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What does (real) number mean?
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wij
Jul 27, 2021, 11:22:02 AM7/27/21
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I just combined my number view into a file. Some issues might be interested:
1. Infinity
2. Real axis
3. 'cardinality' of ℝ is greater than ℵ1,ℵ2,ℵ3,...
4. Infinite series
5. 1=√1=√(-1)*(-1)=i*i=-1.
Possible issues about 0.999...=1 (or lim(x->∞) 1/x=0) had been debated for
years (both are wrong).
https://sourceforge.net/projects/cscall/files/MisFiles/Infinity-zh.txt/download
------
+--------------+
| Nomenclature |
+--------------+
Entity::=
The real object under measurement. We understand and describe the real object
via measurement. What we feel and inside our brain are the results of
measurement, not the real object itself.
Symbol '='::=
x=y means the occurrence of x can be replaced by y, vice versa.
Note that this is a formal definition. Actual usage is normally accompanied by
an 'equivalence relation=='. If "x=y" is a proposition, then a=b iff a==b.
This definition emphasizes the operational meaning (needed in programming).
Trichotomy Property::=
Same as well-ordered property. For any two elements a,b, exactly one of the
relation holds: a<b, a==b, or a>b (elements can be sorted). Explanation for
other symbols (<=,>=) are omitted.
Real Axis::=
A widthless graduated scale for measuring the length or position of an entity.
The division marks on the scale may be referred to as points or
'real numbers'. The marks are the cracks separating the 'continuous space'.
The mark itself has no measure of length, so the point or the number.
Point::=
The same as in Euclidean geometry. A mark of position, same as the marks on a
graduated scale.
Number::=
A symbol(operand or expression) that identifies point.
Number '∞'(infinity)::=
1. ∀n∈ℕ, n<∞
2. The multiplicative inverse of ∞ is 1/∞, the additive inverse is -∞
Distance::=
The absolute differences between two numbers. For two numbers a,b on the real
axis, d=dist(a,b)=|a-b|. (sometimes, length is used for distance)
Interval::=
The set of numbers on the real axis between two numbers. E.g. [a,b]≡
{x| x is a number on the real axis, a<=x<=b}. Definition of other kind of
intervals [),(],() are the same as in average mathematics. Note that a single
point x can construct a single-point interval [x,x], dist(x,x)=len(x,x)=
|x-x|=0.
Density Property::=
For any two different numbers, there exists another different number in
between. Take interval [a,b] for instance: ∀i,j∈[a,b], i<j such that ∃k,
i<k<j.
Extended ℕ::=
ℕ∪{∞}, noted as Eℕ
Extended ℤ::=
ℤ∪{∞}, noted as Eℤ
Extended ℚ::=
Set {p/q; p,q∈Eℤ, q≠0}, noted as Eℚ
Corollary: The maximal(of minimal) number in an open interval can not be
precisely explicitly shown. (prohibited by density property)
+--------------------------------------------------------------+
| Prop5: Non-zero length interval can not be stuffed by points |
+--------------------------------------------------------------+
Let interval A=[a,b], a<b. Because any point p∈A is 1-1 correspondent to
interval [p,p], p∈[p,p], dist([p,p])=|p-p|=0:
But the maximal possible length of all q numbers of sub-intervals containing
a single point in A is q*0=0.
QED.
Corollary: The 'cardinality' of ℝ is greater than ℵ1,ℵ2,ℵ3,...
+------+
| Line |
+------+
Line::=
Line is a set of well-ordered points(if in term of points). Conversely, any
well-ordered set could be called a line.
Straight Line::=
Two different points A,B define a straight line L(A,B)≡ {x| x satisfies
exactly one of the following equations 1:dist(A,x)= dist(A,B)+dist(B,x)
2:dist(x,B)=dist(x,A)+dist(A,B) 3:dist(A,B)=dist(A,x)+dist(x,B)}
I.e. x that satisfies equ-1 is an extended line point from B.
x that satisfies equ-2 is an extended line point from A.
x that satisfies equ-3 is in line [A,B].
Corollary: Real axis is a straight line.
Corollary: The shortest distance between two points is a straight line (this
sentence reads funny).
Corollary: The sum of the length of any two sides of a triangle is greater than
that of the third side.
+-----------------+
| Infinite Series |
+-----------------+
Rule1: Infinite series A,B are equal iff the summand and the index are equal
Rule2: The expanded form of infinite series must list the last addend.
Otherwise, the expanded form is ill-formed (obsecure semantics).
Ex.1 (the last addend is omitted):
A=1+2+3+4+5+...
=(1+2)+(3+4)+5+...
=3+7+5+... // ill-formed, obsecure semantics.
Last addend listed:
A=1+2+3+4+5+...+∞ // well-formed, the exanded form of Σ(n=1,∞) {n}
Ex.2:
S=1+2+4+8+... // ill-formed
<=> S=1+2(1+2+4+8+...)
<=> S=1+2S
<=> S=-1
Last addend listed:
S=1+2+4+8+...+2^∞
<=> S=1+2(1+2+4+...+2^(∞-1))
<=> S=1+2S-2^(∞+1)
<=> S=2^(∞+1)-1 // Lots of similar "magic calculation" deriving the result
// S=-1 can be found in youtube. (the term containing the
// last addend ∞ is ignored)
Ex.3:
"f(n)= Σ(k=0,n) 1/k! => f(∞)=e(The base of natural logarithm)"?
We know for sure ∀n∈ℕ, f(n)∈ℚ. To get the result f(n)=e (f(n)∉ℚ), the only
current option is n=∞. But the issue whether or not f(∞)=e (exact equal by
definition) can only be decided via definition, e.g. e≡f(∞). Otherwise, we
can only say f(∞)≈e. (In considering the definition of the equal sign '=',
other forms of e are likely not mutually replaceable with f(∞))
Ex.4: x= Σ(n=1,∞) 1/n²
A common expression is x= Σ(n=1,∞) 1/n²= π²/6, therefore, π=√(6*x)
The issue here is: Lots of π can be derived from various infinite serieses.
But, according to the definition of '=', the result of mutual substitution
may become inconsistent.
For now, the uncontroversial definition of π is the ratio of the
circumference of a circle to its diameter, it is safer to use '≈'.
Therefore, Σ(n=1,∞) 1/n² ≈ π²/6
+--------------------------------------+
| Equality of multi-value of n-th root |
+--------------------------------------+
From the fundamental theorem of algebra, square root has two solutions, e.g.
√2=±1.414... Therefore, "√2=1.414..." is not really correct and can cause
contradictory results like n=-n. This is a common case, e.g. If √1=1 is
correct, then 1=√1=√(-1)*(-1)=i*i=-1...
∴ The symbol '√' (or nth root of a number) should mean operation, can not be
substituted with any ascertained number.
+-------------------------------------------------------------------------+
| Prop6: Periodic exponential function is equivalent to constant function |
+-------------------------------------------------------------------------+
Let f:T->T be a computational(decisive) function. x,a,t∈ T. function f
satisfies two properties:
P1: f(x)=f(x+n*t) ... f is periodic of t, ∀n∈ ℕ
P2: f(x)^a=f(a*x) ... f has such an exponential arithmetic property
Lemma1: For any a∈ T, f(x)=f(x+a*t)
Proof: f(x)=f(x+a*t)
=> f(x)^(1/a)=f(x+a*t)^(1/a)
<=> f(x/a)= f(x/a+t) ... from P2
<=> f(x/a)= f(x/a) ... from P1
<=> true
Lemma2: For any a,b∈ T, f(a)=f(b)
Proof: f(a)=f(b)
<=> f(a)=f(a+(b-a))
<=> f(a)=f(a+((b-a)/t)*t)
<=> f(a)=f(a) ... From Lemma1
<=> true
∴ If Lemma2 is converted to text: Function that satisfies P1,P2 are equivalent
to a constant function.
+---------------------+
| De Moivre's formula |
+---------------------+
x,θ∈ ℝ, k∈ ℤ, (cos(θ)+i*sin(θ))^x= cos(x*(θ+2kπ))+i*sin(x*(θ+2kπ))
If simplified using "cis": cis(θ)^x= cis(x*(θ+2kπ))
Note: This section contains unsolved issues, but not important for now.
To prevent issues raised by Prop6 (including issues in the identity
x^(a*b)=(x^(1/b))^a=(x^a)^(1/b)):
1. De Moivre's formula only applicable to complex number (Exponential
operation of real number x must first convert x to a complex number of
cis form. But converting complex of cis(x) form to a+b*i form may suffer
downcast problem)
2. cis(θ)≠cis(θ+2kπ)
1. Infinity
2. Real axis
3. 'cardinality' of ℝ is greater than ℵ1,ℵ2,ℵ3,...
4. Infinite series
5. 1=√1=√(-1)*(-1)=i*i=-1.
Possible issues about 0.999...=1 (or lim(x->∞) 1/x=0) had been debated for
years (both are wrong).
https://sourceforge.net/projects/cscall/files/MisFiles/Infinity-zh.txt/download
------
+--------------+
| Nomenclature |
+--------------+
Entity::=
The real object under measurement. We understand and describe the real object
via measurement. What we feel and inside our brain are the results of
measurement, not the real object itself.
Symbol '='::=
x=y means the occurrence of x can be replaced by y, vice versa.
Note that this is a formal definition. Actual usage is normally accompanied by
an 'equivalence relation=='. If "x=y" is a proposition, then a=b iff a==b.
This definition emphasizes the operational meaning (needed in programming).
Trichotomy Property::=
Same as well-ordered property. For any two elements a,b, exactly one of the
relation holds: a<b, a==b, or a>b (elements can be sorted). Explanation for
other symbols (<=,>=) are omitted.
Real Axis::=
A widthless graduated scale for measuring the length or position of an entity.
The division marks on the scale may be referred to as points or
'real numbers'. The marks are the cracks separating the 'continuous space'.
The mark itself has no measure of length, so the point or the number.
Point::=
The same as in Euclidean geometry. A mark of position, same as the marks on a
graduated scale.
Number::=
A symbol(operand or expression) that identifies point.
Number '∞'(infinity)::=
1. ∀n∈ℕ, n<∞
2. The multiplicative inverse of ∞ is 1/∞, the additive inverse is -∞
Distance::=
The absolute differences between two numbers. For two numbers a,b on the real
axis, d=dist(a,b)=|a-b|. (sometimes, length is used for distance)
Interval::=
The set of numbers on the real axis between two numbers. E.g. [a,b]≡
{x| x is a number on the real axis, a<=x<=b}. Definition of other kind of
intervals [),(],() are the same as in average mathematics. Note that a single
point x can construct a single-point interval [x,x], dist(x,x)=len(x,x)=
|x-x|=0.
Density Property::=
For any two different numbers, there exists another different number in
between. Take interval [a,b] for instance: ∀i,j∈[a,b], i<j such that ∃k,
i<k<j.
Extended ℕ::=
ℕ∪{∞}, noted as Eℕ
Extended ℤ::=
ℤ∪{∞}, noted as Eℤ
Extended ℚ::=
Set {p/q; p,q∈Eℤ, q≠0}, noted as Eℚ
Corollary: The maximal(of minimal) number in an open interval can not be
precisely explicitly shown. (prohibited by density property)
+--------------------------------------------------------------+
| Prop5: Non-zero length interval can not be stuffed by points |
+--------------------------------------------------------------+
Let interval A=[a,b], a<b. Because any point p∈A is 1-1 correspondent to
interval [p,p], p∈[p,p], dist([p,p])=|p-p|=0:
But the maximal possible length of all q numbers of sub-intervals containing
a single point in A is q*0=0.
QED.
Corollary: The 'cardinality' of ℝ is greater than ℵ1,ℵ2,ℵ3,...
+------+
| Line |
+------+
Line::=
Line is a set of well-ordered points(if in term of points). Conversely, any
well-ordered set could be called a line.
Straight Line::=
Two different points A,B define a straight line L(A,B)≡ {x| x satisfies
exactly one of the following equations 1:dist(A,x)= dist(A,B)+dist(B,x)
2:dist(x,B)=dist(x,A)+dist(A,B) 3:dist(A,B)=dist(A,x)+dist(x,B)}
I.e. x that satisfies equ-1 is an extended line point from B.
x that satisfies equ-2 is an extended line point from A.
x that satisfies equ-3 is in line [A,B].
Corollary: Real axis is a straight line.
Corollary: The shortest distance between two points is a straight line (this
sentence reads funny).
Corollary: The sum of the length of any two sides of a triangle is greater than
that of the third side.
+-----------------+
| Infinite Series |
+-----------------+
Rule1: Infinite series A,B are equal iff the summand and the index are equal
Rule2: The expanded form of infinite series must list the last addend.
Otherwise, the expanded form is ill-formed (obsecure semantics).
Ex.1 (the last addend is omitted):
A=1+2+3+4+5+...
=(1+2)+(3+4)+5+...
=3+7+5+... // ill-formed, obsecure semantics.
Last addend listed:
A=1+2+3+4+5+...+∞ // well-formed, the exanded form of Σ(n=1,∞) {n}
Ex.2:
S=1+2+4+8+... // ill-formed
<=> S=1+2(1+2+4+8+...)
<=> S=1+2S
<=> S=-1
Last addend listed:
S=1+2+4+8+...+2^∞
<=> S=1+2(1+2+4+...+2^(∞-1))
<=> S=1+2S-2^(∞+1)
<=> S=2^(∞+1)-1 // Lots of similar "magic calculation" deriving the result
// S=-1 can be found in youtube. (the term containing the
// last addend ∞ is ignored)
Ex.3:
"f(n)= Σ(k=0,n) 1/k! => f(∞)=e(The base of natural logarithm)"?
We know for sure ∀n∈ℕ, f(n)∈ℚ. To get the result f(n)=e (f(n)∉ℚ), the only
current option is n=∞. But the issue whether or not f(∞)=e (exact equal by
definition) can only be decided via definition, e.g. e≡f(∞). Otherwise, we
can only say f(∞)≈e. (In considering the definition of the equal sign '=',
other forms of e are likely not mutually replaceable with f(∞))
Ex.4: x= Σ(n=1,∞) 1/n²
A common expression is x= Σ(n=1,∞) 1/n²= π²/6, therefore, π=√(6*x)
The issue here is: Lots of π can be derived from various infinite serieses.
But, according to the definition of '=', the result of mutual substitution
may become inconsistent.
For now, the uncontroversial definition of π is the ratio of the
circumference of a circle to its diameter, it is safer to use '≈'.
Therefore, Σ(n=1,∞) 1/n² ≈ π²/6
+--------------------------------------+
| Equality of multi-value of n-th root |
+--------------------------------------+
From the fundamental theorem of algebra, square root has two solutions, e.g.
√2=±1.414... Therefore, "√2=1.414..." is not really correct and can cause
contradictory results like n=-n. This is a common case, e.g. If √1=1 is
correct, then 1=√1=√(-1)*(-1)=i*i=-1...
∴ The symbol '√' (or nth root of a number) should mean operation, can not be
substituted with any ascertained number.
+-------------------------------------------------------------------------+
| Prop6: Periodic exponential function is equivalent to constant function |
+-------------------------------------------------------------------------+
Let f:T->T be a computational(decisive) function. x,a,t∈ T. function f
satisfies two properties:
P1: f(x)=f(x+n*t) ... f is periodic of t, ∀n∈ ℕ
P2: f(x)^a=f(a*x) ... f has such an exponential arithmetic property
Lemma1: For any a∈ T, f(x)=f(x+a*t)
Proof: f(x)=f(x+a*t)
=> f(x)^(1/a)=f(x+a*t)^(1/a)
<=> f(x/a)= f(x/a+t) ... from P2
<=> f(x/a)= f(x/a) ... from P1
<=> true
Lemma2: For any a,b∈ T, f(a)=f(b)
Proof: f(a)=f(b)
<=> f(a)=f(a+(b-a))
<=> f(a)=f(a+((b-a)/t)*t)
<=> f(a)=f(a) ... From Lemma1
<=> true
∴ If Lemma2 is converted to text: Function that satisfies P1,P2 are equivalent
to a constant function.
+---------------------+
| De Moivre's formula |
+---------------------+
x,θ∈ ℝ, k∈ ℤ, (cos(θ)+i*sin(θ))^x= cos(x*(θ+2kπ))+i*sin(x*(θ+2kπ))
If simplified using "cis": cis(θ)^x= cis(x*(θ+2kπ))
Note: This section contains unsolved issues, but not important for now.
To prevent issues raised by Prop6 (including issues in the identity
x^(a*b)=(x^(1/b))^a=(x^a)^(1/b)):
1. De Moivre's formula only applicable to complex number (Exponential
operation of real number x must first convert x to a complex number of
cis form. But converting complex of cis(x) form to a+b*i form may suffer
downcast problem)
2. cis(θ)≠cis(θ+2kπ)
Peter
Jul 27, 2021, 11:43:07 AM7/27/21
to
The world will little note, nor long remember what we say here
Abraham Lincoln at Gettysburg
wij
Jul 27, 2021, 12:16:22 PM7/27/21
to
wij
Jul 27, 2021, 7:17:37 PM7/27/21
to
On Tuesday, 27 July 2021 at 23:22:02 UTC+8, wij wrote:
> I just combined my number view into a file. Some issues might be interested:
> 1. Infinity
> 2. Real axis
> 3. 'cardinality' of ℝ is greater than ℵ1,ℵ2,ℵ3,...
> 4. Infinite series
> 5. 1=√1=√(-1)*(-1)=i*i=-1.
>
> Possible issues about 0.999...=1 (or lim(x->∞) 1/x=0) had been debated for
> years (both are wrong).
> https://sourceforge.net/projects/cscall/files/MisFiles/Infinity-zh.txt/download
Sorry, my negligence, the correct link is:
> I just combined my number view into a file. Some issues might be interested:
> 1. Infinity
> 2. Real axis
> 3. 'cardinality' of ℝ is greater than ℵ1,ℵ2,ℵ3,...
> 4. Infinite series
> 5. 1=√1=√(-1)*(-1)=i*i=-1.
>
> Possible issues about 0.999...=1 (or lim(x->∞) 1/x=0) had been debated for
> years (both are wrong).
> https://sourceforge.net/projects/cscall/files/MisFiles/Infinity-zh.txt/download
https://sourceforge.net/projects/cscall/files/MisFiles/Infinity-en.txt/download
wij
Jul 29, 2021, 11:18:09 AM7/29/21
to
Symbol '='::=
x=y means the occurrence of x can be replaced by y, vice versa.
an 'equivalence relation=='. If "x=y" is a proposition, then
a=b iff a==b. This definition emphasizes the operational meaning
(needed in programming).
[Note2] We know 1cm and 1m are different. But through the analysis of
a=b iff a==b. This definition emphasizes the operational meaning
(needed in programming).
infinite set theory, card(1cm)=card(1m). Let alone the semantics of
cardinality, the equality will be surely used ambiguously.
Peter
Jul 29, 2021, 1:05:03 PM7/29/21
to
wij wrote:
Entire books have been written on the subject. Landau's is well-known
and shockingly clear.
Entire books have been written on the subject. Landau's is well-known
and shockingly clear.
wij
Jul 29, 2021, 9:37:11 PM7/29/21
to
On Friday, 30 July 2021 at 01:05:03 UTC+8, Peter wrote:
> ...
> ...
> Entire books have been written on the subject. Landau's is well-known
> and shockingly clear.
Would you quote relevant materials, results or links?
> and shockingly clear.
Peter
Jul 30, 2021, 11:05:05 AM7/30/21
to
olcott
Jul 30, 2021, 1:07:41 PM7/30/21
to
On 7/30/2021 10:05 AM, Peter wrote:
> wij wrote:
>> On Friday, 30 July 2021 at 01:05:03 UTC+8, Peter wrote:
>>> ...
>>> Entire books have been written on the subject. Landau's is well-known
>>> and shockingly clear.
>>
>> Would you quote relevant materials, results or links?
>>
>
> Landau, /Foundations of analysis/, AMS Chelsea Publishing.
>
A real(common) number would be any number that is not a fake number, so
> wij wrote:
>> On Friday, 30 July 2021 at 01:05:03 UTC+8, Peter wrote:
>>> ...
>>> Entire books have been written on the subject. Landau's is well-known
>>> and shockingly clear.
>>
>> Would you quote relevant materials, results or links?
>>
>
> Landau, /Foundations of analysis/, AMS Chelsea Publishing.
>
when we use the common (and not the term of-the-art) meaning of the
term: "real" then π is a real number and apple pie is not a real number.
The true nature or actual righteousness results in peace, joy and
contentment. Being a little silly once in a while helps maintain the
proper balance of this.
--
Copyright 2021 Pete Olcott
"Great spirits have always encountered violent opposition from mediocre
minds." Einstein
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