Approximate Method to Transformed Discreet Variables to Continues Variables J.P.D Manoj Sithara Oct/23/2017 Abstract The discreet and continuousness is fundamental things to study problems of nature. However if we examine the neighboring Real numbers with large finite decimal places it appear to be approximate continuity with small amount of gaps.Those gaps can be ignored in large scales evaluations and achieved approximate continuity for many theoretical works in mathematical modeling. Therefore it is possible to create natural numbers in to close neighboring real numbers is also possible. The paper provide pure mathematical background to reduce best approximation to convert discreet-variable to continues variable.

1

Introduction

In many practical cases pure mathematical content of calculus did not provide strong supports to do handle physical problems. As an example it can be consider to discreetness of matter have no meaning to it discreetness in atomic scale to pure mathematics of calculus when it tempt to calculate center of gravity of larger Newtonian objects. Also we can consider the modern computer ability to evaluate many mathematical problem using less pure mathematical contents of real numbers. Computers can draw almost continues curves and makes valuable predictions even without analytical continuation of derivatives explain in algebra of limits. Therefore we must be able to justify the approximate continues transformation using natural numbers with use of pure mathematics.

1

2 2.1

Pure mathematical understanding of discreet variable to continues variable Natural numbers and number systems

The understanding of natural number N can be represent using a number system has the base B .Where the number digit’s satisfying 0 ≤ ai ≤ B to ith number digit ai . Hence any Natural number N in number system can be formed as below

N=

∞ X

ai B i ; [N, a, B, i] ∈ {ℵ}

(1)

i=0

where ℵ is natural number set .As a example we can consider the natural number N = 1502 in decimal base. That is B = 10 and {a3 = 1, a2 = 5, a1 = 0, a0 = 2} with ai |∞ 4 = 0 in wider sense it can be written as a sum of a series as , N = [.. + 0 + 0 + 1 × 103 + 5 × 102 + 0 × 101 + 2 × 100 ] .

2.2

Approximate transformed of Discreet variable to continues variable

It can insert large value of B p (to large p ) in to above natural number N as unaffected to its value as shown in below

N = B +p

∞ X

ai B −(P −i)

(2)

i=0

The value is unaffected but there is something new . Let us consider the P∞ factor xp = i=0 ai B −(P −i) for some real number x ∈ < . Then xp is a rational number have p decimal places. If P is very large then discreetness of xp is undetectably small. Hence we can deduce the transformed to large vale B p . N ≡ B +p xp

(3)

Which gives approximate transformed of Discreet variable N to continues variable x as we take non infinite large value as B p = U . N∼ = Ux

2

(4)

3

Conclusion

Perhaps This can be the grates understanding of natural numbers and it connection real variable. In above steps we have been demonstrate how to compress integers in to finite decimal . For example we can list a sequence of close neighboring real numbers to consecrative integers. Table 1: Natural number compression in to a real variable n- natural number

Variable value with P decimals 0.000......................0 0.000......................1 0.000......................2 0.000......................3 0.000....................... 0.000....................... 0.000....................... 0.000....................... 0.000....................... 0.99999999999.....997 0.99999999999.....998 0.99999999999.....999

0 1 2 3 .. .. .. .. .. 99999999999.....997 99999999999.....998 99999999999.....999

If the integer N ≡ B +p xp has large B p in above Transformation, all the integers have been compress in to variable with P decimal places. In actual practice in mathematical modeling, it is worth to define such a Value to B p = U to approximate thing in to pure variable x to have the formed N ∼ = U x .Where the U is just define value Cant be shown to any real number ,But it will satisfy all the properties of the real number and the value is relatively large number sufficient to consider the Transformed of Discreet to continues.

3

1

Introduction

In many practical cases pure mathematical content of calculus did not provide strong supports to do handle physical problems. As an example it can be consider to discreetness of matter have no meaning to it discreetness in atomic scale to pure mathematics of calculus when it tempt to calculate center of gravity of larger Newtonian objects. Also we can consider the modern computer ability to evaluate many mathematical problem using less pure mathematical contents of real numbers. Computers can draw almost continues curves and makes valuable predictions even without analytical continuation of derivatives explain in algebra of limits. Therefore we must be able to justify the approximate continues transformation using natural numbers with use of pure mathematics.

1

2 2.1

Pure mathematical understanding of discreet variable to continues variable Natural numbers and number systems

The understanding of natural number N can be represent using a number system has the base B .Where the number digit’s satisfying 0 ≤ ai ≤ B to ith number digit ai . Hence any Natural number N in number system can be formed as below

N=

∞ X

ai B i ; [N, a, B, i] ∈ {ℵ}

(1)

i=0

where ℵ is natural number set .As a example we can consider the natural number N = 1502 in decimal base. That is B = 10 and {a3 = 1, a2 = 5, a1 = 0, a0 = 2} with ai |∞ 4 = 0 in wider sense it can be written as a sum of a series as , N = [.. + 0 + 0 + 1 × 103 + 5 × 102 + 0 × 101 + 2 × 100 ] .

2.2

Approximate transformed of Discreet variable to continues variable

It can insert large value of B p (to large p ) in to above natural number N as unaffected to its value as shown in below

N = B +p

∞ X

ai B −(P −i)

(2)

i=0

The value is unaffected but there is something new . Let us consider the P∞ factor xp = i=0 ai B −(P −i) for some real number x ∈ < . Then xp is a rational number have p decimal places. If P is very large then discreetness of xp is undetectably small. Hence we can deduce the transformed to large vale B p . N ≡ B +p xp

(3)

Which gives approximate transformed of Discreet variable N to continues variable x as we take non infinite large value as B p = U . N∼ = Ux

2

(4)

3

Conclusion

Perhaps This can be the grates understanding of natural numbers and it connection real variable. In above steps we have been demonstrate how to compress integers in to finite decimal . For example we can list a sequence of close neighboring real numbers to consecrative integers. Table 1: Natural number compression in to a real variable n- natural number

Variable value with P decimals 0.000......................0 0.000......................1 0.000......................2 0.000......................3 0.000....................... 0.000....................... 0.000....................... 0.000....................... 0.000....................... 0.99999999999.....997 0.99999999999.....998 0.99999999999.....999

0 1 2 3 .. .. .. .. .. 99999999999.....997 99999999999.....998 99999999999.....999

If the integer N ≡ B +p xp has large B p in above Transformation, all the integers have been compress in to variable with P decimal places. In actual practice in mathematical modeling, it is worth to define such a Value to B p = U to approximate thing in to pure variable x to have the formed N ∼ = U x .Where the U is just define value Cant be shown to any real number ,But it will satisfy all the properties of the real number and the value is relatively large number sufficient to consider the Transformed of Discreet to continues.

3