Irrational Numbers - Case Study Example

Is the Irrational Number Normal or Non-normal for Base 3 and 4? Irrational numbers are numbers that lack fractional representations. Deriving their definition from Hippasus, one of Pythogoras students, irrational numbers do not have fractional equivalents. A number that cannot be expressed in form of a fraction or a ratio of two numbers qualifies as irrational number. The interaction between fractions and relevance of irrational numbers indicates that all fractions representing real numbers are rational numbers. Best examples of irrational numbers are the Euler’s Number e, π, and √2. Although elementary mathematics represents π as 22/7, it is incorrect as this fraction presents a real number with a definite end. The irrationally of any number can be presented with regards to the normality of the number. Normal irrational numbers and non-normal irrational numbers differentiate the classes of irrational numbers. For instance, if π was represented as a decimal number and expanded to its billionth decimal place, would the occurrence of 200,000 1s or 2s within this representation mean that π is normal or non-normal irrational number? While considering normality and non-normality of irrational numbers, π and e have been numerically represented to more than trillion decimal places. In these long expansions, patterns or sequences do not exist. The normality of π associates its lack of sequences and patterns to its irrationality. However, while π can be express to base 10 as an irrational number, it is also applicable to other bases such as 3 and 4. Taking normality of irrational numbers to mean the lack of patterns and sequences the expanded form, the representation of irrational numbers may give rise to patterns in other bases. When expressed to base 2, π does not present patterns between the 1s and 0s. Changing the bases on how irrational numbers are presented does not result to rational numbers. However, in their representation to base 10, some irrational numbers become rational numbers if squared. Hence, given for example that √2 is an irrational number, its product is a rational number if multiplied by itself. √2x√2= 2 (rational number) However, Π x Π = Π2 (irrational number) Understandably, in the handling of irrational numbers, chances are that these numbers don’t remain irrational under various conditions. For instance, the above example on the square of π shows that, if multiplied by itself, π remains irrational. This can also be supported if π was expressed in decimal for base two. Nevertheless, the representation of π in any base can be determined by the consideration of conditions making the number to either have or lack sequences and patterns. For instance, in order to get a hold of how irrational numbers relate with other bases or expansion to a certain length, say m-string of decimal places, it is important to comprehend the meaning of sequences. Given that irrational numbers can be expressed to different bases, observations show that regardless of whether irrational numbers remain irrational even in their squares, certain sequences of digits can be identified at least once within the decimal representation of the irrational number for a given string. Additionally, some digits appear much frequently than others – paving way for non-normal irrational numbers or representations of the same. The digits that comprise of irrational numbers are less researched on and thus contemporary scholars spend too much time in expanding the decimals to multibillion places. However, the venture to make the assumptions that irrational numbers are just irrational can be proved through the consideration of a real irrational number α and any positive integer N (Bailey and Crandall, 175-190). Another positive integer, P = P (N), exists which is independent of the positive irrational number α. To show that the possibility of a sequence occurring one time within the decimal places of the number, an integer X satisfies 1 ≥ X ≥ P. The representation of X is such that Xα has an infinite number with possible sequence of N digits 0, 1, 2, 3 …30. However, caution is to be observed as some irrational numbers are constructed to ensure that certain sequences and patterns of N digits do not occur. In this exercise, concern leans on the conditions that make irrational numbers either normal or non-normal. Whether these irrational numbers are perfect irrational number under various bases, especially 2 and 10, this exercise aims at investigating the normality of irrational numbers for bases 3 and 4. Specifically, normality and non-normality for bases 3 and 4 is concerned with proving that these numbers maintain their irrationally with or without sequences and patterns. Normality of Irrational numbers Definition of normal and non-normal numbers can better be expressed through the consideration of examples and proofs. In this case, assume that b denotes any positive integer not less than 2 while Є is a real number represented to its b-ary expansion, = [Є] + 0. α1α2…, Such that α1 and α2 are integers derived from {0, 1, 2… B – 1} while an infinity of the αk are do not equal to b – 1. Considering positive integer N and d as any digit in {0, 1, 2…, b – 1}, sets Ab (d, N, ) := Card {j : 1 ≤ j ≤ N, aj = d}. A more general perspective on the above representation for block Dk = d1…dk of value k digits starting from {0, 1, 2… b ­– 1}, setting Ab(Dk,N, ) := Card{j : 0 ≤ j ≤ N − k, aj+1 = d1, . . . , aj + k = dk}. Integer d1bk −1 + . . . + dk−1 b + dk is essential in the representation of block Dk. Borel (110) initially represented normal numbers in his published seminal paper of 1909. Hence, to define normal numbers, Borel definition is considered with the following example. Take any real number b ≥ 2. Considering the b-ary expansion of real number results to the limit of the sequence (Ab(d, N, ) / N) N ≥ 1 considering the frequency of digit d. However, the above condition is only true if the sequence (Ab(d, N, ) / N) N ≥ 1 converges – otherwise the sequence does not exist. Any real number is normal to base b if all digits 0, 1, 2…, b – 1 occur in the number’s b-ary expansion satisfying the frequency 1/b, under the condition that Whereas, d = 0, 1, 2…, b – 1. If each of Є, bЄ, b2Є, b3Є… is normal to all bases of b, b2, b3; then a number is normal to base b. It is correct to rule out numbers with non-recurring sequences of base b. For instance, a real number with the string, Є, bnЄ, bn+2Є, bn+3Є, bn+5, bn+5… etc may not be considered normal regardless of the fact that the n value of each subsequent integer value has a consecutive prime number added to it. With the consideration of popular irrational numbers such as e, pi, and square root of 2, it is believed that these numbers are normal to base b. According to Borel (194), all real numbers are normal to base b. In addition, Borel believes that all irrational algebraic numbers are normal to base b. However, this belief has not been used proof the normality of the popular irrational constants. Available proofs show that only Champernowne’s constant is normal to base 10 (0.12345678910111213141516171819…..). Taking pi for instance, it is fundamental to query whether pi is normal. Using random theory for the first 2,000,000,000 bits of pi shows that pi is non-normal for base-4 and normal for base 16 considering the first 4 trillion digits. However, the conclusion that pi is normal cannot be proved by the consideration of only 4 trillion decimals (Copeland and Erd¨os, 455 – 473). Normality and Non-normality for Base 3 and 4 Base 3 is also known as ternary number system in which numbers repeat their occurrence after every third digit that is normally a 2. In this number system, 0, 1, and 2 are the only numbers that are valid for use. Hence, in order to write 4 in base three, one would consider how many 3s and ones there are in 4. Hence, 4 can be expressed as 11, with the first 1 representing number of 3s and the second 1 representing the number of ones. While majority of known positive real numbers are constants can be expressed in base three, not all behave the same. For instance, irrational numbers have distinct properties that ensure that these numbers do not terminate under any condition. However, an irrational number is not affected by how the number is represented but rather by its properties. Taking pi or Euler’s constant, e, for example, it is understood that these constants are remarked as normal irrational numbers. In the testing of normality, patterns and sequences are observed to determine the normality state of the number. For instance, the expansion of pi to 2 billion decimal places shows that there are not detectable sequences or repeating digits of the number. Taking 2 billion decimal places as the cut line in ruling pi as a normal irrational number allows pi to be represented in base 2. The fact that pi does not have repeating patterns up to 2 billion digits for base 10, the same is not certain for base 3 or 4. According to the definition or normal and non-normal irrational numbers, repeating sequences are essential considerations. Conceptually, pi is normal for base 10 and thus evidence of its non-normality for the same base is vague. In order to apply theory to practice, consider the representation of 4 as rational number for base 3. 4 to base 3 is 11. On the other hand, to expand pi to its 2 billionth decimal place for base ten the following expression can be used Є, bnЄ, bn+2Є, bn+3Є, bn+5, bn+5… However, it is noted that pi is not represented as b-normal and thus, the sequence and value of n cannot be realized or represented with a formula (Korobov, 33). In addition, the consideration of the above sequence shows that each digit d in the sequence represented a positive real number. Thus, if pi is expressed in terms of positive real digits such as 0.14159265358979323846… up to its 2 billionth decimal place, each digit is expressed and considered an independent real number. Thus to draw the concept of normality and non-normality irrational numbers, base b examples apply below for any positive base-b integer. The representations of natural numbers and irrational constants for base b prove to elicit normality. With reference to an earlier observation, any real number is b-normal, considering the integer b ≥ 2 and accounting each m-long string of normal base-b if represented in the expansion of base b of the integer α with specific precision frequency of (1/bm). Assumptions are made that support that all real numbers are b-normal given that b ≥ 2 is satisfied in the measure theory perspective. In addition, all real numbers are assumed simultaneous b-normal considering all positive integers. While establishing the normality of natural numbers and constants, their representations for bases 3 and 4, are vaguely known. Drawing in to the concept of representing constants and natural numbers in b-base and other bases, consideration of parading explicit and normal natural numbers as inherently difficulty is sought. Champernown number is the only number proven as 10-normal. A study into irrational numbers shows that research on the representation of natural numbers lacks explicit analysis (Calude, 118). Notably, there are no proofs of normality for natural numbers such as π, e, √2, and log2. It cannot be established whether the appearance of the digit 1 consumes ½ of the time, within the limit, for the binary expansion of √2. Hence, a conclusion is drawn from this observation that all irrational algebraic numbers are b-normal to any positive integer base b. However, it is observed that no proof exists to implicate the assumption that all irrational numbers are b-normal to all positive integer bases for any known algebraic number expressed to a specific base. To explore possible representation of irrational numbers as normal or non-normal, take for example y as a real number with algebraic degree of D>1. For a positive number N and C relying on y input, #(|y|, N) of 1-bits within he binary m-string of y via position N that satisfy #(|y|, N) > CN1/d, is needed. However, this method lacks viable establishment of b-normality for irrational algebraic numbers in any particular base b. When the b-normality of α is proved, then rα and r + α normalizes for all non-zero positive rational r. Additionally, it is observed that real constants share normality for base a and b if integer a= bn is satisfied through a ≥ 2 and b ≥ 2. Conversely, Hertling proves that, if n was not present, an uncountable number of instances where irrational constants are a-normal but not b-normal. Discussion For bases 3 and 4, a natural constant, pi, is represented with recurring sequences and patterns. The rationale that all positive real numbers are normal to base b can be proved considering that 0, 1, 2…, b – 1 occur in the number’s b-ary expansion satisfying the frequency 1/b, under the condition that Whereas, d = 0, 1, 2…, b – 1 and; if each of Є, bЄ, b2Є, b3Є… is normal to all bases of b, b2, b3; then a number is normal to base b. However, irrational numbers, although are believed to be normal if presented in algebraic form, do not satisfy this condition for base 4. Normality for irrational numbers means the lack of patterns and sequences. However, non-normality of irrational numbers considers the frequency with which each digit appears. Thus, the expansion of natural constant, pi, for base 3 and 4 also shows that no prove no computational theorem or formula has been able to pinpoint patterns and recurring sequences. It is noted that, if a sequence favors one digit over the others, its normality is questionable, as irrational numbers do not satisfy such conditions if expressed to other bases (Borwein and Borwein, 234-276). Considering the following normality expressions for base b The sequence representations below show that normal numbers elicit such sequences with respect to the frequency of any digit d. These sequence representations fail to proof that irrational numbers can be normal to base 3 and base 4 considering the frequency of digits in base 3 and 4 respectively. = [Є] + 0. α1α2…, Whereas α1 and α2 are integers derived from {0, 1, 2… B – 1} while an infinity of the αk are do not equal to b – 1. Considering positive integer N and d as any digit in {0, 1, 2…, b – 1}, sets Ab (d, N, ) := Card {j : 1 ≤ j ≤ N, aj = d}. A more general perspective on the above representation for block Dk = d1…dk of value k digits starting from {0, 1, 2… b ­– 1}, set Ab(Dk,N, ) := Card{j : 0 ≤ j ≤ N − k, aj+1 = d1, . . . , aj + k = dk}. Integer d1bk −1 + . . . + dk−1 b + dk is essential in the representation of block Dk . These sequences denote the probability of representing any natural number as normal to any base besides 10, such as the Champernown number. Expansion of irrational numbers with respect to concepts and theorems giving possible normality and non-normality only show that normal base 10 is the only base allowing the normality of irrational numbers to be proved- at least to a considerable capacity for pi. Additionally, the proof that irrational numbers are normal to base 10 only holds under the condition that practical expansions have been carried out to a specified number of digits d. Further studies on the normality and non-normality of irrational numbers require a contemporary analysis to all bases. Thus, the normality and non-normality of irrational numbers to base 3 and 4 generate a vital need for further study as it is scholarly to ponder why an irrational real number is normal to base 2 and 10 and not for 3 or 7. Works Cited A. H. Copeland and P. Erd¨os, Note on Normal Numbers. Bulletin of the American Mathematical Society #52 (1). 1946: 857-860. Bailey, David., and Crandall, Richard. On the random character of fundamental constant expansions. Experiment. Math. #10. 2001: 175–190. Beyer, A., Metropolis, N., and Neergaard, J. Statistical study of digits of some square roots of integers in various bases. Math. Comp. #24. 1970: 455–473. Borwein, J. and Borwein, B. Pi and the AGM: A Study in Analytic Number Theory and Computational Complexity. New York: John Wiley, 1987. Calude, C. Borel Normality and Algorithmic Randomness. In: Rozenberg, G., Salomaa, A. (Eds.) Developments In Language Theory, Pp. 113–119. Singapore: World Scientific, 1994. Calude, C., and Dinneen, M. Computing A Glimpse Of Randomness. Exp. Math #11, 2002; 361–370 Francisco, A. et al. Walking on Real Numbers. The Mathematical Intelligencer Vol. 35 (1). Korobov, A. Continued fractions of some normal numbers, Mat. Zametki #47 (2).1990: 28–33. Niven, Ivan., and Zuckerman, H. On the definition of normal numbers. Pacific Journal of Mathematics #1, 1951: 103-109. Shallit, J. Simple continued fractions for some irrational numbers. Number Theory #11, 1979: 209–217. Read More

Another positive integer, P = P (N), exists which is independent of the positive irrational number α. To show that the possibility of a sequence occurring one time within the decimal places of the number, an integer X satisfies 1 ≥ X ≥ P. The representation of X is such that Xα has an infinite number with possible sequence of N digits 0, 1, 2, 3 …30. However, caution is to be observed as some irrational numbers are constructed to ensure that certain sequences and patterns of N digits do not occur.

In this exercise, concern leans on the conditions that make irrational numbers either normal or non-normal. Whether these irrational numbers are perfect irrational number under various bases, especially 2 and 10, this exercise aims at investigating the normality of irrational numbers for bases 3 and 4. Specifically, normality and non-normality for bases 3 and 4 is concerned with proving that these numbers maintain their irrationally with or without sequences and patterns. Normality of Irrational numbers Definition of normal and non-normal numbers can better be expressed through the consideration of examples and proofs.

In this case, assume that b denotes any positive integer not less than 2 while Є is a real number represented to its b-ary expansion, = [Є] + 0. α1α2…, Such that α1 and α2 are integers derived from {0, 1, 2… B – 1} while an infinity of the αk are do not equal to b – 1. Considering positive integer N and d as any digit in {0, 1, 2…, b – 1}, sets Ab (d, N, ) := Card {j : 1 ≤ j ≤ N, aj = d}. A more general perspective on the above representation for block Dk = d1…dk of value k digits starting from {0, 1, 2… b ­– 1}, setting Ab(Dk,N, ) := Card{j : 0 ≤ j ≤ N − k, aj+1 = d1, . . .

, aj + k = dk}. Integer d1bk −1 + . . . + dk−1 b + dk is essential in the representation of block Dk. Borel (110) initially represented normal numbers in his published seminal paper of 1909. Hence, to define normal numbers, Borel definition is considered with the following example. Take any real number b ≥ 2. Considering the b-ary expansion of real number results to the limit of the sequence (Ab(d, N, ) / N) N ≥ 1 considering the frequency of digit d. However, the above condition is only true if the sequence (Ab(d, N, ) / N) N ≥ 1 converges – otherwise the sequence does not exist.

Any real number is normal to base b if all digits 0, 1, 2…, b – 1 occur in the number’s b-ary expansion satisfying the frequency 1/b, under the condition that Whereas, d = 0, 1, 2…, b – 1. If each of Є, bЄ, b2Є, b3Є… is normal to all bases of b, b2, b3; then a number is normal to base b. It is correct to rule out numbers with non-recurring sequences of base b. For instance, a real number with the string, Є, bnЄ, bn+2Є, bn+3Є, bn+5, bn+5… etc may not be considered normal regardless of the fact that the n value of each subsequent integer value has a consecutive prime number added to it.

With the consideration of popular irrational numbers such as e, pi, and square root of 2, it is believed that these numbers are normal to base b. According to Borel (194), all real numbers are normal to base b. In addition, Borel believes that all irrational algebraic numbers are normal to base b. However, this belief has not been used proof the normality of the popular irrational constants. Available proofs show that only Champernowne’s constant is normal to base 10 (0.12345678910111213141516171819…..).

Taking pi for instance, it is fundamental to query whether pi is normal. Using random theory for the first 2,000,000,000 bits of pi shows that pi is non-normal for base-4 and normal for base 16 considering the first 4 trillion digits. However, the conclusion that pi is normal cannot be proved by the consideration of only 4 trillion decimals (Copeland and Erd¨os, 455 – 473). Normality and Non-normality for Base 3 and 4 Base 3 is also known as ternary number system in which numbers repeat their occurrence after every third digit that is normally a 2.

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