' then
for all integers excluding M=0, the following equivalence is valid:
A | ± | B | = | A ± B |
M | M | M |
MULTIPLICATION OF SUBDIVISIONS (fractions)
The preliminary conception of multiplication being repeated additions is not at all obviously applicable to the multiplication of fractions.
If m x n can be interpreted as m repeated additions of n when m and n are integers, what could 2/3 x 4/5 be interpreted as?
How can one achieve 2/3 of a repeated addition of 4/5?
How can one sensibly have 2/3 of a repetition?
Surely an action is repeated or it is not.
Just as modelling the situation geometrically has previously provided an interpretation, so can this present dilemma be resolved.
Consider a square of unit sides that has had one side partitioned into 3 and an adjacent side partitioned into 5.
Such an action divides the area of the square into 15 rectangles, each of which has sides of length 1/3 and 1/5.
2 | x | 4 | = | 2 x 4 | = | 8 |
3 | 5 | 3 x 5 | 15 |
A | x | B | = | A x B |
M | N | M x N |
A x N | = | A | x | N | = | A | x 1 | = | A |
M x N | M | N | M | M |
24 | = | 2x2x2x3 | = | 4x6 | = | 4 | x | 6 | = | 4 | x 1 | = | 4 |
30 | 2x3x5 | 5x6 | 5 | 6 | 5 | 5 |
ADDITIONS (and SUBTRACTIONS) OF FRACTIONS OF ANY DENOMINATORS
By using the strategy of multiplying both numerator and denominator by the same numbers(factors), fractions with different
denominators can be combined by addition and subtraction into a single fraction.
4 | + | 7 | addition of fractions with different denominators |
15 | 12 |
4 | + | 7 | resolve the denominators into (prime) factors |
3 x 5 | 2 x 2 x 3 |
2x2 | x | 4 | + | 7 | x | 5 | multiply top and bottom by missing factors to ensure the denominators are all the same |
2 x 2 | 3x5 | 2x2x3 | 5 |
16 | + | 35 | numerators have been multiplied denominators have been multiplied |
60 | 60 |
16 + 35 | to add fractions with same denominators add the numerators |
60 |
51 | the single fraction equivalent to the sum the two original fractions |
60 |
EXTENDING THE POSITIONAL NOTATION TO REPRESENT SUBDIVISIONS (fractions)
The positional notation that was used to specify the number '3479' was developed to provide a system that could represent any integer, no matter how large.
Moving to the left, each position was counted in packages that were ten times larger than the preceding position.
Moving to the right therefore, each position is counted in packages that are ten times smaller (partitioned into ten) than the preceding position.
For '3479' for example, the '4' is counted in packages that are ten times smaller than the '3' packages,
the '7' packages are ten times smaller than the '4' packages, and
the '9' units are ten times smaller than the '7' packages.
Continuing to use this pattern, would suggest that any digit to the right of the units position would represent packages that were one-tenth
of the size of the units.
Since the units are the reference in this representational system, if we are to consider extending digits to the right then it becomes necessary to
implement a means whereby we know which of the digits is the units count.
There are of course many ways which that could be achieved.
The units digit could be underlined, or circled, or written in italics, or coloured red etc...etc... but the most convenient and common symbolism for most
circumstances is to simply place a 'point' marker immediately after it.
Thus 3479.5 would be taken to mean that the 9 digit was the units digit, and the next digit to the right was 5/10 of a unit.
This representation could then be extended to the right as far as was required with each successive digit being counted in packages that were
ten times smaller than the preceding one. Thus...
3479.568 = 3000 + 400 + 70 + 9 + 5/10 + 6/100 + 8/1000.
3479.568 = 3x10^{3} + 4x10^{2} + 7x10 + 9 + 5/10 + 6/10^{2} + 8/10^{3}.
Such a symbol system thus seems to have the potential to represent any numbers whatsoever, no matter how large or how small.
Surprisingly, some dilemmas are lurking in the symbolic undergrowth, but by and large it works in practice very well.
REPRESENTING PARTITIONS (fractions) IN POSITIONAL NOTATION
If the decimal(base ten) set of numerals is used, then any fractions that have a product of ten as the denominator can be translated into positional form
almost immediately, by using a simple mental arithmetic rule that relocates the decimal point to the left by as many positions as there are zeros in the
denominator.
1789 | = | 1789.0 | = | 1.789 |
1000 | 1000 |
1789 | fractional notation |
1000 |
1000 + 700 + 80 + 9 | expand position notation on top line |
1000 |
1000 | + | 700 | + | 80 | + | 9 | distribute the denominator into each accumulation |
1000 | 1000 | 1000 | 1000 |
1000 | + | 7 | x | 100 | + | 8 | x | 10 | + | 9 | factor each numerator and denominator |
1000 | 10 | 100 | 100 | 10 | 1000 |
1 + | 7 | + | 8 | + | 9 | remove equivalent factors of 1 |
10 | 100 | 1000 |
1 . 7 8 9 | translate directly to positional format |
1 | = | 1 | x | 5 | = | 5 | = 0 . 5 |
2 | 2 | 5 | 10 |
4 | = | 4 | x | 2 | = | 8 | = 0.8 |
5 | 5 | 2 | 10 |
19 | = | 19 | x | 4 | = | 76 | = 0.76 |
25 | 25 | 4 | 100 |
1 | divide 1 into 3 partitions |
3 |
3x0.3 + 0.1 | rewriting the 1 as tenths |
3 |
3x0.3 | + | 0.1 | apply division to both |
3 | 3 |
0.3 | + | 0.1 | after division by 3 |
3 |
0.3 | + | 3x0.03 + 0.01 | rewriting the 0.1 as hundredths |
3 |
0.3 | + | 0.03 | + | 0.01 | after division by 3 |
3 |
0.3333... | after repeating the process and adding forever.. |
base | representation | meaning |
10(decimal) | 0.3333... | 3/10 + 3/10^{2} + 3/10^{3} + 3/10^{4}... |
9 | 0.3000... | 3/9 + 0/9^{2} + 0/9^{3} + 0/9^{4}... |
8(octal) | 0.2525... | 2/8 + 5/8^{2} + 2/8^{3} + 5/8^{4}... |
7 | 0.2222... | 2/7 + 2/7^{2} + 2/7^{3} + 2/7^{4}... |
6 | 0.2000... | 2/6 + 0/6^{2} + 0/6^{3} + 0/6^{4}... |
5 | 0.1313... | 1/5 + 3/5^{2} + 1/5^{3} + 3/5^{4}... |
4 | 0.1111... | 1/4 + 1/4^{2} + 1/4^{3} + 1/4^{4}... |
3 | 0.1000... | 1/3 + 0/3^{2} + 0/3^{3} + 0/3^{4}... |
2(binary) | 0.0101... | 0/2 + 1/2^{2} + 0/2^{3} + 1/3^{4}... |
AN ALGORITHM TO CONVERT FRACTIONS INTO POSITIONAL REPRESENTATION
Whatever base is used
the positional representation of fractions(ratios)
eventually repeats zeros or a group of digits.
A verbal algorithmic procedure for generating a positional representation
of any ratio of the form 'Numerator/Denominator'
might be expressed as follows.
"Starting with the largest (left-hand) digit
partition each with the Denominator
converting any Remainder at each partition into 'units' of the next digit
and adding them to that digit before dividing.
Record the Partitions (Quotient) and Remainder each time
and stop the process when there is the repetition of a Remainder."
Such generalised instructions often stray into the regions of incomprehensibility
and specific examples become the means of establishing procedures.
Written out in detail the example below attempts to illustrate the process:
29/11 | a ratio of integers |
(11x2 + 7)/11 | 29 rewritten as (2 partitions of 11) and 7 |
2 + 7/11 | divide by 11 with remainder 7 not a repeat |
2 + (11x0.6 + 0.4)/11 | remainder 7 converted to 0.1 'units' |
2 + 0.6 + 0.4/11 | remainder 4 digit not repeated |
2 + 0.6 + (11x0.03 + 0.07)/11 | remainder 0.4 converted to 0.01 'units' |
2 + 0.6 + 0.03 + 0.07/11 | remainder 7 digit repeated...STOP |
2.6363... | the '63' digits will repeat forever |
2 | . | 6 | 3 | ... | ← Quotient digits | |||
Denominator → | 11 | 2 | 9 | . | ^{7}0 | ^{4}0 | ^{7}0 | ← Numerator and Remainders |
1 | . | 1 | 4 | 2 | 8 | 5 | 7 | ... | the '142857' group repeats | |
7 | 8 | . | ^{1}0 | ^{3}0 | ^{2}0 | ^{6}0 | ^{4}0 | ^{5}0 | ^{1}0 | all possible remainders used |
NON-REPEATING NUMERICAL PATTERNS
Many other procedures have been devised for the computation of numerical patterns that prove to be arithmetically useful.
When a numeral (symbol for a temporal count recollection) is multiplied by itself there are existing procedures for determining the result.
Thus 3^{2} = 3x3 = (1+1+1)x3 = 3+3+3 = 9, 4^{2} = 16, 5^{2} = 25, and so on.
Eventually the inverse arithmetic question is asked...
'Can any given number be expressed as the product of two identical numbers?'
Simple integer examples would be 36 = 6x6, 81 = 9x9 and 100 = 10x10.
Using ' √ ' to symbolize this 'square root' operation we would write
√36 = 6, √81 = 9, √100 = 10
and even √(7^{2}) = 7
But what about √2, √7, √19 and so on ?
THE POSITIONAL REPRESENTATION OF √2
Whatever it is, √2 must be a numeral such that √2 x √2 = 2
Since 1.0 x 1.0 = 1.00 then √2 must be greater than 1.0
Since 2.0 x 2.0 = 4.00 then √2 must be less than 2.0
The positional representation must be of the form √2 = 1.abcdefg...
where each of the 'abcdefg' is one of the decimal numerals '0 1 2 3 4 5 6 7 8 9'
The simple and direct approach therefore is to successively trial each of the decimal digits in each of
the abcdefg... positions and select that digit which results in the √2 product estimate being less than or equal to 2.0
thus 1.0x1.0 = (1 + 0.0)x(1+0.0) = 1x1 + 1x0.0 + 0.0x1 + 0.0x0.0 = 1.00 is too small
and 1.1x1.1 = (1 + 0.1)x(1+0.1) = 1x1 + 1x0.1 + 0.1x1 + 0.1x0.1 = 1.21 is too small
and 1.2x1.2 = (1 + 0.2)x(1+0.2) = 1x1 + 1x0.2 + 0.2x1 + 0.2x0.2 = 1.44 is too small
and 1.3x1.3 = (1 + 0.3)x(1+0.3) = 1x1 + 1x0.3 + 0.3x1 + 0.3x0.3 = 1.69 is too small
and 1.4x1.4 = (1 + 0.4)x(1+0.4) = 1x1 + 1x0.4 + 0.4x1 + 0.4x0.4 = 1.96 is too small
but 1.5x1.5 = (1 + 0.5)x(1+0.5) = 1x1 + 1x0.5 + 0.5x1 + 0.5x0.5 = 2.25 is too big
The positional representation must be of the form √2 = 1.4bcdefg...
Using the same direct approach for the next digit 'b' we would find
1.40x1.40 = (1 + 0.4 + 0.00)x(1 + 0.4 + 0.00)= 1.9600
1.41x1.41 = (1 + 0.4 + 0.01)x(1 + 0.4 + 0.01)= 1.9881
1.42x1.42 = (1 + 0.4 + 0.02)x(1 + 0.4 + 0.02)= 2.0164 is too large
The positional representation must be of the form √2 = 1.41cdefg...
To obtain 2 correct digits, 4 multiplications and additions were necessary for each trial digit.
To obtain 3 correct digits, 9 multiplications and additions will be necessary for each trial digit.
1.410x1.410 = (1 + 0.4 + 0.01 + 0.00)x(1 + 0.4 + 0.01 + 0.00) = 1.988100
1.411x1.411 = (1 + 0.4 + 0.01 + 0.01)x(1 + 0.4 + 0.01 + 0.01) = 1.990921
1.412x1.412 = (1 + 0.4 + 0.01 + 0.02)x(1 + 0.4 + 0.01 + 0.02) = 1.993744
1.413x1.413 = (1 + 0.4 + 0.01 + 0.03)x(1 + 0.4 + 0.01 + 0.03) = 1.996569
1.414x1.414 = (1 + 0.4 + 0.01 + 0.04)x(1 + 0.4 + 0.01 + 0.04) = 1.999369
1.415x1.415 = (1 + 0.4 + 0.01 + 0.05)x(1 + 0.4 + 0.01 + 0.05) = 2.002225 is too big
The positional representation must be of the form √2 = 1.414defg...
Proceeding with this direct approach, it appears as if we can get the value of the √2 to whatever accuracy we fancy,
provided we are prepared to face up to the rapidly increasing computational effort
required to establish each successive digit.
But maybe it doesn't go on forever at all?
Maybe it will terminate or repeat after a finite number of digits?
Is there any way of determining what is going to be the case?
As it happens, it is possible to come up with a quite plausible argument
that suggests that the decimal positional representation
of √2 will neither terminate nor repeat.
It goes something along these lines.
① if √2 terminates or repeats it can be expressed as a ratio
(as established above, fractions(ratios) repeat or terminate in the positional notation)
② thus √2 = M/N where M and N are integers.
(M and N must be either 'prime'with no factors like 2,3,5,7,11,13,etc
or are the product of 'primes' like 4 =2x2, 6=2x3, 8=2x2x2, etc)
③ If √2 = M/N is true then √2 x √2 = (M/N)x(M/N) must be true
and since √2 x √2 = 2 it must be true that 2 = M^{2}/N ^{2}.
so that we are forced to conclude that M^{2} = 2 x N ^{2}.
④ But if M and N are integers this is not possible.
(when an integer is squared every one of its factors must occur in pairs.
there just cannot be a solitary factor of two included in the M^{2} product.)
⑤ We are obliged to conclude that the original assumption was invalid.
√2 cannot be expressed as a ratio and is hence called 'irrational'
We seem to be confronted by an infinitely partitionable computational abyss...
If we continued the above process for 8 more iterations we would obtain
√2 x √2 = 1.41421356237 x 1.41421356237 = 1.9999999999912458800169
OTHER METHODS OF REPRESENTING AND COMPUTING SQUARE ROOTS
Ingenious methods of representing and computing square roots have been devised, but they are all ultimately constrained by
the rapidly increasing time and energy demands of numerical manipulations.
For instance, since
(√2 -1)x(√2 +1) = 1 | (after multiplying everything out) |
(√2 -1) | = | 1 | (dividing both sides by √2 + 1) |
(√2 + 1) |
(√2 -1) | = | 1 | (since 2 - 1 = 1) |
2 + (√2 - 1) |
1 | + | 1 | ||||||||||
2 | + | 1 | ||||||||||
2 | + | 1 | ||||||||||
2 | + | 1 | ||||||||||
2 | + | 1 | ||||||||||
2 +(√2 - 1) |
COUNTING HYPOTHETICAL 'UNITS' OF LENGTH
In order to measure/estimate distance, it is necessary to define a reference length and then count the number of
these units that could be laid end to end along the distance in question.
Such a process is clearly subject to all sorts of errors and the practical difficulties of comparing a 'straight' standard reference
length with distances that curve thru all the freedoms of space and time.
If the unavoidable metaphysical considerations of 'measuring' a distance are ignored, and hypothetically 'correct' counts of
certain distances are arithmetically manipulated, a consistent non-repeating representation of the ratio of two aspects of
the idea of a 'circle' can be painfully teased out.
SQUARE ROOTS AND THE THEOREM OF PYTHAGORAS
Consider a triangle with side lengths 'a', 'b', 'c' units of length.
The largest
side 'c' opposite the right angle us called the 'hypotenuse'.
Imagine squares to be constructed on each of the sides.
USING SQUARE ROOTS TO ESTIMATE 'PI'
The ratio of the Circumference of a circle to its Diameter appears to be constant and is symbolized by the greek letter for 'pi'.
Circumference | = | π |
Diameter |
# sides | length of side | Circumference | pi_estimate |
2 | 2 | 4 | 2 |
4 | 1.4142135624 | 5.6568542495 | 2.8284271247 |
8 | 0.7653668647 | 6.1229349178 | 3.0614674589 |
16 | 0.390180644 | 6.2428903045 | 3.1214451523 |
32 | 0.1960342807 | 6.2730969811 | 3.1365484905 |
64 | 0.0981353487 | 6.2806623139 | 3.1403311570 |
128 | 0.049082457 | 6.2825545019 | 3.1412772509 |
256 | 0.0245430766 | 6.2830276023 | 3.1415138011 |
512 | 0.0122717693 | 6.2831458807 | 3.1415729404 |
1024 | 0.0061359135 | 6.2831754506 | 3.1415877253 |
2048 | 0.0030679604 | 6.283182843 | 3.1415914215 |
4096 | 0.0015339806 | 6.2831846912 | 3.1415923456 |
COMPUTATION OF THE NUMERICAL CONSTANT 'e'
One other numerical constant that has become intrinsic to much of modern mathematics is designated by the letter 'e'.
Just as 'π' occurs in situations related to modelling space and curvature
'e' tends to
occur in situations which model growth and decay phenomena.
One of the numerically fundamental ways it can be generated
is by computing an ever-expanding product/ratio of integers.
As the positive integer 'n' is increased without limit, the ratio (1 + 1/n)^{n} stabilizes.
n | (1 + 1/n)^{n} | 'e' approximation |
1 | (2/1) | 2.0 |
2 | (3/2)x(3/2) | 2.25 |
3 | (4/3)x(4/3)x(4/3) | 2.37037 |
10 | (1 + 0.1)^{10} | 2.593742460 |
100 | (1 + 0.01)^{100} | 2.704813829 |
100000 | (1 + 0.00001)^{100000} | 2.718268237 |
1000000 | (1 + 0.000001)^{1000000} | 2.718281693 |
THE AMAZING SYMBOLIC LINK 'j'.
When the rules of signs were established and the idea of squares and square roots explored, it was initially assumed
that it was not possible to determine the square root of a negative number.
Because +a x +a = +a^{2} and -a x -a = +a^{2}, it was clear that the square root of any positive number
could either be positive or negative, and so the square root of a negative number was impossible.
The multiplication of two identical numerals could never be negative.
However, the introduction of a symbol 'j' (or 'i') for √-1 allowed the square-roots of negative number to
participate in many of the established arithmetical procedures.
Thus √-16 could be written as '4j' and √-2 written as j√2
Very significantly of course j^{2} = √-1 x √-1 = -1
It has turned out that this symbolic operational entity 'j' has been the means whereby the counting integers, the ideas of
negative numbers, the circular constant 'π' and the exponential number 'e' can all be connected.
There is more than one possibility but the most cryptic and amazing is the relationship
e^{j π } = -1
Probably no symbolic representation encapsulates more succinctly than this one, the reality that
all arithmetic is about the manipulation of numerals and not of numbers.
Numerals may be interpreted as counts in the physical world if that is appropriate to the circumstances,
but they do not...during the process of their manipulations... represent any idealized entities that
inhabit an existential realm.
CONTENTS | 105.5 ARITHMETIC | NEXT |