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.


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 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.


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.


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.


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.


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.


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 = 53 and 10 x 10 = 102 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 103 + 4 x 102 + 7 x 10 + 9


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.

Another geometric approach is to interpret the idea of multiplication to mean a 'product' of two 'one-dimensional' numbers that creates a single 'two-dimensional' number.

2 units x 3 units = 6 units2 (an 'area')

Both perspectives are of great use in different numerical modelling applications.


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.

symbolically       F x (a + b + c + ...) = Fxa + Fxb + Fxc + ...


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'.


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.

The rule of signs that is applied doesn't involve any surprises and conforms to common linguistic expectations. Double positive affirmations are still positive... the statement 'we can positively affirm the report's conclusions' simply emphasizes the positive. The combination of a negative and a positive will end up being construed as negative... whether the words 'not actually possible' or 'actually not possible' are used, we are made aware that the possibilities are negative. Double negatives are alternatives to a positive... claiming that something is 'not impossible' is simply affirming the possible. There are plenty of examples in the world of mathematical modelling as well. For example on a bank account 'adding a deposit' and 'removing a deduction' will both result in the account increasing, whereas 'removing a deposit' and 'adding a deduction' will both result in the account decreasing. Whenever two (or more) signs are encountered juxtaposed, the following rule is applied:

'like signs → + '                   'unlike signs → - '

+(+2) = +2       -(-5) = +5             +(-3) = -3       -(+4) = -4


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

Thus +(-4) - (-5) - (+3) +(+6) -1 = +3

Another approach is to use the fact that adding and subtracting the same amounts will result in zero.
+4 -4 =0             -9 +9 = 0             -57 + 57 = 0 and so on.

+(-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

A third approach would be to work from left to right with resolved signs
forming a running accumulation with each successive term.

+(-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 = +4-1combining result with 4th term
(+4-1) = +3combining result with last term

The different methods tend to suit different circumstances depending upon what modelling is being done and whether the arithmetic is to be carried out 'mentally' or with the use of a calculator. There is no disadvantage in being aware of all possibilities.


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

A fundamental assumption about the division of an accumulation
would seem to be that the same result can be achieved
no matter which of the two following basic computational strategies are chosen.

  the individual units are subdivided first and then the results added or
  the units are added first and then the result subdivided.

Suppose it was necessary to subdivide a collection of 2 entities into 3 subdivisions:

...for different reasons, any of the symbolisms below are equivalent.

 2  two accumulated units
partitioned into three
1   +   1 equivalent because
1 + 1 = 2
1 + 1 equivalent because each unit
is partitioned separately then added
2 x 1 repeated accumulation replaced
by multiplication shorthand

In fact if we allow A to represent any integer, and N to represent any integer other than zero... (trying to subdivide something into zero partitions is too problematical for most of us gardeners)... the following symbolic equivalences are always valid:

A     =     1 + 1 + ... (A repetitions)     =     A x 1

At this stage in the development of arithmetical ideas and procedures, when the entities being considered are apples and sacks of grain and bags of inheritance gold, the point of trying to subdivide an accumulation into a negative number of partitions stretches common sense. How can one divide 5 sacks of grain into -3 partitions? Such dilemmas are resolved by interpretation. A positive (+) partition could be interpreted as a tax refund, so that a negative (-) partition would be interpreted as an amount owed. A tax refund could be computed as a fraction of ones tangible assets of 5 sacks of grain. 5 sacks of grain divided by +3 would be interpreted as 5/(+3)=5/3 sacks refund, whereas 5 sacks of grain divided by -3 would be interpreted as 5/(-3)=-(5/3) sacks... ie 5/3 sacks owed to the Grain Tax Department.


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


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.

Using the idea of areas as a modelling device to interpret the multiplication of fractions, we can see that the rectangle with sides of 2/3 and 4/5 has an area equivalent to 8 of the 1/3 x 1/5 rectangles. When multiplying partitions(fractions) it appears as if the bottom numbers(denominators) must be multiplied together to obtain the total number of partitions in the unit square, and the top numbers(numerators) must be multiplied together to obtain the size of the product rectangle.

2     x     4     =     2 x 4     =     8
3 5 3 x 5 15

As long as the denominators were not zero, a similar area interpretation could be devised to model the multiplication of any fractions constructed with ordinary counting numbers(positive integers). In fact, if the alphabetic symbols A, B, M, N are used to represent any integers at all (+ve or -ve, M, N not zero) the following equivalence is always valid.

A     x     B     =     A x B
M N M x N

This equivalence turns out to be a very useful device for obtaining further simplifications. For instance, multiplication of numerator and denominator of a fraction by the same number does not alter its essential value.

A x N     =     A     x     N     =     A     x     1     =     A
M x N M N M M

Also, if the denominator and numerator can be expressed as multiples of other numbers...(factorizing)... the fraction can be simplified if there are common factors. eg.

24     =     2x2x2x3     =     4x6     =     4     x     6     =     4     x     1     =     4
30 2x3x5 5x6 5 6 5 5


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

51 the single fraction
equivalent to the sum
the two original 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     =     3x103 + 4x102 + 7x10 + 9 + 5/10 + 6/102 + 8/103.

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.


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

(there are 3 zeros in the denominator, so the decimal point after the 9... which is usually omitted for integers... must be moved 3 places to the left.)

Justification of this rules takes several lines of detailed symbolic equivalences.

1789 fractional notation

1000 + 700 + 80 + 9 expand position notation
on top line

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

The fact that 10 = 5 x 2 determines the ease with which certain other fractions can be converted to positional notation. Any fraction which has a denominator made up of the factors 2 and/or 5 can be easily converted by multiplying both numerator and denominator by whatever 2 and/or 5 factors are necessary in order to convert the denominator into a product of ten's. Thus

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

Any fractions which only have factors of 2 and/or 5 in their denominator... 2, 4, 5, 8, 10, 16, 20, 25, 32, 40, and so on... can all thus be relatively easily converted into a positional formal version which has a finite number of digits.

3679/1600 = 2.299375 is an exact and terminating decimal.
( 1600 = 2x2x2x2x2x2x5x5 = 26 x 52 )

Fractions which have prime factors other than 2 or 5 in their denominator... 1/3, 4/7, 8/11, 2/13, and so on... create presentation problems for the positional notation, because it is logically impossible to express such partitions exactly in terms of subdividing units into 10's.
There are numerous methods that can be devised to demonstrate this situation. Without indicating the reason for every minute transformation, the following sequence should illustrate the phenomenon.

1 divide 1
into 3 partitions

3x0.3 + 0.1 rewriting the 1
as tenths

3x0.3     +     0.1 apply division to both
3 3

0.3     +     0.1 after division by 3

0.3     +     3x0.03 + 0.01 rewriting the 0.1
as hundredths

0.3     +     0.03     +     0.01 after division by 3

0.3333... after repeating the process
and adding forever..

This non-terminating representation of 1/3 depends entirely on the fact that ten was used as the base(count packages). If other bases are used, the representation may or may not terminate with zeros.

Positional representation of 1/3 using different counting bases
base representation meaning
10(decimal) 0.3333... 3/10 + 3/102 + 3/103 + 3/104...
9 0.3000... 3/9 + 0/92 + 0/93 + 0/94...
8(octal) 0.2525... 2/8 + 5/82 + 2/83 + 5/84...
7 0.2222... 2/7 + 2/72 + 2/73 + 2/74...
6 0.2000... 2/6 + 0/62 + 0/63 + 0/64...
5 0.1313... 1/5 + 3/52 + 1/53 + 3/54...
4 0.1111... 1/4 + 1/42 + 1/43 + 1/44...
3 0.1000... 1/3 + 0/32 + 0/33 + 0/34...
2(binary) 0.0101... 0/2 + 1/22 + 0/23 + 1/34...

Most individuals become comfortable with establishing numerical relationships in base ten arithmetic and slip into automated interpretations of positional representation.
Working in bases other than ten can be a challenging exercise.
'100' represents a 'hundred' in base ten, but 'four' in base 2.
'12.4' represents 'twelve and four-tenths' in base ten, but 'ten and one half' in octal.
For many symbolic purposes the positional notation is not as efficient or clear as a fractional representation, but proves its worth in computerized calculations, where the individual numerals are held in precise and ordered memory locations.


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

Provided the given ratio was relatively simple, and the individual attempting the arithmetical operation was adequately drilled in rote operations with numerals, the following tabulated layout can compact the operation considerably.

      2 . 6 3 ... ← Quotient digits
Denominator → 11 2 9 . 70 40 70 ← Numerator and Remainders

Such a procedure will necessarily eventually produce a repeated remainder. For any finite Denominator M say, the maximum Remainder is M-1. There are therefore only M-1 possible different integer Remainders. As a consequence of this, there are only M-1 possible digits in the positional representation before a repeated Remainder will occur if it is carried over into a similar pattern of partial quotient integers. Once this happens of course, the entire pattern of Quotient digits will be repeated once again if the procedure is continued.

For any numerical ratio A/M (a rational number)... the above algorithmic iteration procedure for producing a positional representation will eventually repeat either zeros or groups of digits no matter what base is used for the counting process.

Some end up with repeated (usually omitted) zeros:

1000/512 = 1.953125000...

Some require every possible remainder before they repeat:

8/7 = 1.142857 142857 ...

  1 . 1 4 2 8 5 7 ... the '142857' group repeats
7 8 . 10 30 20 60 40 50 10     all possible remainders used

The example below repeats after only 6 of the possible 12 remainders have been used:

15/13 = 1.153846 153846 ...

The one below uses every one of the 58 possible remainders before a repeat occurs:

100/59 = 1.6949152542372881355932203389830508474576271186440677966101 6...

In some situations the positional notation is ridiculous.

Further difficulties also become evident as the denominator becomes larger. The seemingly simple repetitive procedure of obtaining the next positional digit for fractions with 'small' denominators becomes increasingly impractical for manual derivations of 'large' denominators, and mechanized computational approximations become the only pragmatic option. If it is necessary for some reason to partition a unit up in the order of billionths, few humans would have the ability or willingness to carry out a manual computation of the sort indicated below.

1/3569125834 = 0.00000000028018065109216880583650500...

Such a decimal representation will eventually repeat but it might take 3569125833 remainders before it occurs. The pragmatic strategy is to simplify the situation with approximations, estimations and error predictions, and to facilitate the whole process by using computing machines.


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 32 = 3x3 = (1+1+1)x3 = 3+3+3 = 9,         42 = 16,         52 = 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         √(72) = 7

But what about         √2,         √7,         √19 and so on ?


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.

if √2 terminates or repeats it can be expressed as a ratio
(as established above, fractions(ratios) repeat or terminate in the positional notation)

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)

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 = M2/N 2.
so that we are forced to conclude that M2 = 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 M2 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


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)

Thus by repeated substitution
it is possible to represent √2 - 1 as a 'continued fraction' thus:

2 +(√2 - 1)

At first glance, such a representation looks rather like some sort of tautology
where (√2 - 1) is being expressed as a complicated calculation of itself.
It we don't know what (√2 - 1) is to begin with, how can a more complex version be of any help?

Remarkably, if we replace the (√2 - 1) in the 'continued fraction' by an estimation...
(0.4 would be good but in fact any positive value would do)...
and then repeatedly add 2 and divide the result into 1 (take the reciprocal)
the calculations 'converge' relentlessly towards the value of 0.41421356237
which of course allows us to conclude that √2 = 1.41421356237

It is tempting to imagine that such a process is a great improvement over the original 'trial and error' method indicated above, but the computational demands of dividing increasingly accurate estimations into 1 (finding the inverse) become prohibitive, without the assistance of automated calculation power. Eventually of course, pragmatism takes over, and the increasing effort of computations is balanced against the requirements for accuracy.

[ Historically, considerable effort was put into presentations such as this, but the relative obscurity and awkwardness of the 'continued fraction' symbolism has been surplanted by more efficient modern mathematical methods. The above 'continued fraction' scenario is considerably demystified when it is restructured into a graphical analysis situation where the value of (√2 - 1) is the intersection of the two functional equations y = x and y = 1/(x+2) ]


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.


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.

Construct lines parallel to sides 'a' and 'b' thru all corners of the 'c' square.

A square of side length (a+b) will be formed, consisting of the 'c' square plus four identical triangles... they are all identical because they are equiangular and each has its side 'c' opposite a right angle
The area of each of the four triangles is 0.5x(base)x(height) = 0.5ab
The area of the 'c' square is therefore equal to the area of the largest '(a+b)' square minus the area of the four identical triangles.

c2= (a + b)2 - 4 x 0.5ab
c2= a2 +2ab+b2 - 2ab
c2= a2 +b2

Thus...         c = √( a2 +b2)

This relationship has an enormous variety of calculation applications.

For example:
If a = 3 and b = 4 then c = √( 32 + 42) = √(25) = 5
If a = 1.5 and b = 2.3 then c = √( 1.52 + 2.32) = √(7.54) = 2.7459
and if c = 17 and a = 5 then 52 + b2 = 172 so that b = √(172 - 52) = 16.248


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    

The earliest attempts to estimate π were geometric of course
and these methods can still provide a useful working value
although as with estimates of √2
the computational effort increases dramatically as increased accuracy is striven for.

One approach would be to use an even-sided polygon to approximate the circumference
and calculate the implications if the number of sides were doubled.

Suppose P1P2 of length 'x' was one side of a regular even-sided polygon
inscribed inside a circle of radius 1 unit.
Then P1P3 of length 'y' would be one side of a polygon with twice as many sides.
The Theorem of Pythagoras can be used on the two coloured right-angled triangles.
and value of y can be calculated from the value of x.

for triangle OMP1             OM2 = 12 - (½ x)2
i.e     OM2 = 4/4 - (1/4) x2     =     (1/4)(4 - x2)
so that             OM = ½√(4 - x2)

for triangle MP1P3            y2 = (½ x)2 + (1 - ½√(4 - x2))2
which multiplies out and simplifies to             y2 = 2 - √(4 - x2)

so that             y = √(2 - √(4 - x2))

This means that if the length 'x' of one side of a polygon of 2n sides is known... (where n is any positive integer)... then the length 'y' of one side of a polygon of 4n sides can be calculated. The sum of all the sides of each polygon can be used as an estimate of the circumference of the circle and thus and estimate of the value of   'π'   can be calculated. The table below shows the sorts of values obtained when the accuracy undertaken is just a few decimal places.

# sideslength of side Circumferencepi_estimate
22 42
41.4142135624 5.65685424952.8284271247
80.7653668647 6.12293491783.0614674589
160.390180644 6.24289030453.1214451523
320.1960342807 6.27309698113.1365484905
640.0981353487 6.28066231393.1403311570
1280.049082457 6.28255450193.1412772509
2560.0245430766 6.28302760233.1415138011
5120.0122717693 6.28314588073.1415729404
10240.0061359135 6.28317545063.1415877253
20480.0030679604 6.2831828433.1415914215
40960.0015339806 6.28318469123.1415923456

Working to 10 decimal places, the amount of computational labour involved... without any assistance from calculating machines... is considerable. Even with a polygon of over one thousand sides, the accuracy of the estimate for   'π'   is only reliable to about 5 significant figures. Clearly, the computing power that is necessary to enable estimations greater than this will become far beyond most individuals.

One alternative method, which was derived from ideas of using series to represent functions, suggests that   'π'   could be calculated from the infinite expression

π/4     =   1 - 1/3 + 1/5 - 1/7 + 1/9 - 1/11 + 1/13 - ...

Whilst there may be some theoretical satisfaction in contriving such a calculation, the practicalities are less than useful. Attempting to compute   'π'   manually is prohibitively labour intensive. After 1000 terms, a value of 3.14 is about all that can be relied upon. After 1000000 terms, we may perhaps be confident about 3.14159... Without assistance, no single human could perform those calculations.
Many other methods have been devised, so that accuracies of millions of places have been computed... for no other reason than to test the computational speed and capacity of the latest 'super-computer', or simply as a competitive sport...
All computers and calculators adopt a cut-off point for the representation of 'pi' but for general purposes, most don't specify more than about a dozen or so digits.

  π   = 3.14159265358979

would be about as 'accurate' as could normally be expected.


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

Clearly, the manual calculations become so onerous so very rapidly, that the longer computations can only be reasonably undertaken with the help of calculating machines. As with the calculation of 'pi', other methods of computing 'e' have been devised but the greatest 'accuracy' provided for a number which does not have a repeating positional representation is about a dozen digits.

  e   = 2.71828182854904

is more than adequate for most purposes.


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 = +a2 and -a x -a = +a2, 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     j2     =     √-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

ej 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.