e^x → 1 + x (as x → 0).
Because log e = 1, this could be written as.
e^x → 1 + x(log e) as x → 0.
Interestingly if we now generalise for n^x,
n^x → 1 + x(log n) as x → 0.
This therefore provides a ready means of approximating the value of log n (for any value of n).
So log n → (n^x - 1)/x as x → 0.
So for example if we set x = .000001,
log 2 → (2^.000001 - 1)/.000001 = (1.000000693147... - 1)/.000001 = .693147...
So this approximation is already correct to 6 decimal places with respect to its true value and the relative accuracy of the approximation can be continually increased through taking a smaller value of x.
Monday, June 18, 2012
Interesting Relationship
As n becomes larger Cos (2π/n) → 1.
So for example when n = 1000, Cos (2π/n) = .99998026...
Also when n becomes larger Sin (2π/n) → 2π/n.
Thus again when n = 1000, Sin (2π/n) = .0062831439... = (6.2831439.../1000)
(and 2π = 6.2831853...),
and with respect to Euler's Identity,
e^(iπ) = - 1;
therefore
e^(2iπ) = 1 (i.e. 1^1)
Also,
e^(2iπ) = Cos (2π) + i Sin (2π).
Therefore e^{(2iπ)/n} = Cos {(2π)/n} + i Sin {(2π)/n}.
So as n → ∞,
e^{(2iπ)/n} = 1^(1/n) → 1 + (2iπ)/n
So for example when n = 1000, Cos (2π/n) = .99998026...
Also when n becomes larger Sin (2π/n) → 2π/n.
Thus again when n = 1000, Sin (2π/n) = .0062831439... = (6.2831439.../1000)
(and 2π = 6.2831853...),
and with respect to Euler's Identity,
e^(iπ) = - 1;
therefore
e^(2iπ) = 1 (i.e. 1^1)
Also,
e^(2iπ) = Cos (2π) + i Sin (2π).
Therefore e^{(2iπ)/n} = Cos {(2π)/n} + i Sin {(2π)/n}.
So as n → ∞,
e^{(2iπ)/n} = 1^(1/n) → 1 + (2iπ)/n
Tuesday, January 24, 2012
1 and 0 (Type 1 and Type 2 Mathematical Interpretations)
Basic to everything that follows is that each mathematical symbol that can be given a quantitative meaning in conventional (Type 1) terms can equally be given a qualitative meaning in holistic (Type 2) terms.
In Type 1 terms 1 and 0 are the two most important numbers. Indeed the binary system based on these digits can be potentially used to encode all information systems.
1 is implicit in the recognition of phenomenal form. Thus to distinguish an object such as a cup implicitly the number 1 (i.e. as a unit) is involved. So in a sense multiple units (entailing the other natural numbers) can be viewed as successive "1"s.
0 in any context is then the absence or negation of form. So if there is no chocolate left in the box this implies a prior positing of form.
So in Type 1 terms 1 and 0 have a somewhat static finite meaning.
Thus once again 1 relates to the positing of (unit) form in finite terms; 0 relates to the corresponding negation of such form again in finite terms so that 1 - 1 = 0.
Again in Type 2 terms 1 and 0 are the most important numbers; however this time they have a generalised holistic meaning.
So 1 in this context relates to oneness (as the quality embracing all form). Not surprisingly in the West - where emphasis on form is so prominent, the highest spiritual experience is typically seen in terms of union (universal oneness).
By contrast 0 again in this holistic context relates to the quality of emptiness or nothingness. In the East where emphasis on form historically has not been so pronounced the highest spiritual experience is more often seen in terms of emptiness (i.e. the general absence of all form).
Now Type 1 is based on an either/or logic of clear separation. So in this context
1 ≠ 0.
However Type 2 is based on a both/and logic of complementary opposites. So in this case, ultimately 1 = 0.
In other words in complementary terms 1 - 1 = 0. So Oneness in this holistic sense implies its negation so as to be simultaneously = 0.
Again in mystical literature the most balanced spiritual experience is recognised as a plenum-void which represents the ultimate holistic expression of the unity of form (1) with emptiness (0).
So whereas the Type 1 approach is based on a static fixed logic, Type 2 by contrast is based on a dynamic interactive logic which ultimately is purely intuitive in nature.
The line is 1-dimensional; linear logic on which Type 1 Mathematics is based is likewise 1-dimensional.
The circle is - literally - 0-dimensional; likewise the circular logic on which Type 2 Mathematics is based is ultimately purely intuitive (0-dimensional). In fact as we shall outline in more detail later, froam a very valid perspective the Euler Identity can be seen as an exploration on the ultimate meaning of 0 (as a dimension). However before this ultimate stage can be reached, there are various multi-dimensional interpretations possible of an increasingly refined nature where linear (rational) and circular (intuitive) notions interpenetrate.
Type 1 and Type 2 meanings are necessarily involved in all mathematical experience. However in Type 1 terms, the qualitative aspect of symbolic meaning is inevitably reduced in a merely quantitative manner.
What continually astonishes me is how realisation is so limited in the mathematics profession of this fundamental fact. It is as if the almost unanimous consensus it maintains with respect to its specialised methods, provides a considerable - though false - sense of security with respect to its interpretation of truth.
For example with respect to number (which we are dealing with here) quantitative and qualitative aspects are always necessarily involved.
Particular number objects always entail a more general holistic dimension. Therefore though distinct perceptions of number may indeed be given a - relatively distinct quantitative meaning - the general concept of number (with which they necessarily interact) is - relatively - of a qualitative and potentially infinite nature.
However this fact becomes quickly lost in Type 1 terms where the concept becomes reduced (in quantitative terms) as applying to all actual numbers.
One important result of all this is that the infinite notion in turn becomes reduced - inaccurately - through an attempted extension of finite meaning.
And this is inevitable in a discipline that again attempts to incorporate intuitive understanding (which is qualitatively distinct) in a reduced rational manner.
So with respect to 1 and 0 we cannot hope to understand these in a merely quantitative manner, which would entail number objects (corresponding to mental perceptions) independent of their general conceptual meaning.
So properly understood as I have outlined above, we have two distinct aspects of interpretation (quantitative and qualitative respectively) corresponding to two distinctive Mathematics i.e. Type 1 and Type 2.
And one cannot hope to avoid this issue by attempting to sideline it as a philosophical issue without any direct bearing on mathematical procedures.
In fact - again though it is not properly realised - every number that can be used has both a quantitative and qualitative aspect.
For example if I refer to the number 1, this strictly has no meaning until a dimensional number (as power or exponent is assigned). And relative to the base quantity, the number dimension is of a qualitative holistic nature.
Now in Type 1 terms this dimensional number will always be reduced to 1, illustrating graphically what is - literally - a 1-dimensional approach (in qualitative terms).
For example in Type 1 terms 2^3 = 8 (i.e. 8^1).
So even though a qualitative transformation (in 3-dimensional terms) is involved here, the result is expressed in a merely reduced (quantitative) manner i.e. with respect to the 1st dimension.
To conclude just as 1 and 0 are so important in our digital age through providing a means to - potentially - encode all information processes (in Type 1 terms), 1 and 0 are equally important - though not yet recognised - as a means for potentially encoding all transformation processes (in Type 2 terms).
So because of my firm conviction in this regard, I have been working on the development of a new type of science (which combines both linear and circular type logical interpretation).
Finally Type 3 mathematical understanding (which is the most comprehensive) entails the balanced interplay of both Type 1 and Type 2 interpretations.
Ultimately true appreciation of the Euler Identity requires such Type 3 understanding!
In Type 1 terms 1 and 0 are the two most important numbers. Indeed the binary system based on these digits can be potentially used to encode all information systems.
1 is implicit in the recognition of phenomenal form. Thus to distinguish an object such as a cup implicitly the number 1 (i.e. as a unit) is involved. So in a sense multiple units (entailing the other natural numbers) can be viewed as successive "1"s.
0 in any context is then the absence or negation of form. So if there is no chocolate left in the box this implies a prior positing of form.
So in Type 1 terms 1 and 0 have a somewhat static finite meaning.
Thus once again 1 relates to the positing of (unit) form in finite terms; 0 relates to the corresponding negation of such form again in finite terms so that 1 - 1 = 0.
Again in Type 2 terms 1 and 0 are the most important numbers; however this time they have a generalised holistic meaning.
So 1 in this context relates to oneness (as the quality embracing all form). Not surprisingly in the West - where emphasis on form is so prominent, the highest spiritual experience is typically seen in terms of union (universal oneness).
By contrast 0 again in this holistic context relates to the quality of emptiness or nothingness. In the East where emphasis on form historically has not been so pronounced the highest spiritual experience is more often seen in terms of emptiness (i.e. the general absence of all form).
Now Type 1 is based on an either/or logic of clear separation. So in this context
1 ≠ 0.
However Type 2 is based on a both/and logic of complementary opposites. So in this case, ultimately 1 = 0.
In other words in complementary terms 1 - 1 = 0. So Oneness in this holistic sense implies its negation so as to be simultaneously = 0.
Again in mystical literature the most balanced spiritual experience is recognised as a plenum-void which represents the ultimate holistic expression of the unity of form (1) with emptiness (0).
So whereas the Type 1 approach is based on a static fixed logic, Type 2 by contrast is based on a dynamic interactive logic which ultimately is purely intuitive in nature.
The line is 1-dimensional; linear logic on which Type 1 Mathematics is based is likewise 1-dimensional.
The circle is - literally - 0-dimensional; likewise the circular logic on which Type 2 Mathematics is based is ultimately purely intuitive (0-dimensional). In fact as we shall outline in more detail later, froam a very valid perspective the Euler Identity can be seen as an exploration on the ultimate meaning of 0 (as a dimension). However before this ultimate stage can be reached, there are various multi-dimensional interpretations possible of an increasingly refined nature where linear (rational) and circular (intuitive) notions interpenetrate.
Type 1 and Type 2 meanings are necessarily involved in all mathematical experience. However in Type 1 terms, the qualitative aspect of symbolic meaning is inevitably reduced in a merely quantitative manner.
What continually astonishes me is how realisation is so limited in the mathematics profession of this fundamental fact. It is as if the almost unanimous consensus it maintains with respect to its specialised methods, provides a considerable - though false - sense of security with respect to its interpretation of truth.
For example with respect to number (which we are dealing with here) quantitative and qualitative aspects are always necessarily involved.
Particular number objects always entail a more general holistic dimension. Therefore though distinct perceptions of number may indeed be given a - relatively distinct quantitative meaning - the general concept of number (with which they necessarily interact) is - relatively - of a qualitative and potentially infinite nature.
However this fact becomes quickly lost in Type 1 terms where the concept becomes reduced (in quantitative terms) as applying to all actual numbers.
One important result of all this is that the infinite notion in turn becomes reduced - inaccurately - through an attempted extension of finite meaning.
And this is inevitable in a discipline that again attempts to incorporate intuitive understanding (which is qualitatively distinct) in a reduced rational manner.
So with respect to 1 and 0 we cannot hope to understand these in a merely quantitative manner, which would entail number objects (corresponding to mental perceptions) independent of their general conceptual meaning.
So properly understood as I have outlined above, we have two distinct aspects of interpretation (quantitative and qualitative respectively) corresponding to two distinctive Mathematics i.e. Type 1 and Type 2.
And one cannot hope to avoid this issue by attempting to sideline it as a philosophical issue without any direct bearing on mathematical procedures.
In fact - again though it is not properly realised - every number that can be used has both a quantitative and qualitative aspect.
For example if I refer to the number 1, this strictly has no meaning until a dimensional number (as power or exponent is assigned). And relative to the base quantity, the number dimension is of a qualitative holistic nature.
Now in Type 1 terms this dimensional number will always be reduced to 1, illustrating graphically what is - literally - a 1-dimensional approach (in qualitative terms).
For example in Type 1 terms 2^3 = 8 (i.e. 8^1).
So even though a qualitative transformation (in 3-dimensional terms) is involved here, the result is expressed in a merely reduced (quantitative) manner i.e. with respect to the 1st dimension.
To conclude just as 1 and 0 are so important in our digital age through providing a means to - potentially - encode all information processes (in Type 1 terms), 1 and 0 are equally important - though not yet recognised - as a means for potentially encoding all transformation processes (in Type 2 terms).
So because of my firm conviction in this regard, I have been working on the development of a new type of science (which combines both linear and circular type logical interpretation).
Finally Type 3 mathematical understanding (which is the most comprehensive) entails the balanced interplay of both Type 1 and Type 2 interpretations.
Ultimately true appreciation of the Euler Identity requires such Type 3 understanding!
Thursday, January 19, 2012
Introduction
The Euler Identity which is commonly expressed as e^(iπ) + 1 = 0 has long been recognised as perhaps the most beautiful formula (equation) in all of Mathematics.
Now the conventional explanation of this beauty is generally given with reference that the simple Identity combines five of the most important mathematical constants
i.e. e, i, π, 1 and 0 with the three mathematical operations addition, exponentiation and multiplication (which occur just once in the Identity).
All of this is of course true, but somehow it misses out on the true significance of the Euler Identity.
In other words as the conventional approach to Mathematics is merely of a (reduced) quantitative nature - in what I refer to as Type 1 Mathematics - it can only interpret the Identity in quantitative terms.
However its real significance is that it combines both quantitative and qualitative type interpretation in a truly remarkable manner.
Therefore to highlight this neglected qualitative aspect of interpretation we must employ Holistic i.e. Type 2, Mathematics.
Then finally in the blending of both quantitative and qualitative aspects of interpretation we finally move to - what I term - Radial i.e. Type 3, Mathematics.
In my own development - psychological and mathematical - the Euler Identity has proved of enormous significance.
Firstly it helped to resolve a long standing issue I had with the Number Spectrum.
So some 20 years ago I realised that the most scientific manner of interpreting the full spectrum of development was in terms of appropriate use of the holistic interpretation of the number types. This spectrum entailed pre-rational, rational, and post-rational (contemplative) type structures. Now for all the main stages (pre-rational, rational and post-rational) I was able to uniquely identify each stage with a number type (the holistic interpretation of which defined the key structural aspects of the stage). However the final most developed stages required incorporating the mature use of rational with contemplative type understanding (i.e. radial).
Putting it briefly the qualitative interpretation of the Euler Identity provided the essential link in explaining the move from contemplative to radial development.
More recently the Euler Identity has played a major role in enabling me to discover - what I consider - is the true meaning of the Riemann Hypothesis.
So just as the Euler Identity properly requires both quantitative and qualitative aspects of mathematical interpretation, the Riemann Hypothesis ultimately relates to the central condition necessary for the consistency of both aspects.
So the attempt to prove (or disprove) the Riemann Hypothesis through conventional (quantitative) means is thereby strictly futile.
In fact without acceptance of the Hypothesis (as the fundamental axiom) we would have no reason to trust all the accepted axioms of Conventional Mathematics.
Now the conventional explanation of this beauty is generally given with reference that the simple Identity combines five of the most important mathematical constants
i.e. e, i, π, 1 and 0 with the three mathematical operations addition, exponentiation and multiplication (which occur just once in the Identity).
All of this is of course true, but somehow it misses out on the true significance of the Euler Identity.
In other words as the conventional approach to Mathematics is merely of a (reduced) quantitative nature - in what I refer to as Type 1 Mathematics - it can only interpret the Identity in quantitative terms.
However its real significance is that it combines both quantitative and qualitative type interpretation in a truly remarkable manner.
Therefore to highlight this neglected qualitative aspect of interpretation we must employ Holistic i.e. Type 2, Mathematics.
Then finally in the blending of both quantitative and qualitative aspects of interpretation we finally move to - what I term - Radial i.e. Type 3, Mathematics.
In my own development - psychological and mathematical - the Euler Identity has proved of enormous significance.
Firstly it helped to resolve a long standing issue I had with the Number Spectrum.
So some 20 years ago I realised that the most scientific manner of interpreting the full spectrum of development was in terms of appropriate use of the holistic interpretation of the number types. This spectrum entailed pre-rational, rational, and post-rational (contemplative) type structures. Now for all the main stages (pre-rational, rational and post-rational) I was able to uniquely identify each stage with a number type (the holistic interpretation of which defined the key structural aspects of the stage). However the final most developed stages required incorporating the mature use of rational with contemplative type understanding (i.e. radial).
Putting it briefly the qualitative interpretation of the Euler Identity provided the essential link in explaining the move from contemplative to radial development.
More recently the Euler Identity has played a major role in enabling me to discover - what I consider - is the true meaning of the Riemann Hypothesis.
So just as the Euler Identity properly requires both quantitative and qualitative aspects of mathematical interpretation, the Riemann Hypothesis ultimately relates to the central condition necessary for the consistency of both aspects.
So the attempt to prove (or disprove) the Riemann Hypothesis through conventional (quantitative) means is thereby strictly futile.
In fact without acceptance of the Hypothesis (as the fundamental axiom) we would have no reason to trust all the accepted axioms of Conventional Mathematics.
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