Universal Numbers - The Universe Of Eternity
“Every elementary number is respective with certain reference or standards and there exists no absolute elementary number that forms a basic building block of all numbers. Any set of
elementary entities can be always broken down into smaller elementary entities, but this is all with the certain perspectives like ”urge to, keep on finding deep or more”. However, with the balanced perspective, its duality, which is zero and one in our mathematics. And same is followed by the universe too”
Balram A. Shah
Abstract The urge to understand the universe and solve some existing problems has led us to discover a special set of numbers that isn’t just a higher set of real and complex numbers but also handles zero and infinity in its true existence. This number system articulates a wide and very basic understanding of mathematics in its natural form with respect to the universe. In real numbers, dividing by zero results in multiple solutions so it is best practice to avoid dividing by zero, but what if dividing by zero has a unique solution? Universal numbers carry additional accuracy about every number and produces unique results for every indeterminate form. Related to this number system, theories, frameworks, axioms, theorems, and formulas are established. Also, some problems are solved which had no confirmed solutions in the past. Problems solved in this article gives more understanding about the imaginary number, calculus, infinite summation series, negative factorial, Euler's number e, and mathematical constant π from a very new perspective. With these numbers, we also understand that zero, one, and mathematical constant e are standard reference points playing a very important role in mathematics. Universal number system simply opens a new horizon for entire mathematics and its property to holds additional details allows us to deal with every number precisely but may require computation intelligence and power for evaluation in some cases. The three major aspects of the universal number are the endless horizon of number scale, infinite precision of every number and the reference point (zero, one, e and π).
Keywords: Zero is one minus one, Complex numbers subset of universal numbers, division by zero, division by infinity, new and universal concept of zero and infinity, theory of universal numbers, negative factorial, alternative of infinitesimals, alternative of calculus, indeterminate form, equation of zero, alternative form of zero, Subadd framework, Muldive framework, indexing scale, indexing scale, universal forms and universal numbers.
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1. Introduction The existence of any natural quantity in the universe are in the form of non-terminating decimal digits rather than terminating numbers but for various reasons, most of the time non-terminating numbers are used in the form of terminating numbers at the closest approximation. The human has no necessity to know every quantity in its full accurate manner in daily life rather to be in approximation. Generally, the sun doesn’t spend the same amount of time every single day nor two apples weighing the same have the same number of atoms. And if two atoms are of the same weight, it is expected both may not have the same number of subatomic particles in it. It's just a matter of fact that at what precision a person is dealing with numbers. On other hand, real numbers or complex are not enough to express these non-terminating numbers as this is not enough.
When we say, this is one mango, we also know that mango is made up of many atomic particles and every mango has different numbers of atoms. So, saying one mango is generally correct but according to universe number is incorrect as, if we bring another single mango, it will not have same numbers of atomic particles. Similarly, it is very expected that the two same atoms will not have same number of building blocks entities that makes atom. We live in certain range of precision in this universe and we calculate the scale only in and around that range of precision but universe precision is not limited.
The way, a recurring non-termination rational number cannot be described in a single sequence but requires a fraction which is a combination of numbers and arithmetic operator to describe its true existence, similarly, there exists an entirely new number system that requires a combination of numbers, arithmetic operators and the concept of infinity.
Universal number maps to a specific real number but the number can change its virtue relatively to another number or type of mathematical operation it is involved in if those numbers are expressed in real number. Let us consider ab= but depending on some condition, ab and vice versa. For example, 0. 99 1= but that same 0. 99 act as a different number in different conditions and it happens relatively to another number’s effect and type of operation it is involved in. There also exist certain conditions that a smaller number can suppress the existence of a larger number including infinitely larger number.
2. Universal Theory
Theorem 1: There always exists a home for everything including the home itself. Irrespective of what basic system, framework, idea and things we are trying to create or assume, it always has a home in which it exists. Even a basic home(system) has its home. But we will never know all the aspects, property and nature of a home unless the home has the same property as the system in an infinite reparative process (infinite loop) or finite cyclic system (cyclic loop). Theorem 2: In the mathematical perspective, every change always remains balanced in the universe with reference to the neutral point. And if not, then some quantity has been ignored.
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Theorem 3: Any change gives rise to counter change and this counter change holds the potential to neutralize the self system or this counter change also holds the potential to create some changes in another surrounding system. However, someone may view this changes as positive aspects or anegative aspects based on consideration of reference point. It is just a matter of perspective. Knowingly or unknowingly, its viewers wish or mindset to see certain things in certain ways. Theorem 4: If any number is yielding multiple unique solutions then that number is the combination of multiple unique numbers.
Everything system In the single dimensional universe, the only concept that exists is the endless number line made of an infinite number of points where it is inappropriate to assume any blank space rather there are only points everywhere on the number line. And any point on the number line is identical to any other point without any difference in any property and prospective among the points. Similarly, in three- dimensional, the entire space is made of points where the three-dimensional horizon is endless and the concept of center point does not it exist in any particular.
Nothing system From the perspective of nothingness, there exist nothing, it simply means there is nothing to imagine. But this nothing is also part of the home if you know anything about it, including the void.
Theorem 5: Nothing is nothing only in a particular system. But that nothing always resembles something if we consider that nothing from its home system or the higher home of the system.
Nothing is something that makes no difference nor it resembles anything nor it makes sense in that system, but that nothing always resembles something from its home system or the higher home of the system. Example: Subadd system is the home for Muldive where unit one is nothing in Muldive system (real numbers) but one represents something in Subadd system (real numbers).
Neutral State or a Triple state system. This can be imagined in the Subadd system as, there is a number zero that resembles nothing in the system and every time there arises any number, the system creates a counter number simultaneously in that system keeping the system as zero in the same moment. In other words, the system creates a number and counter number at the same moment keeping the entire system zero or balanced. So, the entire system as a whole is nothing and in a balanced state and represents the whole system.
3. Principle Theory of Universal number One of the ways, universal numbers can be understood as the consideration of the rate at which the
Universal Numbers 4 of 49 real number was formed to its final precise value that is denoted in concise and precise value. These numbers are dynamic, it has infinite precision and cannot be represented without infinity or zero. The precise value of universal numbers remains unknown but its accuracy converges toward a specific real number and it tends to possess some properties and behaviors of that specific number. In this number system there exist three major points that are neutral point and two free ends which never ends. So, if there exists any number on this number line, there exists a counter number (inverse number) on the same number line except for a neutral number. This neutral point is different with respect to different frameworks of numbers and two simple frameworks are addition-subtraction (Subadd) and multiplication-division (Muldive).
Definition [1]: In various number systems, all numbers are balanced from a specific reference point that does not have its own inverse or it acts as its own inverse and also its presence creates no effect or meaning in that number system. These numbers are termed as a neutral number of that system. Universal numbers system is established with the help of two major number frameworks that exist in day to day mathematics with common attributes. These two frameworks are Subadd and Muldive and its common elements are,
• A neutral point (neutral number)
• Two end points (end numbers) Every number that exists in Subadd and Muldive are emerged from a neutral point in a certain perspective. And any numbers that exist in this framework are arising with one condition taken from the neutral number of the system. This condition is,
• The number maintains balance in the system with reference to the neutral numbers. According to the above condition, a number is formed which expresses some value but simultaneously there forms another number that expresses its counter value and when both these numbers meet together, it forms a neutral value again. It can be also viewed as every number exist in the state of neutral number but its half part expresses a certain imbalance value in the system. A number expresses a specific value and its counter number also expresses the exact same value according to the framework. These values and counter values are obtained by a certain specific operation and that is
• Operator (functions) Based on the above elements, condition and operation, below are the two-basic framework with its mathematical expression and representation.
Subadd Zero – Neutral point 0
Positive infinity – End point +n : n =
Negative infinity – End point − n : n =
Positive and Negative – Operation type +−,
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Its corresponding number line along with other numbers can be represented as below
Figure [1]
The number exists in a neutral state and it is expressed as
{ 0nn }−+ where it is read from the center and not from LHS to RHS It can be also expressed as below in its different forms where all are same {nn+− 0 }
{0|0}nn−−++ ; {0|0}nn++−−
{0|0}nn−+ ; {0|0}nn+−
{ 0+− | 0 } ; { 0−+ | 0 }
{ 0 }
Muldive One – Neutral point 1
Infinity – End point n : n =
1 Zero – End point : n = n
Multiplication and division – Operation type ,
Its corresponding number line along with other numbers can be represented as below
Figure [2]
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The number exists in a neutral state and it is expressed as
{nn 1 } ; {nn 1 }
It can be also expressed as below in its different forms where all are same
{nn 1 | 1 } ; {nn 1 | 1 }
{ 1 }
Subadd and Muldive The zero in the Subadd framework is formed by the combination of two end points from Muldive framework therefore the zero have the possibility to hold either property of the end point expressed 1 1 by and − . Muldive is a complete balanced framework according to the multiplication-division operation point of view, but there exists another major operation and that is addition-subtraction. According to this operation, the entire system is duplicated such that another set of Subadd framework represents the counter value for one Subadd framework. And both of these frameworks can be merged together with different merging configurations. And each such configuration has its own uniqueness in attributes, characteristics, applications. Some of the merging configurations are illustrated in figure [3], [4] and [5].
In this topic, we will discuss about figure [5] as it is the standard configuration followed in current mathematics that represents x axis− . Figure [3] The zero in the Subadd framework is formed by the combination of two end points from the Muldive framework, therefore, the zero holds both properties of the end points which is mathematically 1 1 1 1 expressed by and − . However, zero can be either or − but cannot be both in any given equation.
Figure [4]
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Figure [5]
4. Universal Numbers Depending on the various configuration of mathematical expressions from the Subadd and Muldive framework, many different mathematical frameworks are formed with their unique perspective, properties and its associated application. Out of many such frameworks, a simple framework is formed that possesses universal properties that align with all the frameworks of current mathematics. And this framework forms the basic grounds for the number system that is termed as universal numbers. The framework of the universal number is expressed as 1 k : kn=; n The way zero is formed by the combination of two universal numbers approaching towards each other, every real number or imaginary number is formed by a combination of two universal numbers meeting at a single point and that single meeting point is mapped by a specific real number or an imaginary number. Similarly, imaginary numbers are expressed as
1 im n So, the general expression for the complex number is given by
11 k + i m : n = nn
1 Therefore, any real number given by k holds the properties and characteristics of both k + and n 1 k − but in any given equation, k represent any one of those and not both. n However, this framework is futile unless the proper standards of infinity and zero are used according to Subadd and Muldive frameworks.
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Zero and Infinity According to the Subadd framework, the positive infinity and negative infinity remains conceptual with no proper validation about its specific value which are essential points of the Subadd framework. Similarly, the infinity and inverse of infinity in Multive system also remain conceptual without its particular value defined. But, when the negative and positive Muldive frameworks are viewed together, the counter value of infinity tends to end at the counter value of negative infinity which is illustrated in figure 6. And these two end points are collectively known as point zero on the Subadd framework. On the other hand, zero serves as the neutral point on the Subadd framework making this point more interesting than ever.
Figure [6]
The counter of infinity in the Muldive framework is defined by a function that has the following expression and properties. Its expression is given as 1 : n = n where is the function with the following properties:
• It increments continuously. • Its maximum value never saturates. • P(A) = 1; A is the probability that x keeps incrementing in an incremental function 1 1 fx+ ()= until it touches the function fx− ()=− . x x
fx+ () and fx− () are functions that represent positive zero and negative zero respectively.
5. Standard infinity and standard zero Standard infinity and standard zero are mathematically expressed by and 1 respectively. And any infinity represents by represents the exact same infinity in every equation. Similarly, any zero represented by 1 represents the exact same zero in every equation. The variation in the strength of infinity and zero is denoted by the indexing scale which is discussed in the next subtopic.
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1 1 1 1 Note: is equal to zero but saying zero is equal to is a wrong description as − and collectively called as zero. The set of universal numbers is represented by the . This is combination of capital letter U and N.
Remark [1]: Any number which is described with the help of infinity forms the member of universal numbers Example (1): Euler’s constant “e” is an approaching number. Some of the other simple existing examples of approaching numbers are
1 1 1 (2) l im (3) l im 1+ (4) l im 1− x → 0 x x →x x →x
1 (− 1)n+1 1 (5) (6) (7) 2n n n! n=1 n=1 n=0
Universal numbers have three critical pieces of information that must be taken into account. 1) Where number is approaching 1 1 Example: A = lim11−= ; B = lim11−= x →x x →x 2) From which direction it is approaching In the above example, A is approaching from LHS and B is approaching from RHS. 3) At what rate it is approaching
1 1 Example: A = lim10.999−= ; B = lim10.999−= x →2 x →x x
Zero Zero being the neutral numbers, it associates itself with every number. The way, one is associated with every number in form of multiplication or division, zero is associated with every number in form of addition or subtraction.
Example (8): y=+ a x bc2 can we also expressed as yaxbc=+(0)(0) (0)(0) (2 0)
Axiom [1]: In the context of a real number, zero and unity are numbers that co-exist with all the numbers with the help of some operator. Lemma [1]: Zero co-exists with any numbers with the help of the addition or subtraction operator. Lemma [2]: Unity co-exist exists with any numbers with help of a multiplication or division operator. Below are mathematical expressions where numbers in RHS have less detail of itself which will be understood throughout this paper. Example (9): a( b−= b ) 0
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111111 Example (10): 11...0−=−+++++= 2248162n n=1
Example (11): lim (ln ) 0 x = x→1
(1)− n 2 Example (12): +=0 22 n=1 n 12(2)
Axiom [2]: Positive zero and negative zero collectively makes a neutral zero. Describing a zero in a form of infinite summation series also gives a full detail about the existence of zero. But not all zero are the same nor all infinity are same and it differs by the scale at which it reaches to their respective point. The summation series not only describes the number is trying to reach zero but it also describes at what scale the number is trying to reach to the point zero. Indexing scale is the term used to measure the scale of increment of infinity or a zero.
Remark [2]: In the context of infinite series, the indexing scale in the term is used to express the overall closeness of an approaching number that is trying to get to its final value with each additional term in the infinite series. Axiom [3]: Universal numbers can be best described in the form of an equation incorporated with the concept of infinity. In examples (13) & (14), both are approaching to unity but its indexing scales are different. So, with the context of a universal number, both unities are not the same but are the same in the context of the real number. And the indexing scale is dependent on the equation of the summation series.
111111 Example (13): =+++= ...1 n(1)261220(1) nn++ n n=1
1111111 Example (14): =++++++= ...1 224816322nn n=1
In example (13), the first four indexing scale is 0.8 and in example (14), the first four indexing scale is 0.9375. 1111 1111 +++= 0.8 +++= 0.7 93 5 261220 2481 6
As mentioned in axiom [3], a detailed zero can be expressed in the form of a summation series. But, there exist many series that can represent the zero and every series with a different indexing scale represent a different zero or approaching zero, therefor the approaching zero or zero is denoted by a subscript to the zero notation to differentiate these various zeros. These subscripts are as follows.
0xy ,0 → Zero with unknown indexing scale.
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0a ,0 b → Zero with known indexing scale where a and b denotes the specific equation which describes about zero value.
However, its less often that a summation series can obtain zero with a simple summation equation and one of the difficulties arises due to corollary [1] based on axiom [4]. Axiom [4]: In any summation series, if all terms are of the same sign polarity then the overall summation value can never be less than any known value of the indexing scale. Corollary [1]: A summation series with all elements in the same sign can never obtain a zero except from the null summation series. Corollary [2]: Any sequence of all positive real numbers can never yield an overall sum of negative values in the cartesian coordinates system. Similarly, any sequence of all negative real numbers can never yield an overall sum in negative value in the cartesian coordinates system.
Proof: Let us consider an infinite series where n represents an equation, S represents the total sum of the series and a, b, c & d be are some random terms in the summation series. Also, let this series be of all positive terms.
n =++++=abcdS ... n=1
Now, assume that we know some indexing scale value b c+= k and we have to prove Sk . So, proving by contradiction, let us assume Sk akdk+++ ...
Now consider that we have taken a random term from the summation series to add with known indexing value k . So, let that random term be a . This implies kak+
a 0
The above result implies a is a negative value but we know that there are no negative value terms in the entire summation series. So, this contradicts a has to be a negative value to satisfy the above equation which is an impossible event on any trial as summation series have only positive terms. Hence, the assumption Sk is false therefore Sk is true.
Similarly, it is true for negative real numbers.
Closeness Theorem: Let a and fx() be some function. Also, a− and a+ be a universal number approaching from LHS and RHS to real value a respectively. So, if the difference between fa()−
Universal Numbers 12 of 49 and fa()+ is equivalent to zero in the context of a real number, then fa() is equivalent to fa()− and fa()+ in the context of a real number.
Proof: Let a and fx()be some function. Also, a− and a+ be a universal number approaching from LHS and RHS to real value a respectively.
1 1 aa− =− and aan+ =+= : n n
− + aan=−=0:xx and aan=−=0:yy
aa− and aa−
So, the function fa() can be also expressed as fa()− or fa()+ as a a−+ a . And, evaluating fafa()()0−+− validates that the given function has a negligible difference of infinitesimal value before and after the given real point. It also validates that the function is not stepping or breaking at that point with the context of a real number.
Infinity Infinity is a diverse number where it can be expressed in various forms having various properties and considering it approaches to unknowingly very large numbers, they all are categorized into one term known as infinity and symbolically represented by . However, the more specific and detailed infinity can be represented in the form of a summation series as it also gives the indexing scale of the infinity. So, this summation series can be algebraically represented by
n = (1) nk= where, n denotes the equation in the summation series where it sums to an infinite number. Similarly, there exist negative infinity and detailed negative infinity in summation series can be expressed in the form of
n = − (2) nk=
Similar to zero, infinity can be also expressed in multiple ways using the infinite summation series. So, to distinguish this infinite series with different indexing scale, infinity is denoted by a subscript to the infinity notation. These subscripts are as follows.
xy, → Infinity with unknown indexing scale.
a , b → Infinity with known indexing scale where a and b denotes the specific equation which describes about infinite value.
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6. Arithmetic operation with Approaching numbers Universal numbers behave the same as real numbers and any arithmetic operations are applicable to these numbers. The additional advantage for this number is that we can apply arithmetic operations even on infinitesimals, zero and infinity with standard rules and principles without ending up with multiple solutions. Axiom [5]: Infinitesimals, approaching zero and infinities are quantitatively additive, subtractive, multiplicative and divisional. So, by expressing an approaching number in form of summation series, all the arithmetic operations can be used with the rules and properties of the summation series system.
Addition Let us consider the following summation series
=+++=aaa ... n 123 a nk=
By the additive property of summation series, we can write the sum of two same series as
=++++=+++aaaaaa ...... nn123123 n== kn k
= =()()()...a + a + a + a + a + a + n 1 1 2 2 3 3 nk=
=+++222...aaa n 123 nk=
2... =+++aaa n 123 nk=
This implies
a + a =2 a (3)
Similarly,
002aaa+= 0 (4)
Subtraction For subtraction, let the infinite series be given by
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=+++=aaa ... n 123 a nk=
Using the subtractive property of the summation series, we can write
=+++−=+++=aaaaaa ...... 0 nn123123 nknk== This implies
0 0aa 0−= (5)
Similarly
−aa = 0 (6)
Multiplication If the infinite series of an approaching number is given by
=+++=aaa ... n 123 a nk= then using multiplicative property, we can write
=+++=+++aaaaaa ...... nn123123 n== kn k
2 ==+++ aaa ... n 123 nk= this implies
2 a a =( a ) (7)
Similarly
2 0a 0 a= ( 0 a ) (8)
Division Similar to multiplication, if the infinite series of an approaching number is given by
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=+++=aaa ... n 123 a nk= then using division property, we can write
=+++aaa ... n 123 nk= = 1 =+++aaa ... n 123 nk= this implies a = 1 (9) a
Similarly 0 a = 1 (10) 0a
Taking equation (9), we can express infinity in its inverse form as
1 =a 1 a
=aa(01) (11)
Similarly, taking equation (10), we can express zero in its inverse form as
1 01a = 0a
01aa(=) (12)
In the above examples, arithmetic operations are conducted between same scaling index numbers. However, any arithmetic operations between different indexing scales are also possible as long as such arithmetic operations are possible with summation and product series.
7. Indexing scale Indexing scale is an infinite incremental numerical value that is used to denote the intenseness of a number to be itself. It expresses the scale at which infinity or zero approaches to its final value. It also expresses the relative strength of the infinity or zero compared to standard zero or infinity that allows us to understand the rate at which infinity or zero approaches its corresponding value. This scale may
Universal Numbers 16 of 49 not useful for many real numbers but it gets useful when a single real number tends to have multiple solutions in any given equation. This indexing scale has no specific real end so the mathematical analysis is performed while the value is incrementing and not when it gets to its final value. The indexing scale is defined differently for different forms of equation. If infinity or zero is represented by infinite summation series, the indexing scale is defined by the overall closeness achieved by the zero or infinity to its final value with each additional term of the infinite series. If infinity or zero is represented by the algebraic equation, the indexing scale is defined by the overall closeness achieved by the zero or infinity to its final value by each new incremented variable in the equation. Since standard infinity and standard zero is represented by x and (1 ) respectively, x represents the mathematical expression to express the scale at which it approaches its corresponding value. However, the indexing scale of infinity or zero may differ with respect to another infinity or zero. So, indexing scale difference is the term used that differentiates the indexing scale of zero or infinity with respect to another zero or infinity respectively.
Common Indexing scale In the equation, there may or may not exist an approaching number, but it may be convenient due to some reasons to consider an approaching number in place of a real number, so common indexing scale is the term used to state that same numbers are been replaced with a same approaching number with same indexing scale (with same numbers of infinite terms if we want to consider it in the form of summation series). For example, if we take the equation (13) and want to consider dx as an approaching zero, then all the dx are approaching zero with the same indexing scale. And in the equation, a common indexing scale is abbreviated as subscript "C" to approaching the number Note: In any given equation, if the indexing scale is assumed to be some indexing scale and their indexing scale does not change till the end, such a number's subscript can be left blank. So, any approaching zero or infinity whose indexing scale is not given, it can be considered with any indexing scale as best fit for the entire equation.
()xdxx+−22 (13) dx Remark [3]: Adding or subtracting any number except zero with infinity changes the indexing scale of that infinity and that infinity shouldn’t be considered as same as earlier.
Example (15): From reference [1] Ballu equation is expressed as
n n k ()xk− x (−= 1) 1 : (14) ()!!n− k k n k =0
Isolating the positive and negative terms, we can express equation (14) when xn as
n nn(xk− 2 )n ( xk−+(2 1)) −=1 (15) (n− 2)!(2)! k k n − (2 k + 1)!(2 k + 1)! kk==00( )
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So, when n =y or x =y , equation (15) will have infinite single polarity terms in both the equation and it results in as
−a = b 1 (16)
Remark [4]: Multiplying or dividing any number except absolute true unity with infinity or zero changes the indexing scale of that infinity or zero and that infinity or zero shouldn’t be considered as same as earlier. Remark [5]: Two zeros are not equivalent to the same approaching zero unless it has the same indexing scale. Example (16):
=aa
−aax = 0
−ayx( = 1 1 ) 0
0x 0x =a or (11)−=y (1− 1) y a
=ayx 0 or 000yax= (17)
Now, it may be appealing to consider =yx01 and therefore =a 1 but it is a wrong conclusion. Equation (17) is true in its form and will be discussed such concepts in an uneven indexing scale.
Axiom of Infinity [6]: If x =x then x will change its overall indexing scale but it will still remain infinity for following arithmetic operation
• Addition with any constant except negative infinities. • Subtraction with any constants except positive infinities. • Multiplied by any constants except zeros. • Divided by constants except for infinities. • All the above arithmetic operations are also valid for periodic functions as long as the function includes the mentioned constants and does not result in values that are mentioned to exclude for respective arithmetic operations within its given period range in their own equation.
Axiom of Zero [7]: If x = 0x then 0x will change its overall indexing scale but it will still remain zero for following the arithmetic operation.
• Addition and subtraction only with zero. • Multiplied by any constants except infinities. • Divided by any constants except zero.
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• All the above arithmetic operations are also valid for periodic functions as long as the function includes the mentioned constants and does not result in values that are mentioned to exclude for respective arithmetic operation within its given period range in their own equation.
8. Numerical analysis of Univeral number Based on the indexing scale, any equation can be solved accurately by assuming any progressive incremental indexing scale in place of zero or infinity. This test can be performed with the following conditions. 1) This is performed when there occurs multiplication between infinity and zero or its other forms like the division between infinity and zero and division between zero and zero. 2) It can be done by taking 3-5 trial numbers in place of a variable of universal number and these trial numbers should be taken as the power of ten. 3) So, if the difference between two consecutive trials is more than one then the total product is infinity. And if less than one then it will converge to a specific number which is present in the form of common repetitive numbers in the last 2 trials and its decimal accuracy is determined by the numbers of zeros present in the value obtained from the difference between the last two trial's result. 4) Any output number can be rounded off to the closest real number if and only if the indexing scale of output is higher than the assumed indexing scale for our trial. And no rounding of output value should be done if indexing scale of output is same or less compared to assumed indexing scale. Not following this will cause instability of output value in certain condition.
Below are some examples whose data can be obtained from the data logbook.
1 Example (17): x ln1+ (18) x
10 100 1000 10000 0.9531 0.9950 0.9995 0.9999
Ans: 1 −
1 Example (18): x ln1− (19) x
10 100 1000 10000 - 1.0536 -1.0050 - 1.0005 -1.0000
Ans: −1+
1 Example (19): x ln (20) x
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10 100 1000 10000
- 23.025 - 460.51 - 6907.76 - 92103.40
Ans: −
11 Example (20): ln (21) xx
10 100 1000 10000
- 0.2302 - 0.0460 - 0.0069 - 0.0009
Ans: − 0
1 Example (21): ln ( )x (22) x
10 100 1000 10000 0.2302 0.0460 0.0069 0.0009
Ans: + 0
9. Predictive Analysis Infinity and zero are such entities that have no specific value but rather it keeps on incrementing its value with no end which makes it impossible to solve these numbers in any equation. However, the solution can be predicted while these numbers are incrementing. The way infinity or zero approaches to its final value, in the same manner, the solution will approach to its final value. So, the higher the value of zero or infinity is taken, the more precise the solution value will be. And taking such multiple solutions and analyzing where the solution is approaching, it allows us to predict the final solution. The substitution of the incremental value process is the same as approximation analysis but the substitution value should be very larger than 10,000 for better accuracy.
1 Note: and are standard zero and infinity so any incremental value substation must be exactly the same in every standard zero and standard infinity. And its strength is defined by its indexing scale but not the standard infinity or zero itself.
10. Framework of zero and infinity The general framework of infinity and infinitesimals in any equation is given as
1 Infinity, Zero = nn,:=N n
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The indexing scale variable of zero and infinity can be taken anything based on our choice or requirement. But, if the initial framework for infinity is taken differently, then the model of zero also must be also taken accordingly and vice versa. This is illustrated below
2 1 Infinity Ze, ro = nn,:=N n 2
n 1 Infinity Ze, ro = en,:n =N e
So, in any given equation, both infinity and zero are expressed such that it forms inverse of each other. Similarly, the numbers which approach to any given real constant is given by
1 Approaching constant number = kkn= :;N n
1 where k + is approaching from RHS towards the real constant k which can be also expressed as n
1 k + . Similarly, k − is approaching from LHS towards the real constant k and is expressed as k −. n So, according to this framework, the muldive system number line can be represented in figure [7]. Also, the precise region number line is represented in figure [8].
Figure [7] 1 In this number line, is an approaching zero equal to 0x , x approaching infinity, and subscript x x represents the indexing scale of those approaching numbers.
11. Formulas and problems
A problem where L’hopital’s is not applicable. Example (22): 3xx+ sin when x = (23) 2xx+ cos
Universal Numbers 21 of 49
Solution: 3sin3(sin)xxxx++ / 2cos2(cos)xxxx++ /
Using the axiom of zero, where the value of sin x and cos x fluctuates from −1 to 1, we can write the above equation as 3 0 3 2 0 2
Example of derivative proof using zero.
Example (23): Prove the derivative of x 3 where dx is zero on any indexing scale.
()(33)xdxxxdxxdxxdxx+−+++−3333223 (24) dxdx
dxx322++ dxxdx33 ++dxxxdx2233 dx Using axioms of zero, we get
3x 2
Factorial of negative integers. Example (24): The general factorial expansion is given by nnn(1)!!−= (25)
So, when n = 0
0(01)!0!−= (26)
0! (1)!−= 0 when n =−1 nnn(1)!!−=
−1( − 1 − 1)! = ( − 1)! (27)
−1( − 2)! = ( − 1)!
(− 2)! = − ( − 1)!
Universal Numbers 22 of 49
Or ( 1− )! ( 2)!−= (28) −1 when n =−2 n n( 1n )!−= !
−−−=−2(21)!(2)! (29)
( 2− )! ( 3−= )! (30) −2 when n =−3 with additional expansion. nnnnnnnn(1)(2)(3)(4)(4)(5)!!−−−−−−= (31)
−−−−−−−−−−−−−=−3(31)(32)(33)(34)(34)(35)!(3)!
−−−−−−−=−3(4)(5)(6)(7)(8)(9)!(3)!
(3)!− (9)!−= −−−−−−3(4)(5)(6)(7)(8)
(− 1)! (−= 9)! (32) (−−−−−−−− 1)( 2)( 3)( 4)( 5)( 6)( 7)( 8)
Note: Zero is such a number that can be both positive and negative if it is unknown, but it cannot be both at the same time, so we consider zero to be negative zero in the above equation as the rest of all the numbers are negative and there exists no positive number. With this pattern including zero to be negative zero, we conclude that the value of negative factorial is given by (1)!− 1 ()!(1)−=−n n || ()!(1)−=−n n+1 ()!(1)−=−n n+1 c (33) (1)!n − 0(1)!c n − (1)!n − However, if zero is not considered as negative zero, then the value of negative one factorial can be left in the equation without any substitution for negative one factorial.
Problems on the indeterminant form.
Example (25): 1 If one is treated as a universal number, its basic form is given by
n 1 1 : n =N (34) n
Universal Numbers 23 of 49 so, a number approaching from the positive side towards one
1 n ln1+ n ee= 1 : from example (17) (35) number approaching from the negative side
nl n 1−1 n ee= −1 : from example (18) (36)
But if we consider one as an absolute true unity which is a neutral number of muldive framework, then absolute true unity raised to infinity is absolute true unity and this complies with the definition of neutral number with multiplication operation. So, absolute true unity raised to infinity is absolute true unity is by definition and requires no proof. Therefore,
eif 11= + 111= have PISif −1 − eif 11=
*PIS is explained and its definition is available in definition [3].
Example (26): 00 Considering numbers in the above equation as universal numbers, the basic form can be expressed as
1 1 n : n =N (37) n This gives four simple possibilities
1111 −− 1111nnnn ;;;: −−= n N (38) nnnn First form
1 11ln n nn 1 −0 − =ee = =1 from example (20) (39) n
Second form
− 1 1n 1 1 = = =1+ from example (20) (40) −0 n 11 e ln nn e
Universal Numbers 24 of 49
So, the first two forms can be also expressed as
010 = (41)
Third and fourth form
11 11 11nn −=−=− ( 111) nn( ) (42) nn
1 111cossin(−=+) n i nn when 0then sin ( θ)=θ and cos(θ) 1=
1 i 1111(−=) n (43) n
1 1ln(1)n i − −= = 11 : n =N (44) nnn We can conclude the above result in simple form as
0 0 ( 01) = : (−01) = i π0 (45)
So, for the same indexing scale of zero and infinity, we can express the above equations for positive zero power as
0 i π (−+01) =
Example (27): 0 Above equation can be expressed in the basic form of universal numbers as
1 n (n) : n =N
Expressing the above equation in exponential form, we get
1ln(n ) een ==0 1 from example (21)
12. Uneven indexing scales The indexing scale in any given equation exists in form of natural indexing scale or the indexing scale which we have assumed. In such conditions, the equation remains simple but once the equation goes under various arithmetic operations, its output may have a different indexing scale compared to the
Universal Numbers 25 of 49 initial indexing scales taken in the equation. The missing information of the indexing scale can be neglected and approximated to the nearest real number value except for neutral numbers or free end numbers. This is due to, zero, one and infinity tends to have multiple solutions in some equations, so it gets necessary to know the indexing scale of such numbers.
Major categories of Indexing scale difference Variation in the indexing scale between two approaching numbers changes the way these numbers interact with each other. The interaction differs in infinite ways but according to the major application, we will list 5 categorize. Below are the categories where x1 and x 2 represent the indexing scale of two different numbers and xx12= . Also kk& 1 is used to denote the scale at which one indexing scale is varying with respect to another where k& k11 : k & k 1.
Abbreviation [1]: Indexing scale difference is abbreviated as ISD
1) Equality difference xx12:
x1 2) Linear Difference xkx: || : x 12 k 2
k 1 k 3) Power difference xx12: ( ) || ( xx12) :
x ln ()x1 4) Exponential difference xk: 2 || : x 1 ln ()k 2
x ln ()x 2 1 5) Higher exponential difference xx12:ln () || : x 2 lnln () x1
However, there are many other ISD exist in the middle of these categories but based on major application, only a few ISD are mentioned above. For example, we can obtain many precise ISD categories between exponential difference and higher exponential difference as some are given below
x ln (xk11 ) xkk : ( ) 2 || : x 1 1 ln (k ) 2
ln ()x kx12 1 xk1 : ( ) || : x 2 kk1 ln ()
Note: ISD of a number is measured with respect to the indexing scale of another number or with respect to a common indexing scale in the equation.
Universal Numbers 26 of 49
Based on the above ISD, various ISD are plotted on the Muldive system as figure [8]. In figure [8], R is the real number and other elements are its indexing scale where n is approaching infinity that is used to approach to that real number.
Figure [8]
Note: If the indexing scale of every number in the equation is the same, then writing the indexing scale of every number should be considered as same as not writing the indexing scale of any number in that equation. This is just to make it clean and easy while writing. Based on the above mentioned ISD between two universal numbers, an example equation is given below.
Example (28):
kn nn k 1 kk 1 k e =+1 || ee=+=+11 : n =x (46) n nkn(1/)
Similarly
kn n −k 1 −k k e =−1 || e =−1 : n =x (47) n n Above equation can be also written as
n k k lnln1(e ) =+ n
nk+ kn= ln : n =x (48) n
13. Universal Form and Universal Entities There are certain forms where the precision of numbers is very sensitive to its output. This means even a small change in the value makes a very huge difference in its output. So, such equations get easily noticed in the real number system. In the real number domain, some equations form indeterminant forms and it is unknown why it does so until universal numbers are discovered.
Universal Numbers 27 of 49
Definition [1]: Universal forms are those forms where two numbers represent specific numbers in the real number domain and these numbers combine with certain mathematical operators and results in any real number depending on its ISD. So, any form that leads to obtaining the entire real number is termed as a universal form. These forms are also known as indeterminate forms in existing mathematics.
Definition [2]: Universal Entities are those number which forms the universal form and also these numbers are such that it changes its nature of existence based on the indexing scale of another number. The universal entities are zero, one and infinity. As infinity can be also represented with the help of inverse zero, we can reduce it to zero and one. So, generally, we can say that universal entities are zero and one. These numbers are such that these numbers can align itself in some configuration with the help of arithmetic operators to produce any real number value and its equation resembles the exact same real number for any output value. Axiom [8]: The total value of any universal form remains the same even if the indexing scale changes as long as ISD remains the same.
Universal form and transition between zero and infinity (multiplicative) The universal form may occur between two numbers of any indexing scale or between a specific indexing scale range. Below are the given universal forms between infinity and zero. y 0 = xyx
0 0 === kx x k (49) x kx x 0kx
The above equation is also true for any ISD, so whichever value replaces k is also the final solution. The number which forms the universal form in specific indexing scale undergoes can be viewed as the transition of indexing scale that yields various specific solutions within a specific range indexing scale. However, the zero or infinity dominates with its existence in such operation if its ISD goes greater than the value of k .
Zero Reduction theorem: Let 00xy be a division between two zeros of different indexing scales.
While the elimination of indexing scale value between 0x in the numerator and 0 y in the denominator, if indexing scale of 0x gets totally eliminated and indexing scale of 0 y remains in some form, then the overall value of the equation is infinity and if, indexing scale 0 y gets totally eliminated and indexing scale of 0x remains in some form, then the overall value is zero.
Proof: Zero Reduction theorem is based on the first and second points of the axiom of infinity and the first point of the axiom of zero.
Zero reduction theorem also applicable for infinity fractions where the fraction is yx in the equation and due to inverse property between zero and infinity, it can be substituted as 00xy .
Universal Numbers 28 of 49
Universal form and transition between unity and infinity Unity raised to infinity yields any real value with a difference in the indexing scale between them. The overall value of these forms transcends from zero to infinity with a specific indexing scale as given below from equation (50) to (54). The main objective of transition is to express the interaction between two specific numbers with different indexing scales where they remain the same with respect to the real number domain. Example: Equation from (50) to (54) denotes only one form with respect to real number domain and that is unity raised to infinity and yet it leads to form any real number value.
x 11++== x k (50) ( x1/k ) ( x )
++x kx k (11xk/ ) ==( x ) e (51)
++xx (11xx) ==( ) e (52)
++x xk/ 1/k (11kx) ==( x ) e (53)
x 111++==x1/k (54) ( x k ) ( x )
In the above transition of indexing scale and values, any indexing scale of infinity higher than power ISD will result in infinity and any ISD of unity lower than power difference will result in unity. In terms of mathematical expression
+ x 1/k (1 y ) = : yx (55)
+ x + k (11y ) = : yx (56)
Similarly,
x 111−−==x1/k (57) ( x k ) ( x )
−−x xk/ −1/k (11kx) ==( x ) e (58)
−−xx−1 (11xx) ==( ) e (59)
−−x kx −k (11xk/ ) ==( x ) e (60)
x 1−−== 1x k 0 (61) ( x1/k ) ( x )
Universal Numbers 29 of 49
As similar as earlier, for indexing scale of infinity higher than x k will result in zero and indexing scale of unity higher than x k will result in unity. In terms of mathematical expression
− x − k (11y ) = : yx (62)
− x 1/k (10y ) =+ : yx (63)
Universal form and transition between zero and infinity (exponential) The transition of indexing scale and output values of universal form between zero and infinity occur within a very narrow range of indexing scale differences compared to the transition between unity and infinity. But, zero is such a number that can be represented both by positive zero approaching number and negative zero approaching number. So, in this transition, the negative zero approaching number will be represented by a negative sign and zero without any sign is a positive zero approaching number and k 1.
The transition between infinity and zero are
0x −0x = ; =0 (64) ( ()ln()x x ) ( ()ln()x x )
0x −0x 1 =x k ; =x (65) ( k ) ( k ) k
0x −0x =1 ; =1 (66) ( x k ) ( x k )
Every ISD higher than the higher exponential difference of infinity will result in infinity and zero in LHS and RHS of the equation respectively. Similarly, every ISD lower than the power difference of infinity will result in 1+ and 1− in LHS and RHS of the equation respectively. Its equation expression is given by
0 x x 2 (y ) = : yx ln () 2 (67)
−0 x x 2 (=y ) 0 : yx ln (2 ) (68)
0x + k (=y ) 1 : yx (69)
− 0x − k (=y ) 1 : yx (70)
The transition between zero and zero
0x −0x 01= ; 01= (71) ( x k ) ( x k )
Universal Numbers 30 of 49
0x 1 −0x 0 x = ; 0 x = k (72) ( k ) k ( k )
0x −0x 00= ; 0 = (73) ( ()ln()x x ) ( ()ln()x x )
As similar to the above transition of ISD and output values, every ISD higher than the higher exponential difference of base zero will result in zero and infinity in LHS and RHS of the equation respectively. Similarly, the ISD lower than power difference of zero will result in 1− and 1+ in LHS and RHS of the equation respectively and equationally it is expressed as
0x − k (01y ) = : yx (74)
− 0x + k (01y ) = : yx (75)
0 x x 2 (00y ) =+ : yx ln ( ) 2 (76)
−0 x x 2 (0 y ) =− : yx ln () 2 (77)
Note: Transition mentions in the RHS may turn appealing to represent itself by writing inverse of its base and output value while removing negative signs of exponent zero as both are the same. But both the forms are different and such forms occur in different conditions which turns the same after undergoing its respective mathematical operation. There are also other many universal forms that can be obtained and some are mentioned below.
0 −0 ln/lnxk ln/lnxk (=x ) k ; (01/x ) = k (78)
0kx/ln k −0kx/ln −k (=x ) e ; (0x ) = e (79)
Universal form and transition between infinity and infinity (subtractive) The universal form between infinity and infinity with subtraction operation can be derived in many forms using the above three universal forms. One of the expressions is given by
x k xxek − = (80)
Or the above equation can be also written as
1 xk/ xx(ek) − = (81)
1 k x Note: x (e ) is infinity for any positive value of k.
Universal Numbers 31 of 49
The transition of output value is directly dependent on k value. The output value is exactly the same as ISD and k value can be negative too.
14. Real numbers and Prime indexing scale All real numbers are a subset of universal numbers. And every real number can be expressed in the form of approaching numbers with a very large indexing scale. Also, there exists no real number between real numbers and approaching number approaching to that real number. However, in a certain condition, a universal number approaching to the number doesn’t behave like a real number. Example (29): One to the power infinity yields output as zero at a certain indexing scale which implies one is less than one and not equivalent to one and this creates difficulties to fix a general solution. So, commonly, we consider two general indexing scales, one is a common indexing scale and another is the prime indexing scale.
Definition [3]: Prime indexing scale is the term used for indexing scale of number whose indexing scale has the highest value and it creates very large ISD with other numbers, even larger than higher exponential difference.
Abbreviation [2]: Prime indexing scale is abbreviated as PIS.
Abbreviation [3]: Common indexing scale is abbreviated as CIS. In the context of real numbers, CIS and PIS are the terms that will be generally used to specify the indexing scale as real numbers are considered with a high indexing scale (PIS) or a number with an equivalent indexing scale (CIS) with respect to all other indexing scales in the equation. In general conditions, PIS and CIS refer to the base number's indexing scale and not the exponent number's unless specified. Also, PIS and CIS refer to the real number’s indexing scale and not zero’s or infinity’s unless specified. Using the above definition and general condition of PIS
0 00= (82)
11 = (83)
Similarly, the general condition with CIS are
0 01= (84)
1 = e & 1 = e −1 (85)
ln(0) ln( ) ln(0)+ ln( ) = 0 & = = −1 (86) ln( ) ln(0)
Universal Numbers 32 of 49
A real number can be represented by many approaching numbers and if those approaching number with different indexing scale yields a different output in a given equation then its corresponding real number will also have all those multiple solutions. From all the above observations of this section, we can state the following axiom. Axiom [9]: In any exponential operation between multiple numbers, the number with the highest indexing scale sets the last infinite value and all required accuracy of decimal digits to be real numbers are relatively followed by other numbers. However, all numbers may be independently real numbers.
For relative indexing scale and from universal form and transition between unity and infinity, it can be concluded that
Axiom [10]: Any positive number less than one raised to the infinity of any indexing scale is equal to zero.
15. Zero in Numericals According to the subadd system, one is the neutral number that is eligible to define in the algebraic system without fluctuating the stability of the system. And before, any term turns into standard zero, it has to get rid of the entity that is present along with the zero. This entity is the horizon of that world. The standard zero is numerically defined as
(1-1)=0 (87)
And this is known as zero’s unity identity. This identity, we can generally express as,
nn(1-1)(0)= (88)
However negative zero is defined differently compared to positive zero and so negative zero is numerically defined as
(-1 +1)=0− (89)
And zero’s unity identity for negative zero is generally expressed as
nn(-1+= 1) (-0) (90)
Zero’s unity identity is more sophisticated than it seems as this equation is made of only neutral numbers and we may end up ignoring few rules as if we are handling other numbers. Here, precision is important at infinitely small scale levels and so the mathematical rules. The difference between positive zero and negative zero is illustrated in the image.
Universal Numbers 33 of 49
Figure [9]
This difference must be also balanced in the LHS and RHS of the equation. So, it is incorrect to write n(-1+1)=n(+1-1) as placement of every number too matters.
The zero doesn’t have its quantitative value but it always makes difference in how zero was formed. 1 The standard decimal zero is where x is infinity and this stands as a reference point to understand x the quality of zero.
1 1 aaa−= : where is a standard zero. x x
Notation for zero can be also written as 0a
All the mathematical operations are applicable on standard zero. So, while inserting any zero in the equation, we must also balance the quality of the zero in that equation. Eg: Let us assume the given equation is xyzm++= and if we like to add aa− in the LHS of the equation then it is mandatory to add 0a in the RHS of the equation. xyzm++=
x+ y + a − a + z = m + 0a
Zero Matters
Zero is nothing but yet it counts, it is also a unit that is part of the home. Not including zero may create unrealistic results as we may unknowingly end up in not balancing the positive or negative aspects of the zero. So, any positive zero must be balanced with negative zero and vice versa. And these zeros must be balanced using zero’s unity identity.
Every zero must be balanced by its counter zero value in the equation by following zero’s unity identity and this is called zero’s balance rule.
Universal Numbers 34 of 49
n(1-1)=n(1-1) (91)
n(1-1)= n(0) (92)
Or
n(-1+1)=n(-1+1) (93)
n (-1+ 1) =n (-0) (93)
Example (30):
3 5− 6 + 4 =
9 5− 6 + 4 − 0 = x
9 9−= 0 x
9( 1 1−= ) 0 9
One of the common equations is ababab22−=−+ ()() that yields inequality result when ab= but according to universal numbers, this formula is invalid due to ignorance of zero in the equation. Due to such ignorance, the equation will yield an unbalanced result.
Eg: When ab= in the equation ababab22−=−+ ()()
aaaaaa22−=−+ ()()
aaaaaaa()()()−=−+
a()()() aaaaaa−−+ = ()()aaaa−−
a=+() a a
aa= 2
Real prove for equation ab22−
a2− b 2 + abab − = a 2 + abab − − b 2
a22− b + ab(1 − 1) = a ( a + b ) − b ( a + b )
Universal Numbers 35 of 49
ababaabbab22−+=+−+ (0)()()
a-b+ab(0)=(a22 + b)(a -b) (94)
When ab= in abababab22−+=+− 0()()()
aaaaaaaa22−+=+− 0(.)()()
aaaaaaaaa()0(.)()()−+=+−
aaaaaaaaa()0(.)()()−+− += ()()()aaaaaa−−−
0(.)aa aaa+=+ () a(0)
aaaa+=+ ()
Special case of subtraction for zero The addition, multiplication and division between zero are as usual but the subtraction of zero yields a special result.
Note: BODMAS rule is very important in this operation. Case 1:
0− 0 = 0(1 − 1)
As we know (11)0−=
0−= 0 0(0)
0−= 0 02 (95)
Case 2: As we know 0=− (1 1)
00(11)(11)−=−−−
Taking common out (11)− from RHS
0− 0 = (1 − 1)(1 − 1)
0−= 0 0(0)
Universal Numbers 36 of 49
0 0−= 0 2
Case 3: Considering the pattern in a decremental way.
3(0) 0−= 2(0)
2(0)−= 0 1(0)
1(0) 0−= 0(0)
This implies
0 0−= 0 2
Special case of subtraction for infinity The addition, multiplication and division between infinity are as usual but the subtraction of infinity yields a special result. Case 1:
− = −(11)
As we know (11)0−=
− = (0)
−= 1 (96)
Zero’s balance rule Any given equation is associated with zero that has to be balanced in that equation and this equation contains the information about zero that at what scale the zero is turning into a zero which is known as indexing scale. And this indexing scale of zero is given by the maximum positive or negative magnitude of that equation.
xx−=0x
Or xxx−=(0)
If equation a++= b c d then this equation with its indexing scale of the zero is given by
a+ b + c − d = 0d or a+ b + c − d = 0a++ b c . And if any magnitude is added to the equation, then it should be taken care that magnitude is also added to the indexing scale of the zero. So, if an equation
Universal Numbers 37 of 49
is a b+ c + d − = 0d and if we want to add mm− in that equation then that equation will be expressed as abcmdm+++−−= 0dm+ .
Number line beyond Real numbers The number line beyond the real numbers is not necessary as it is the same repetition of the entire real number line system multiple times. But, it turns necessary to understand the subtraction operation between zero to zero and infinity to infinity. Figure 10 is the representation of universal number lines with some of the major mathematical operation where n belongs to universal numbers.
Figure [10]
In the subtraction operation of nnn−:1 will yield 0:0 if n as all real numbera are very small compaired to universal numbers. And same for nnnn−:1, . So, if we want to yield output greater than zero for then we have to use universal number, example: equation (80) which is in the form of − that can be manuplated to be less then infinity that will yield any value for 0 to on the real number line system.
x k xxe − = k: k
The value of , 11− and 00− is undefine so it is respective and can change with various respective until we define value for atleast one operation and that followed by all the subtraction operation. So, generally, we defined 1−= 1 0 and using this definition, rest operation yields the specific outcome. However, if we choose to define − = 0 then 110−=2 and 0−= 0 03 . So, based on the specific definition for any one operation will shift the outcome accordingly for rest of the operations, but the sequenceial series of outcomes on the number line will remain same.
16. Miscellaneous problem
Problem [1]: Below given are the two generalize pattern formulas, justify which one is true when
Universal Numbers 38 of 49
X0= .
X10 = and 00X = : X
Solution:
010 = is true when ISD of base zero is equal or less than the power difference.
000 = is true when ISD of base zero is equal or more than the higher exponential difference. But both cannot be true in the same mathematical operation.
1 1 Problem [2]: There are given two equations. A = 2 0 0 ( x ) or 20 and B = 8 0 ( x ) or 8 x x where A and B are zero in terms of the real number system. Solve the following:
B 1 1) A + B 2) A – B 3) A × B 4) A ÷ B 5) A B 6) A A/B Solution:
1) A + B
20( 08xxx) 028+===( 000) ( ) x/28
2) A – B
20( 08xxx) 012−===( 000) ( ) x/12
3) A × B 200 80= 1600== 0 0 ( xx) ( ) ( xx22) /160
4) A ÷ B
20 5 20( 0) 8( 0) = = = 2.5 xx82
5) A B 81 2020xx 1 0x === ( 01xk/ ) xx8
B 6) A 1 / B / A
Universal Numbers 39 of 49
8 8 x x 82 1 x 2020520205 B / A= 8 / 20 === 20/ x 8 / 20 882
Problem [3]: Prove the following formulas available in approximation analysis are true where LHS of the equation have CIS.
A: =ln 1( 1+ ) B: =ln − 0( ) C: 0 ln ( ) 0= D: 0 ln 0( 0 ) =
Solution:
A: =ln( 1+ ) 1 ln( 1+ ) == lne 1 from equation (52)
B: =ln − 0( ) ln0ln0( ) ==−( ) from axiom [9]
C: 0 ln ( ) 0= ln ()ln==0 (1)0 from equation (69)
D: 0ln00( ) = ln (0)ln0 (1)0== from equation (74)
( 3− )! Problem [4]: Evaluate using formula obtained from example (24). ( 8− )!
Solution:
(−−− 3)!( 1)!( 1)! 37 =−−( 1)( 1) from equation (33) (7)!(31)!(71)!−−−
(−−−− 1)!( 1)!720 ( 1)! 720 −−== 27202 ( 1)!2 −−
(− 3)! = 360 (− 8)!
Problem [5]: Evaluate the Game function and Pi function for negative integers. Solution:
Gama function is given by
=()n e−xn x−1 dx 0
Where Pi function can be written in form Gama function as
Universal Numbers 40 of 49
=()n +( 1n = ) e x− dxxn 0
Integrating the above expression individually for ()n − , we get
For n =−4
e−x dx x4 Integration by parts: u v' u=− v u' v 11 ueu'e=⎯⎯→=−−−xx ; v'= ⎯⎯→ =v xx433
ee−−xx e−x 1 e−x −− dx − − dx ….. A 33xx33 3x 3 3 x 3 Integration of reaming integral part separately from equation A
e−x dx x 3 Integration by parts 11 ueu'e=⎯⎯→=−−xx − ; v'v=⎯⎯→= − xx322
ee−−xx e−x 1 e−x −− dx − − dx ….. B 22xx22 2x 2 2 x 2 Integration of reaming integral part separately from equation B
e−x dx x 2 Integration by parts 1 1 ueu'e=⎯⎯→=−−xx − ; v' =⎯⎯→ v =− x 2 x
ee−x −x −− dx ….. C xx Exponential integral is given by
e−x Ei()−= x dx ….. D x
Universal Numbers 41 of 49
Putting D in C and then C in B, we get
e−x e−x 1 e−x − − − E i x() − − −−− Eix() x 2x 2 2 x
Putting B in A, we get
−x −x −x e 1 e 1 e − − − −−− Eix() + C 3 − 2 3x 3 2x 2 x
2e−x e−x e Ei− x x ()− − + −− + C ….. E 6 x 3 6x 2 66x
e−x Taking − common for equation E, we get 6 x 3
eexEixxx−xx( 32()2−+−+ ) −+C 6 x 3
Similarly, integration of Pi function for negative factorial of 1, 2 and 5 are obtained as For n =−1
e−x dx=− Ei() x x For n =−2
−xx e−x eex( Eix ()1−+) dx =− xx2 For n =−5
−xx4 3 2 e−x e( e x Ei(− x ) + x − x + 2 x − 6) dx = xx5424
Calculating for other negative factorials, we get a solution such that it can be generalized in a below given pattern,
(− 1)nx+1 e− n e−−xn x2 dx = Ei(−+ xxk )! (− 1)kkn− n+1 (1)!nx+ k =0
Now using upper bound and lower bound of integration as zero to infinity, we get
Universal Numbers 42 of 49
(1)− n+1 1 n Eixxk()!−+ (1)− kkn− (1)!nex+ xn+1 k =0 0
So, when x = ,
Using zero reduction theorem, the entire equation goes to zero as e x is the term with the highest and dominant indexing scale which cannot be reduced by any other existing term indexing scale and also Ei ( )− 0 =
0 (1)− n+1 1 n Eixxk()!−+ (1)− kkn− where x = xn+1 (1)!nex+ k =0
When x = 0 ,
Using a zero reduction theorem, Ei ( 0)− can be neglected as 1/ x n+1 = and Ei ( 0)− = but indexing scale of Ei ( 0)− follows logarithmic function whereas 1/ 0 n+1 follows power function so is a relatively a constant entity if it is compared to 1/ 0 n+1 . Also, e 0 =1.
Therefore
n+1 n (− 1) C 1 −−Eix()+ (− 1)kkxkn− ! where x = 0 1 (1)!n + x n+1 k =0 e x
(1)− n+1 1 n − (1)− kkxkn− ! n+1 (1)!nx+ k =0
Expanding the summation series, we get
(− 1)n+1 1 −−+−+ xxxxnnnnnn x n−−−−123 26...! n+1 ( ) (1)!nx+
Again, 1/ x n+1 = is the highest indexing scale compared to other power terms, so all the terms behave as constant with respect to1/ x n+1
C (− 1)n+1 1 1 2 6 n! − − + − +... where (n+ 1)! x x2 x 3 x 4 x n+1 The last term is negative when n is an odd integer and positive if n is an even integer. So
x n+1
n +1
Universal Numbers 43 of 49
Therefore, we can express the above solution in a general form as
n+1 −−xn2 x +( 3n = ) +(n = 2) e x dx ; x = , n− 0 n +1
n −−xn1 x (n + 2) = +(n = 1) e x dx ; , 0 n
n−1 − xn x +( 1n = ) =(n) e x dx ; , 0 n −1
−−xn2 1 (n +1) = e x dx ; x = 0 , 0 nxn
Using the formula of derivatives and integration, we can write the above equation as
1 d n =1xx ; , x n xn−1 dx
−−xn2 e x dx x dxn ; , 0
n 1 d n dx 1xx ; , x dx
Problem [6]: Evaluate the Game function for positive integers. Solution: The integration of positive integers in the same manner as above for negative factorial and its general solution is obtained as n n! +=(1)n e− xn x dx =−ex−−xn k 0 ()!nk− k =0 0
When x =, the entire equation goes zero as e−x = 0 is of highest ISD when . So,
0 n n! =−ex−−x n k k =0 ()!nk−
When x = 0,
Universal Numbers 44 of 49 expanding the summation series, we get the following
n nnnn!!!! xxxxxnknnnn−−−−−=++++ n 1 23... k =0 ()!(1)!(2)!()!nknnnn−−−−
Every term with x is zero as x = 0
0 n n!!! n n n! xn− k= x n −1 + x n −23 + x n − +... + x nn− k =0 (n− k )! (1)! n − (2)! n − ()!nn−
nn!! +( 1n = ) e x− dxxn xxnnnnn−−==! 0 ()!()!nnnn−−
For positive integer, Gama function obtains a precise factorial value given by, +( 1nn = ) ! ; n+
e x −1 Problem [7]: Prove lim1 = . x→0 x Solution:
Let us consider x = 0 as the limit is x → 0
1 ee0 −−11 1 ==− e 0 1
1 11 +−1111 = +− = + − = Therefore
d Problem [8]: (eexx) = . Evaluate whether it is precisely true with respect to universal numbers. dx Solution:
x If it is not precise then evaluate how and when e starts losing its precision.
Solution:
Universal Numbers 45 of 49
n x dx fe'1( ) =+ dxn
n−1 n−1 xdx x 1 n 11++ =+n 1 ndxn nn
n−1 d x x (e ) =+1 dx n
n dnx x (e ) =+1 dxnxn + Similarly,
n x nn(1)− x f''e( ) =+1 ()nxn+ 2
Deriving the function multiple times, we get the general solution as
n kx n! x fe( ) =+1 ()!nknxn−+ () k
n! feekxx( ) = ()!nknx−+ () k where k is the number of times the equation has been derived.
Analyzing general solution when k is a constant real number value Using general solution to cover n number of derivatives,
d k n! (eexx) = : n = dxk ()!() n−+ k n x k
n! 1 1 Simplifying in the form of 1− where is the indexing scale of unity, we ()!nknx−+ () k n n get the following approximate equation.
d k k2 +− k(2 x 1) eexx=−1 : n = & nk k ( ) dx2 n
k 2 d xx1 k+− k(2 x 1) (ee) =−1 : c = dxk n/ c 2
Universal Numbers 46 of 49
k d xxx − (eee) ==(1 ) m dx k nc/ where m is the indexing scale of e x
k d xxx (eee) ==m : In the context of Real numbers dx k
So, in the context of universal numbers, every time we differentiate the f() x= e x , e x depletes its ddkk+1 precision of being itself. We can also state that, (eexx) ( ) . However, the loss in dxdxkk+1 precision can be noticed only in context the of universal numbers but in the context of a real number, d k we cannot track it. So, in the context of a real number, (eexx) = . dx k
Analyzing general solution when k is infinite value The general solution can be written as
d k n! (eexx) = : n = dxnknxk ()!−+ () k
n! So, if k the value increases to the infinity of any indexing scale, the overall value of ()!nknx−+ () k will not remain one but it starts decrementing from unity to zero. Therefore, if we differentiate infinite times, the function will go to zero as nnn ! when kn= .
n d xxn! (ee) ==0y : n = dxnknxn ()!−+ () k
n d x (e ) = 0y : n = dx n where y is the indexing scale of zero which is way higher than the indexing scale of infinity represented by n .
Problem [9]: Find the stability of Euler's identity ei =−1. Solution: Euler's formula is given by
eix =+cos( x ) i sin ( x )
At x =
Universal Numbers 47 of 49
eii =+cos( ) sin ( )
We can write LHS as
in i 1 e =+1 : n =x n We know that by the relative difference
kn n k 1 k e =+1 =+1 : n =x n n
So,
n i i e =+1 n Therefore,
n i 1cos()sin+=+ () i n
i 0 1cos()sin+=+ ()( i ) : n =x n
i 0 1(1)+=−
(1)1−−0 = i = (1 − ( − 1)0 )i
17. Results Using the basic analogy and principles of mathematics, we have built the foundation with theories, axioms, theorems and formulas of universal numbers that have let us understand in detail about every number including zero, unity and infinities. With the built framework, every zero and infinity can be uniquely used in any given equation without collapsing the equations. Also, many new formulas and detailed solutions are obtained which were unknown or inappropriate until the current date.
18. Conclusions and Discussions Zero is not just real number.
Universal Numbers 48 of 49
With the perspective of Universal numbers, Subadd system can be considered to be comprised of 0, +1 and -1 values only on the number line and the rest of the values are established by the muldive system. In future works, this system can be also merged with calculus for better possibilities of applications and it can be an alternative in certain fields of calculus as the entire calculus is based on the limit and infinitesimal, So, advancement in this number can be alternative to solve some calculus problems without knowing calculus. Also, it can increase the efficiency of calculus problems for computers as this can be programmed using the concept of algebra and not calculus. We know that zero, one and infinity with respect to a real number system have the probability to form indeterminate forms, but every number has the probability to form indeterminate. We only know these numbers as we get such numbers results in real number domain frequently whereas other numbers yield its multiple solutions in a form of infinity or zero and we treat all zero and infinity in the same manner so it always remains hidden. Considering this fact, universal numbers will be used more frequently once we start using various specific infinities and zeros in our equation.
We can prove that 0.99 1= but it is not true always as at some indexing scale 10 = (from equation 55) and unity in this equation can be also expressed as 0 .9 9 . This implies that 0 .9 9 is less than one and not one in this condition. Similarly, there are also occur other situations and numbers that change the number's state of existence depending on the operator or other interaction number. However, the universal number system can easily handle such conditions. The possible explanation to the above condition can be justified as: The indexing scale of both infinity and unity will reach to last infinite value (inverse of zero) which means unity should not be less than one. But due to the difference in indexing scale of both the numbers that is infinity have very large indexing scale than the unity, so the indexing scale of infinity reaches to last point faster and tends to detect that the indexing scale of unity has not reached to last infinite value which means that the numbers of 9’s present in the value of 0 .9 9 are not enough to justify that it is not equal to one. And therefore, in the output, it hints that one is less than one in the reference of that entire equation. The intuitive way to explain is, whenever an indexing scale of unity reaches to last infinite value, the indexing scale of infinity increase the horizon of the end value of infinite and so the indexing scale of infinite exists to a new larger value position and that is set as last infinite value for that equation. On the other side, the required accuracy for 0 .9 9 to be as one is not satisfied. It is important to know that, in certain mathematical operation, the number with the highest indexing scale sets the last infinite value and all required accuracy are relatively followed by another number. Looking into the final equation obtained from problem [4] and general formulas and relations related to e x in the context of derivatives and integration, it indicates that maybe some sort of neutral number for the calculus system. The zero was thought that be non-qualitative. It may not be quantitative in a general dimension, but it is quantitative in the zero dimension. Zero’s unity identity gives the relation between Muldive and the Subadd system. The zero balance rule is a very general rule that all mathematical equations must follow in the universal number system.
Universal Numbers 49 of 49
Whenever a number has multiple unique values, it simply means that number is the combination of multiple unique values. It is just that we are unaware of its exact unique value so we assume that it may be existing in all states. This may be also true in Quantum physics where it is believed that mattes exist in multiple states or at multiple places.
Acknowledgment
This research was independently conducted and no help was taken from any particulars, however, for certain analyses and verification of results, computational knowledge engine by Wolfram Alpha LLC was of great use.
Universal Numbers