Some special numbers

Jai Ganesh · 2021-11-25 02:44:02

37) Harshad number

In mathematics, a harshad number (or Niven number) in a given number base is an integer that is divisible by the sum of its digits when written in that base. Harshad numbers in base n are also known as n-harshad (or n-Niven) numbers. Harshad numbers were defined by D. R. Kaprekar, a mathematician from India. The word "harshad" comes from the Sanskrit harṣa (joy) + da (give), meaning joy-giver. The term "Niven number" arose from a paper delivered by Ivan M. Niven at a conference on number theory in 1977.

Definition

Stated mathematically, let X be a positive integer with m digits when written in base n, and let the digits be

(It follows that must be either zero or a positive integer up to . X can be expressed as

X is a harshad number in base n if:

A number which is a harshad number in every number base is called an all-harshad number, or an all-Niven number. There are only four all-harshad numbers: 1, 2, 4, and 6. The number 12 is a harshad number in all bases except octal.

Examples

The number 18 is a harshad number in base 10, because the sum of the digits 1 and 8 is 9 (1 + 8 = 9), and 18 is divisible by 9.
The Hardy–Ramanujan number (1729) is a harshad number in base 10, since it is divisible by 19, the sum of its digits (1729 = 19 × 91).
The number 19 is not a harshad number in base 10, because the sum of the digits 1 and 9 is 10 (1 + 9 = 10), and 19 is not divisible by 10.

Jai Ganesh · 2021-11-28 21:35:38

38) Hypercomplex numbers

Some number systems that are not included in the complex numbers may be constructed from the real numbers in a way that generalize the construction of the complex numbers. They are sometimes called hypercomplex numbers. They include the quaternions H, introduced by Sir William Rowan Hamilton, in which multiplication is not commutative, the octonions, in which multiplication is not associative in addition to not being commutative, and the sedenions, in which multiplication is not alternative, neither associative nor commutative.

In mathematics, hypercomplex number is a traditional term for an element of a finite-dimensional unital algebra over the field of real numbers. The study of hypercomplex numbers in the late 19th century forms the basis of modern group representation theory.

A definition of a hypercomplex number is given by Kantor & Solodovnikov (1989) as an element of a finite-dimensional algebra over the real numbers that is unital but not necessarily associative or commutative. Elements are generated with real number coefficients

for a basis . Where possible, it is conventional to choose the basis so that . A technical approach to hypercomplex numbers directs attention first to those of dimension two.

Jai Ganesh · 2021-11-28 21:50:06

39) Transfinite numbers

For dealing with infinite sets, the natural numbers have been generalized to the ordinal numbers and to the cardinal numbers. The former gives the ordering of the set, while the latter gives its size. For finite sets, both ordinal and cardinal numbers are identified with the natural numbers. In the infinite case, many ordinal numbers correspond to the same cardinal number.

In mathematics, transfinite numbers are numbers that are "infinite" in the sense that they are larger than all finite numbers, yet not necessarily absolutely infinite. These include the transfinite cardinals, which are cardinal numbers used to quantify the size of infinite sets, and the transfinite ordinals, which are ordinal numbers used to provide an ordering of infinite sets. The term transfinite was coined by Georg Cantor in 1895, who wished to avoid some of the implications of the word infinite in connection with these objects, which were, nevertheless, not finite. Few contemporary writers share these qualms; it is now accepted usage to refer to transfinite cardinals and ordinals as "infinite". Nevertheless, the term "transfinite" also remains in use.

Definition

Any finite natural number can be used in at least two ways: as an ordinal and as a cardinal. Cardinal numbers specify the size of sets (e.g., a bag of five marbles), whereas ordinal numbers specify the order of a member within an ordered set (e.g., "the third man from the left" or "the twenty-seventh day of January"). When extended to transfinite numbers, these two concepts become distinct. A transfinite cardinal number is used to describe the size of an infinitely large set, while a transfinite ordinal is used to describe the location within an infinitely large set that is ordered. The most notable ordinal and cardinal numbers are, respectively:

(Omega): the lowest transfinite ordinal number. It is also the order type of the natural numbers under their usual linear ordering.

(Aleph-null): the first transfinite cardinal number. It is also the cardinality of the natural numbers. If the axiom of choice holds, the next higher cardinal number is aleph-one, . If not, there may be other cardinals which are incomparable with aleph-one and larger than aleph-null. Either way, there are no cardinals between aleph-null and aleph-one.

The continuum hypothesis is the proposition that there are no intermediate cardinal numbers between

and the cardinality of the continuum (the cardinality of the set of real numbers): or equivalently that is the cardinality of the set of real numbers. In Zermelo–Fraenkel set theory, neither the continuum hypothesis nor its negation can be proven.

Some authors, including P. Suppes and J. Rubin, use the term transfinite cardinal to refer to the cardinality of a Dedekind-infinite set in contexts where this may not be equivalent to "infinite cardinal"; that is, in contexts where the axiom of countable choice is not assumed or is not known to hold. Given this definition, the following are all equivalent:

is a transfinite cardinal. That is, there is a Dedekind infinite set such that the cardinality of is

There is a cardinal

such that

Although transfinite ordinals and cardinals both generalize only the natural numbers, other systems of numbers, including the hyperreal numbers and surreal numbers, provide generalizations of the real numbers.

a) In set theory, an ordinal number, or ordinal, is one generalization of the concept of a natural number that is used to describe a way to arrange a (possibly infinite) collection of objects in order, one after another.

Any finite collection of objects can be put in order just by the process of counting: labeling the objects with distinct natural numbers. The basic idea of ordinal numbers is to generalize this process to possibly infinite collections and to provide a "label" for each step in the process. Ordinal numbers are thus the "labels" needed to arrange collections of objects in order.

An ordinal number is used to describe the order type of a well-ordered set (though this does not work for a well-ordered proper class). A well-ordered set is a set with a relation < such that:

* (Trichotomy) For any elements x and y, exactly one of these statements is true:

* x < y
* y < x
* x = y

* (Transitivity) For any elements x, y, z, if x < y and y < z, then x < z.

* (Well-foundedness) Every nonempty subset has a least element, that is, it has an element x such that there is no other element y in the subset where y < x.

Two well-ordered sets have the same order type, if and only if there is a bijection from one set to the other that converts the relation in the first set, to the relation in the second set.

b) In mathematics, cardinal numbers, or cardinals for short, are a generalization of the natural numbers used to measure the cardinality (size) of sets. The cardinality of a finite set is a natural number: the number of elements in the set. The transfinite cardinal numbers, often denoted using the Hebrew symbol

followed by a subscript, describe the sizes of infinite sets.

Cardinality is defined in terms of bijective functions. Two sets have the same cardinality if, and only if, there is a one-to-one correspondence (bijection) between the elements of the two sets. In the case of finite sets, this agrees with the intuitive notion of size. In the case of infinite sets, the behavior is more complex. A fundamental theorem due to Georg Cantor shows that it is possible for infinite sets to have different cardinalities, and in particular the cardinality of the set of real numbers is greater than the cardinality of the set of natural numbers. It is also possible for a proper subset of an infinite set to have the same cardinality as the original set—something that cannot happen with proper subsets of finite sets.

There is a transfinite sequence of cardinal numbers:

This sequence starts with the natural numbers including zero (finite cardinals), which are followed by the aleph numbers (infinite cardinals of well-ordered sets). The aleph numbers are indexed by ordinal numbers. Under the assumption of the axiom of choice, this transfinite sequence includes every cardinal number. If one rejects that axiom, the situation is more complicated, with additional infinite cardinals that are not alephs.

Cardinality is studied for its own sake as part of set theory. It is also a tool used in branches of mathematics including model theory, combinatorics, abstract algebra and mathematical analysis. In category theory, the cardinal numbers form a skeleton of the category of sets.

Jai Ganesh · 2021-11-28 22:01:40

40) Nonstandard numbers

Hyperreal numbers are used in non-standard analysis. The hyperreals, or nonstandard reals (usually denoted as *R), denote an ordered field that is a proper extension of the ordered field of real numbers R and satisfies the transfer principle. This principle allows true first-order statements about R to be reinterpreted as true first-order statements about *R.

Superreal and surreal numbers extend the real numbers by adding infinitesimally small numbers and infinitely large numbers, but still form fields.

In mathematics, the system of hyperreal numbers is a way of treating infinite and infinitesimal (infinitely small but non-zero) quantities. The hyperreals, or nonstandard reals, *R, are an extension of the real numbers R that contains numbers greater than anything of the form

(for any finite number of terms).

Such numbers are infinite, and their reciprocals are infinitesimals. The term "hyper-real" was introduced by Edwin Hewitt in 1948.

The hyperreal numbers satisfy the transfer principle, a rigorous version of Leibniz's heuristic law of continuity. The transfer principle states that true first-order statements about R are also valid in *R. For example, the commutative law of addition, x + y = y + x, holds for the hyperreals just as it does for the reals; since R is a real closed field, so is *R. Since

for all integers n, one also has for all hyperintegers H. The transfer principle for ultrapowers is a consequence of Łoś' theorem of 1955.

Concerns about the soundness of arguments involving infinitesimals date back to ancient Greek mathematics, with Archimedes replacing such proofs with ones using other techniques such as the method of exhaustion. In the 1960s, Abraham Robinson proved that the hyperreals were logically consistent if and only if the reals were. This put to rest the fear that any proof involving infinitesimals might be unsound, provided that they were manipulated according to the logical rules that Robinson delineated.

The application of hyperreal numbers and in particular the transfer principle to problems of analysis is called nonstandard analysis. One immediate application is the definition of the basic concepts of analysis such as the derivative and integral in a direct fashion, without passing via logical complications of multiple quantifiers. Thus, the derivative of f(x) becomes

for an infinitesimal , where st(·) denotes the standard part function, which "rounds off" each finite hyperreal to the nearest real. Similarly, the integral is defined as the standard part of a suitable infinite sum.

Jai Ganesh · 2021-12-07 22:43:29

41) Harmonic number

In mathematics, the n-th harmonic number is the sum of the reciprocals of the first n natural numbers:

Starting from n = 1, the sequence of harmonic numbers begins:

Harmonic numbers are related to the harmonic mean in that the n-th harmonic number is also n times the reciprocal of the harmonic mean of the first n positive integers.

Harmonic numbers have been studied since antiquity and are important in various branches of number theory. They are sometimes loosely termed harmonic series, are closely related to the Riemann zeta function, and appear in the expressions of various special functions.

The harmonic numbers roughly approximate the natural logarithm function and thus the associated harmonic series grows without limit, albeit slowly. In 1737, Leonhard Euler used the divergence of the harmonic series to provide a new proof of the infinity of prime numbers. His work was extended into the complex plane by Bernhard Riemann in 1859, leading directly to the celebrated Riemann hypothesis about the distribution of prime numbers.

When the value of a large quantity of items has a Zipf's law distribution, the total value of the n most-valuable items is proportional to the n-th harmonic number. This leads to a variety of surprising conclusions regarding the long tail and the theory of network value.

Bertrand's postulate implies that, except for the case n = 1, the harmonic numbers are never integers.

Jai Ganesh · 2021-12-16 18:26:04

42) Split-complex number

In algebra, a split complex number (or hyperbolic number, also perplex number, double number) has two real number components x and y, and is written

, where . The conjugate of z is . Since , the product of a number z with its conjugate is , an isotropic quadratic form, .

The collection D of all split complex numbers z = x + y j for x, y ∈ R forms an algebra over the field of real numbers. Two split-complex numbers w and z have a product wz that satisfies N(wz) = N(w)N(z). This composition of N over the algebra product makes (D, +, ×, *) a composition algebra.

A similar algebra based on

and component-wise operations of addition and multiplication, , where xy is the quadratic form on , also forms a quadratic space. The ring isomorphism

relates proportional quadratic forms, but the mapping is not an isometry since the multiplicative identity (1, 1) of R2 is at a distance √2 from 0, which is normalized in D.

Split-complex numbers have many other names;

In mathematics, a function of a motor variable is a function with arguments and values in the split-complex number plane, much as functions of a complex variable involve ordinary complex numbers. William Kingdon Clifford coined the term motor for a kinematic operator in his "Preliminary Sketch of Biquaternions" (1873). He used split-complex numbers for scalars in his split-biquaternions. Motor variable is used here in place of split-complex variable for euphony and tradition.

For example,

Functions of a motor variable provide a context to extend real analysis and provide compact representation of mappings of the plane. However, the theory falls well short of function theory on the ordinary complex plane. Nevertheless, some of the aspects of conventional complex analysis have an interpretation given with motor variables, and more generally in hypercomplex analysis.

Definition

A split-complex number is an ordered pair of real numbers, written in the form

where x and y are real numbers and the quantity j satisfies

Choosing

results in the complex numbers. It is this sign change which distinguishes the split-complex numbers from the ordinary complex ones. The quantity j here is not a real number but an independent quantity.

The collection of all such z is called the split-complex plane. Addition and multiplication of split-complex numbers are defined by

This multiplication is commutative, associative and distributes over addition.

Jai Ganesh · 2022-01-07 21:12:34

43) Narcissistic Numbers

Narcissistic Number is a number that is the sum of its own digits each raised to the power of the number of digits:

Examples :

153

Explanation:

1634

Explanation:

In number theory, a narcissistic number (also known as a pluperfect digital invariant (PPDI), an Armstrong number (after Michael F. Armstrong) or a plus perfect number) in a given number base b is a number that is the sum of its own digits each raised to the power of the number of digits.

Definition

Let n be a natural number. We define the narcissistic function for base

to be the following:

where is the number of digits in the number in base b, and

is the value of each digit of the number. A natural number n is a narcissistic number if it is a fixed point for , which occurs if . The natural numbers are trivial narcissistic numbers for all b, all other narcissistic numbers are nontrivial narcissistic numbers.

For example, the number 153 in base

is a narcissistic number, because k=3 and .

Jai Ganesh · 2022-01-15 02:29:11

44) Permutable prime

A permutable prime, also known as anagrammatic prime, is a prime number which, in a given base, can have its digits' positions switched through any permutation and still be a prime number. H. E. Richert, who is supposedly the first to study these primes, called them permutable primes, but later they were also called absolute primes.

In base 10, all the permutable primes with fewer than 49,081 digits are known

Of the above, there are 16 unique permutation sets, with smallest elements

Note

is a repunit, a number consisting only of n ones (in base 10). Any repunit prime is a permutable prime with the above definition, but some definitions require at least two distinct digits.

All permutable primes of two or more digits are composed from the digits 1, 3, 7, 9, because no prime number except 2 is even, and no prime number besides 5 is divisible by 5. It is proven that no permutable prime exists which contains three different of the four digits 1, 3, 7, 9, as well as that there exists no permutable prime composed of two or more of each of two digits selected from 1, 3, 7, 9.

There is no n-digit permutable prime for

which is not a repunit. It is conjectured that there are no non-repunit permutable primes other than those listed above.

Jai Ganesh · 2022-03-04 22:26:48

45) Sierpiński number

In number theory, a Sierpiński number is an odd natural number k such that

is composite for all natural numbers n. In 1960, Wacław Sierpiński proved that there are infinitely many odd integers k which have this property.

In other words, when k is a Sierpiński number, all members of the following set are composite:

If the form is instead

, then k is a Riesel number.

Known Sierpiński numbers

The sequence of currently known Sierpiński numbers begins with:

78557, 271129, 271577, 322523, 327739, 482719, 575041, 603713, 903983, 934909, 965431, 1259779, 1290677, 1518781, 1624097, 1639459, 1777613, 2131043, 2131099, 2191531, 2510177, 2541601, 2576089, 2931767, 2931991, ...

The number 78557 was proved to be a Sierpiński number by John Selfridge in 1962, who showed that all numbers of the form

have a factor in the covering set {3, 5, 7, 13, 19, 37, 73}. For another known Sierpiński number, 271129, the covering set is {3, 5, 7, 13, 17, 241}. Most currently known Sierpiński numbers possess similar covering sets.

Jai Ganesh · 2022-03-04 22:34:46

46) Riesel number

In mathematics, a Riesel number is an odd natural number k for which

is composite for all natural numbers n (sequence A101036 in the OEIS). In other words, when k is a Riesel number, all members of the following set are composite:

If the form is instead

, then k is a Sierpinski number.

Known Riesel numbers

The sequence of currently known Riesel numbers begins with:

509203, 762701, 777149, 790841, 992077, 1106681, 1247173, 1254341, 1330207, 1330319, 1715053, 1730653, 1730681, 1744117, 1830187, 1976473, 2136283, 2251349, 2313487, 2344211, 2554843, 2924861, ...

Jai Ganesh · 2022-03-07 02:14:08

47) Wieferich prime

In number theory, a Wieferich prime is a prime number p such that

divides therefore connecting these primes with Fermat's little theorem, which states that every odd prime p divides . Wieferich primes were first described by Arthur Wieferich in 1909 in works pertaining to Fermat's Last Theorem, at which time both of Fermat's theorems were already well known to mathematicians.

Since then, connections between Wieferich primes and various other topics in mathematics have been discovered, including other types of numbers and primes, such as Mersenne and Fermat numbers, specific types of pseudoprimes and some types of numbers generalized from the original definition of a Wieferich prime. Over time, those connections discovered have extended to cover more properties of certain prime numbers as well as more general subjects such as number fields and the abc conjecture.

As of March 2021, the only known Wieferich primes are 1093 and 3511

Jai Ganesh · 2022-05-23 01:23:43

48) Perfect totient number

In number theory, a perfect totient number is an integer that is equal to the sum of its iterated totients. That is, we apply the totient function to a number n, apply it again to the resulting totient, and so on, until the number 1 is reached, and add together the resulting sequence of numbers; if the sum equals n, then n is a perfect totient number.

For example, there are six positive integers less than 9 and relatively prime to it, so the totient of 9 is 6; there are two numbers less than 6 and relatively prime to it, so the totient of 6 is 2; and there is one number less than 2 and relatively prime to it, so the totient of 2 is 1; and 9 = 6 + 2 + 1, so 9 is a perfect totient number.

The first few perfect totient numbers are

3, 9, 15, 27, 39, 81, 111, 183, 243, 255, 327, 363, 471, 729, 2187, 2199, 3063, 4359, 4375, .....

In symbols, one writes

for the iterated totient function. Then if c is the integer such that

one has that n is a perfect totient number if

In number theory, Euler's totient function counts the positive integers up to a given integer n that are relatively prime to n. It is written using the Greek letter phi as

or , and may also be called Euler's phi function. In other words, it is the number of integers k in the range 1 ≤ k ≤ n for which the greatest common divisor gcd(n, k) is equal to 1. The integers k of this form are sometimes referred to as totatives of n.

For example, the totatives of n = 9 are the six numbers 1, 2, 4, 5, 7 and 8. They are all relatively prime to 9, but the other three numbers in this range, 3, 6, and 9 are not, since gcd(9, 3) = gcd(9, 6) = 3 and gcd(9, 9) = 9. Therefore, φ(9) = 6. As another example, φ(1) = 1 since for n = 1 the only integer in the range from 1 to n is 1 itself, and gcd(1, 1) = 1.

Euler's totient function is a multiplicative function, meaning that if two numbers m and n are relatively prime, then φ(mn) = φ(m)φ(n). This function gives the order of the multiplicative group of integers modulo n (the group of units of the ring

).

Jai Ganesh · 2022-06-07 19:32:19

49) Smarandache–Wellin number

In mathematics, a Smarandache–Wellin number is an integer that in a given base is the concatenation of the first n prime numbers written in that base. Smarandache–Wellin numbers are named after Florentin Smarandache and Paul R. Wellin.

The first decimal Smarandache–Wellin numbers are:

2, 23, 235, 2357, 235711, 23571113, 2357111317, 235711131719, 23571113171923, 2357111317192329, ...

Smarandache–Wellin prime

A Smarandache–Wellin number that is also prime is called a Smarandache–Wellin prime. The first three are 2, 23 and 2357 (sequence A069151 in the OEIS). The fourth is 355 digits long: it is the result of concatenating the first 128 prime numbers, through 719.

The primes at the end of the concatenation in the Smarandache–Wellin primes are

2, 3, 7, 719, 1033, 2297, 3037, 11927, ...

The indices of the Smarandache–Wellin primes in the sequence of Smarandache–Wellin numbers are:

1, 2, 4, 128, 174, 342, 435, 1429, ...

The 1429th Smarandache–Wellin number is a probable prime with 5719 digits ending in 11927, discovered by Eric W. Weisstein in 1998. If it is proven prime, it will be the eighth Smarandache–Wellin prime. In March 2009, Weisstein's search showed the index of the next Smarandache–Wellin prime (if one exists) is at least 22077.

Jai Ganesh · 2022-07-06 22:05:12

50) p-adic number

In mathematics, the p-adic number system for any prime number p extends the ordinary arithmetic of the rational numbers in a different way from the extension of the rational number system to the real and complex number systems. The extension is achieved by an alternative interpretation of the concept of "closeness" or absolute value. In particular, two p-adic numbers are considered to be close when their difference is divisible by a high power of p: the higher the power, the closer they are. This property enables p-adic numbers to encode congruence information in a way that turns out to have powerful applications in number theory – including, for example, in the famous proof of Fermat's Last Theorem by Andrew Wiles.

These numbers were first described by Kurt Hensel in 1897, though, with hindsight, some of Ernst Kummer's earlier work can be interpreted as implicitly using p-adic numbers. The p-adic numbers were motivated primarily by an attempt to bring the ideas and techniques of power series methods into number theory. Their influence now extends far beyond this. For example, the field of p-adic analysis essentially provides an alternative form of calculus.

More formally, for a given prime p, the field

of p-adic numbers is a completion of the rational numbers. The field is also given a topology derived from a metric, which is itself derived from the p-adic order, an alternative valuation on the rational numbers. This metric space is complete in the sense that every Cauchy sequence converges to a point in . This is what allows the development of calculus on , and it is the interaction of this analytic and algebraic structure that gives the p-adic number systems their power and utility.

The p in "p-adic" is a variable and may be replaced with a prime (yielding, for instance, "the 2-adic numbers") or another expression representing a prime number. The "adic" of "p-adic" comes from the ending found in words such as dyadic or triadic.

Jai Ganesh · 2022-08-04 20:56:09

51) Friendly number

In number theory, friendly numbers are two or more natural numbers with a common abundancy index, the ratio between the sum of divisors of a number and the number itself. Two numbers with the same "abundancy" form a friendly pair; n numbers with the same "abundancy" form a friendly n-tuple.

Being mutually friendly is an equivalence relation, and thus induces a partition of the positive naturals into clubs (equivalence classes) of mutually "friendly numbers".

A number that is not part of any friendly pair is called solitary.

The "abundancy" index of n is the rational number

, in which denotes the sum of divisors function. A number n is a "friendly number" if there exists m ≠ n such that . "Abundancy" is not the same as abundance, which is defined as .

"Abundancy" may also be expressed as

where denotes a divisor function with equal to the sum of the k-th powers of the divisors of n.

The numbers 1 through 5 are all solitary. The smallest "friendly number" is 6, forming for example, the "friendly" pair 6 and 28 with "abundancy"

the same as . The shared value 2 is an integer in this case but not in many other cases. Numbers with "abundancy" 2 are also known as perfect numbers. There are several unsolved problems related to the "friendly numbers".

In spite of the similarity in name, there is no specific relationship between the friendly numbers and the amicable numbers or the sociable numbers, although the definitions of the latter two also involve the divisor function.

Jai Ganesh · 2022-09-02 02:37:52

52) Pythagoras' constant

The square root of 2, often known as root 2, radical 2, or Pythagoras' constant, and written as

, is the positive algebraic number that, when multiplied by itself, gives the number 2. It is more precisely called the principal square root of 2, to distinguish it from the negative number with the same property.

Geometrically the square root of 2 is the length of a diagonal across a square with sides of one unit of length; this follows from the Pythagorean theorem. It was probably the first number known to be irrational. Its numerical value truncated to 65 decimal places is:

1.41421356237309504880168872420969807856967187537694807317667973799...

The square root of 2.

Alternatively, the quick approximation 99/70 (≈ 1.41429) for the square root of two was frequently used before the common use of electronic calculators and computers. Despite having a denominator of only 70, it differs from the correct value by less than 1/10,000

Jai Ganesh · 2022-10-04 03:05:12

53) Square root of 3

The square root of 3 is the positive real number that, when multiplied by itself, gives the number 3. It is denoted mathematically as

or . It is more precisely called the principal square root of 3, to distinguish it from the negative number with the same property. The square root of 3 is an irrational number. It is also known as Theodorus' constant, after Theodorus of Cyrene, who proved its irrationality.

As of December 2013, its numerical value in decimal notation had been computed to at least ten billion digits. Its decimal expansion, written here to 65 decimal places, is :

1.732050807568877293527446341505872366942805253810380628055806

The fraction

(1.732142857...) can be used as a good approximation. Despite having a denominator of only 56, it differs from the correct value by less than
1/10,000
(approximately with a relative error of ). The rounded value of 1.732 is correct to within 0.01% of the actual value.

The fraction

(1.73205080756...) is accurate to

Archimedes reported a range for its value:

.

The lower limit 1351/780 is an accurate approximation for

to 1/608400
(six decimal places, relative error ) and the upper limit 265/153 to 2/23409
(four decimal places, relative error).

Jai Ganesh · 2022-10-12 21:11:02

54) Sublime Numbers

In number theory, a sublime number is a positive integer which has a perfect number of positive factors (including itself), and whose positive factors add up to another perfect number.

The number 12, for example, is a sublime number. It has a perfect number of positive factors (6): 1, 2, 3, 4, 6, and 12, and the sum of these is again a perfect number: 1 + 2 + 3 + 4 + 6 + 12 = 28.

There are only two known sublime numbers: 12 and

The second of these has 76 decimal digits:

6,086,555,670,238,378,989,670,371,734,243,169,622,657,830,773,351,885,970,528,324,860,512,791,691,264.

Jai Ganesh · 2022-11-03 21:11:47

55) Natural logarithm of 2

The decimal value of the natural logarithm of 2 is approximately

The logarithm of 2 in other bases is obtained with the formula

The common logarithm in particular is

The inverse of this number is the binary logarithm of 10:

.

By the Lindemann–Weierstrass theorem, the natural logarithm of any natural number other than 0 and 1 (more generally, of any positive algebraic number other than 1) is a transcendental number.

Jai Ganesh · 2023-01-03 01:05:10

56) Square Root of 5

The square root of 5 is the positive real number that, when multiplied by itself, gives the prime number 5. It is more precisely called the principal square root of 5, to distinguish it from the negative number with the same property. This number appears in the fractional expression for the golden ratio. It can be denoted in surd form as:

It is an irrational algebraic number. The first sixty significant digits of its decimal expansion are:

2.23606797749978969640917366873127623544061835961152572427089...
which can be rounded down to 2.236 to within 99.99% accuracy. The approximation
161/72 (≈ 2.23611) for the square root of five can be used. Despite having a denominator of only 72, it differs from the correct value by less than
1/10,000
approx.

As of January 2022, its numerical value in decimal has been computed to at least 2,250,000,000,000 digits.

The successive partial evaluations of the continued fraction, which are called its convergents, approach

:

Their numerators are 2, 9, 38, 161, … , and their denominators are 1, 4, 17, 72, … .

Each of these is a best rational approximation of

; in other words, it is closer to than any rational with a smaller denominator.

Jai Ganesh · 2023-10-08 22:28:28

Silver ratio

In mathematics, two quantities are in the silver ratio (or silver mean) if the ratio of the smaller of those two quantities to the larger quantity is the same as the ratio of the larger quantity to the sum of the smaller quantity and twice the larger quantity. This defines the silver ratio as an irrational mathematical constant, whose value of one plus the square root of 2 is approximately 2.4142135623. Its name is an allusion to the golden ratio; analogously to the way the golden ratio is the limiting ratio of consecutive Fibonacci numbers, the silver ratio is the limiting ratio of consecutive Pell numbers. The silver ratio is denoted by δS.

Mathematicians have studied the silver ratio since the time of the Greeks (although perhaps without giving a special name until recently) because of its connections to the square root of 2, its convergents, square triangular numbers, Pell numbers, octagons and the like.

or equivalently,

The silver ratio can also be defined by the simple continued fraction [2; 2, 2, 2, ...]:

are ratios of consecutive Pell numbers. These fractions provide accurate rational approximations of the silver ratio, analogous to the approximation of the golden ratio by ratios of consecutive Fibonacci numbers.

The silver rectangle is connected to the regular octagon.

Jai Ganesh · 2024-02-03 17:42:34

58) Wedderburn–Etherington number

The Wedderburn–Etherington numbers are an integer sequence named for Ivor Malcolm Haddon Etherington and Joseph Wedderburn that can be used to count certain kinds of binary trees. The first few numbers in the sequence are

0, 1, 1, 1, 2, 3, 6, 11, 23, 46, 98, 207, 451, 983, 2179, 4850, 10905, 24631, 56011, ...

Combinatorial interpretation

These numbers can be used to solve several problems in combinatorial enumeration. The nth number in the sequence (starting with the number 0 for n = 0) counts

* The number of unordered rooted trees with n leaves in which all nodes including the root have either zero or exactly two children. These trees have been called Otter trees, after the work of Richard Otter on their combinatorial enumeration. They can also be interpreted as unlabeled and unranked dendrograms with the given number of leaves.
* The number of unordered rooted trees with n nodes in which the root has degree zero or one and all other nodes have at most two children. Trees in which the root has at most one child are called planted trees, and the additional condition that the other nodes have at most two children defines the weakly binary trees. In chemical graph theory, these trees can be interpreted as isomers of polyenes with a designated leaf atom chosen as the root.
* The number of different ways of organizing a single-elimination tournament for n players (with the player names left blank, prior to seeding players into the tournament). The pairings of such a tournament may be described by an Otter tree.
* The number of different results that could be generated by different ways of grouping the expression

for a binary multiplication operation that is assumed to be commutative but neither associative nor idempotent. For instance

can be grouped into binary multiplications in three ways, as

, or

. This was the interpretation originally considered by both Etherington and Wedderburn. An Otter tree can be interpreted as a grouped expression in which each leaf node corresponds to one of the copies of

and each non-leaf node corresponds to a multiplication operation. In the other direction, the set of all Otter trees, with a binary multiplication operation that combines two trees by making them the two subtrees of a new root node, can be interpreted as the free commutative magma on one generator

(the tree with one node). In this algebraic structure, each grouping of

has as its value one of the n-leaf Otter trees.

Formula

The Wedderburn–Etherington numbers may be calculated using the recurrence relation

beginning with the base case

In terms of the interpretation of these numbers as counting rooted binary trees with n leaves, the summation in the recurrence counts the different ways of partitioning these leaves into two subsets, and of forming a subtree having each subset as its leaves. The formula for even values of n is slightly more complicated than the formula for odd values in order to avoid double counting trees with the same number of leaves in both subtrees.

Jai Ganesh · 2024-05-25 19:03:41

59) Algebraic number

An algebraic number is a number that is a root of a non-zero polynomial in one variable with integer (or, equivalently, rational) coefficients. For example, the golden ratio,

, is an algebraic number, because it is a root of the polynomial
. That is, it is a value for x for which the polynomial evaluates to zero. As another example, the complex number is algebraic because it is a root of .

All integers and rational numbers are algebraic, as are all roots of integers. Real and complex numbers that are not algebraic, such as π and e, are called transcendental numbers.

The set of algebraic numbers is countably infinite and has measure zero in the Lebesgue measure as a subset of the uncountable complex numbers. In that sense, almost all complex numbers are transcendental.

Jai Ganesh · 2024-05-26 20:36:48

60) Factorial prime

A factorial prime is a prime number that is one less or one more than a factorial (all factorials greater than 1 are even).

The first 10 factorial primes (for n = 1, 2, 3, 4, 6, 7, 11, 12, 14) are:

2 (0! + 1 or 1! + 1), 3 (2! + 1), 5 (3! − 1), 7 (3! + 1), 23 (4! − 1), 719 (6! − 1), 5039 (7! − 1), 39916801 (11! + 1), 479001599 (12! − 1), 87178291199 (14! − 1), ...
n! − 1 is prime for:

n = 3, 4, 6, 7, 12, 14, 30, 32, 33, 38, 94, 166, 324, 379, 469, 546, 974, 1963, 3507, 3610, 6917, 21480, 34790, 94550, 103040, 147855, 208003, ... (resulting in 27 factorial primes)
n! + 1 is prime for:

n = 0, 1, 2, 3, 11, 27, 37, 41, 73, 77, 116, 154, 320, 340, 399, 427, 872, 1477, 6380, 26951, 110059, 150209, 288465, 308084, 422429, ... (resulting in 24 factorial primes - the prime 2 is repeated)

No other factorial primes are known as of October 2022.

When both n! + 1 and n! − 1 are composite, there must be at least 2n + 1 consecutive composite numbers around n!, since besides n! ± 1 and n! itself, also, each number of form n! ± k is divisible by k for 2 ≤ k ≤ n. However, the necessary length of this gap is asymptotically smaller than the average composite run for integers of similar size.

Jai Ganesh · 2024-05-28 22:07:42

61) Sphenic number

In number theory, a sphenic number (from Greek: 'wedge') is a positive integer that is the product of three distinct prime numbers. Because there are infinitely many prime numbers, there are also infinitely many sphenic numbers.

Definition

A sphenic number is a product pqr where p, q, and r are three distinct prime numbers. In other words, the sphenic numbers are the square-free 3-almost primes.

Examples

The smallest sphenic number is 30 = 2 × 3 × 5, the product of the smallest three primes. The first few sphenic numbers are

30, 42, 66, 70, 78, 102, 105, 110, 114, 130, 138, 154, 165, ...

As of 2020 the largest known sphenic number is

It is the product of the three largest known primes.

Divisors

All sphenic numbers have exactly eight divisors. If we express the sphenic number as

, where p, q, and r are distinct primes, then the set of divisors of n will be:

The converse does not hold. For example, 24 is not a sphenic number, but it has exactly eight divisors.

Math Is Fun Forum

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