Cantore Arithmetic must increase the parameter to the depth as the width has been taken by the restrictions of many mathematicians. In regards to think on their candor of restricted I must balance the first to thought as that is more to the balances of infinity, to declare by name for Cantor Arithmetic to the advancement of math is dot dot dot Leonardo Boracic, known as Fibonacci.
The easy sequence will enable Cantore Arithmetic to outline in word via the conch. By this spiral staircase the immediate comprehension to d.n.a. is poured to the stout. The Loretto Chapel in New Mexico is a standing outline? The Fibonacci sequence has advanced Cantore Arithmetic to what physics has now and Cantore Arithmetic is still in word not worldly. A real text to the mill.
Harbour Island is the basis for this decision as the conch is known to invite a world of plenty to the appetite of both mind and body. The conch is so incredible that the outline can follow Gordon Ramsay in preparation and to the diner the conch can be a provision as staples to the store.
Parameter
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A parameter (from Ancient Greek παρά (pará) 'beside, subsidiary', and μέτρον (métron) 'measure'), generally, is any characteristic that can help in defining or classifying a particular system (meaning an event, project, object, situation, etc.). That is, a parameter is an element of a system that is useful, or critical, when identifying the system, or when evaluating its performance, status, condition, etc.
Parameter has more specific meanings within various disciplines, including mathematics, computer programming, engineering, statistics, logic, linguistics, and electronic musical composition.
In addition to its technical uses, there are also extended uses, especially in non-scientific contexts, where it is used to mean defining characteristics or boundaries, as in the phrases 'test parameters' or 'game play parameters'.[1]
Modelization[edit]
When a system is modeled by equations, the values that describe the system are called parameters. For example, in mechanics, the masses, the dimensions and shapes (for solid bodies), the densities and the viscosities (for fluids), appear as parameters in the equations modeling movements. There are often several choices for the parameters, and choosing a convenient set of parameters is called parametrization.
For example, if one were considering the movement of an object on the surface of a sphere much larger than the object (e.g. the Earth), there are two commonly used parametrizations of its position: angular coordinates (like latitude/longitude), which neatly describe large movements along circles on the sphere, and directional distance from a known point (e.g. "10km NNW of Toronto" or equivalently "8km due North, and then 6km due West, from Toronto" ), which are often simpler for movement confined to a (relatively) small area, like within a particular country or region. Such parametrizations are also relevant to the modelization of geographic areas (i.e. map drawing).
Mathematical functions[edit]
Mathematical functions have one or more arguments that are designated in the definition by variables. A function definition can also contain parameters, but unlike variables, parameters are not listed among the arguments that the function takes. When parameters are present, the definition actually defines a whole family of functions, one for every valid set of values of the parameters. For instance, one could define a general quadratic function by declaring
- ;
Here, the variable x designates the function's argument, but a, b, and c are parameters that determine which particular quadratic function is being considered. A parameter could be incorporated into the function name to indicate its dependence on the parameter. For instance, one may define the base-b logarithm by the formula
where b is a parameter that indicates which logarithmic function is being used. It is not an argument of the function, and will, for instance, be a constant when considering the derivative .
In some informal situations it is a matter of convention (or historical accident) whether some or all of the symbols in a function definition are called parameters. However, changing the status of symbols between parameter and variable changes the function as a mathematical object. For instance, the notation for the falling factorial power
- ,
defines a polynomial function of n (when k is considered a parameter), but is not a polynomial function of k (when n is considered a parameter). Indeed, in the latter case, it is only defined for non-negative integer arguments. More formal presentations of such situations typically start out with a function of several variables (including all those that might sometimes be called "parameters") such as
as the most fundamental object being considered, then defining functions with fewer variables from the main one by means of currying.
Sometimes it is useful to consider all functions with certain parameters as parametric family, i.e. as an indexed family of functions. Examples from probability theory are given further below.
Examples[edit]
- In a section on frequently misused words in his book The Writer's Art, James J. Kilpatrick quoted a letter from a correspondent, giving examples to illustrate the correct use of the word parameter:
- A parametric equaliser is an audio filter that allows the frequency of maximum cut or boost to be set by one control, and the size of the cut or boost by another. These settings, the frequency level of the peak or trough, are two of the parameters of a frequency response curve, and in a two-control equaliser they completely describe the curve. More elaborate parametric equalisers may allow other parameters to be varied, such as skew. These parameters each describe some aspect of the response curve seen as a whole, over all frequencies. A graphic equaliser provides individual level controls for various frequency bands, each of which acts only on that particular frequency band.
- If asked to imagine the graph of the relationship y = ax2, one typically visualizes a range of values of x, but only one value of a. Of course a different value of a can be used, generating a different relation between x and y. Thus a is a parameter: it is less variable than the variable x or y, but it is not an explicit constant like the exponent 2. More precisely, changing the parameter a gives a different (though related) problem, whereas the variations of the variables x and y (and their interrelation) are part of the problem itself.
- In calculating income based on wage and hours worked (income equals wage multiplied by hours worked), it is typically assumed that the number of hours worked is easily changed, but the wage is more static. This makes wage a parameter, hours worked an independent variable, and income a dependent variable.
Mathematical models[edit]
In the context of a mathematical model, such as a probability distribution, the distinction between variables and parameters was described by Bard as follows:
- We refer to the relations which supposedly describe a certain physical situation, as a model. Typically, a model consists of one or more equations. The quantities appearing in the equations we classify into variables and parameters. The distinction between these is not always clear cut, and it frequently depends on the context in which the variables appear. Usually a model is designed to explain the relationships that exist among quantities which can be measured independently in an experiment; these are the variables of the model. To formulate these relationships, however, one frequently introduces "constants" which stand for inherent properties of nature (or of the materials and equipment used in a given experiment). These are the parameters.[2]
Analytic geometry[edit]
In analytic geometry, a curve can be described as the image of a function whose argument, typically called the parameter, lies in a real interval.
For example, the unit circle can be specified in the following two ways:
- implicit form, the curve is the locus of points (x, y) in the Cartesian plane that satisfy the relation
- parametric form, the curve is the image of the function
with parameter As a parametric equation this can be written
The parameter t in this equation would elsewhere in mathematics be called the independent variable.
Mathematical analysis[edit]
In mathematical analysis, integrals dependent on a parameter are often considered. These are of the form
In this formula, t is the argument of the function F, and on the right-hand side the parameter on which the integral depends. When evaluating the integral, tis held constant, and so it is considered to be a parameter. If we are interested in the value of F for different values of t, we then consider t to be a variable. The quantity x is a dummy variable or variable of integration (confusingly, also sometimes called a parameter of integration).
Statistics and econometrics[edit]
In statistics and econometrics, the probability framework above still holds, but attention shifts to estimating the parameters of a distribution based on observed data, or testing hypotheses about them. In frequentist estimation parameters are considered "fixed but unknown", whereas in Bayesian estimation they are treated as random variables, and their uncertainty is described as a distribution.[citation needed][3]
In estimation theory of statistics, "statistic" or estimator refers to samples, whereas "parameter" or estimand refers to populations, where the samples are taken from. A statistic is a numerical characteristic of a sample that can be used as an estimate of the corresponding parameter, the numerical characteristic of the population from which the sample was drawn.
For example, the sample mean (estimator), denoted , can be used as an estimate of the mean parameter (estimand), denoted μ, of the population from which the sample was drawn. Similarly, the sample variance (estimator), denoted S2, can be used to estimate the variance parameter (estimand), denoted σ2, of the population from which the sample was drawn. (Note that the sample standard deviation (S) is not an unbiased estimate of the population standard deviation (σ): see Unbiased estimation of standard deviation.)
It is possible to make statistical inferences without assuming a particular parametric family of probability distributions. In that case, one speaks of non-parametric statistics as opposed to the parametric statistics just described. For example, a test based on Spearman's rank correlation coefficient would be called non-parametric since the statistic is computed from the rank-order of the data disregarding their actual values (and thus regardless of the distribution they were sampled from), whereas those based on the Pearson product-moment correlation coefficient are parametric tests since it is computed directly from the data values and thus estimates the parameter known as the population correlation.
Probability theory[edit]
In probability theory, one may describe the distribution of a random variable as belonging to a family of probability distributions, distinguished from each other by the values of a finite number of parameters. For example, one talks about "a Poisson distribution with mean value λ". The function defining the distribution (the probability mass function) is:
This example nicely illustrates the distinction between constants, parameters, and variables. e is Euler's number, a fundamental mathematical constant. The parameter λ is the mean number of observations of some phenomenon in question, a property characteristic of the system. k is a variable, in this case the number of occurrences of the phenomenon actually observed from a particular sample. If we want to know the probability of observing k1occurrences, we plug it into the function to get . Without altering the system, we can take multiple samples, which will have a range of values of k, but the system is always characterized by the same λ.
For instance, suppose we have a radioactive sample that emits, on average, five particles every ten minutes. We take measurements of how many particles the sample emits over ten-minute periods. The measurements exhibit different values of k, and if the sample behaves according to Poisson statistics, then each value of k will come up in a proportion given by the probability mass function above. From measurement to measurement, however, λ remains constant at 5. If we do not alter the system, then the parameter λ is unchanged from measurement to measurement; if, on the other hand, we modulate the system by replacing the sample with a more radioactive one, then the parameter λ would increase.
Another common distribution is the normal distribution, which has as parameters the mean μ and the variance σ².
In these above examples, the distributions of the random variables are completely specified by the type of distribution, i.e. Poisson or normal, and the parameter values, i.e. mean and variance. In such a case, we have a parameterized distribution.
It is possible to use the sequence of moments (mean, mean square, ...) or cumulants (mean, variance, ...) as parameters for a probability distribution: see Statistical parameter.
Computer programming[edit]
In computer programming, two notions of parameter are commonly used, and are referred to as parameters and arguments—or more formally as a formal parameter and an actual parameter.
For example, in the definition of a function such as
- y = f(x) = x + 2,
x is the formal parameter (the parameter) of the defined function.
When the function is evaluated for a given value, as in
- f(3): or, y = f(3) = 3 + 2 = 5,
3 is the actual parameter (the argument) for evaluation by the defined function; it is a given value (actual value) that is substituted for the formal parameterof the defined function. (In casual usage the terms parameter and argument might inadvertently be interchanged, and thereby used incorrectly.)
These concepts are discussed in a more precise way in functional programming and its foundational disciplines, lambda calculus and combinatory logic. Terminology varies between languages; some computer languages such as C define parameter and argument as given here, while Eiffel uses an alternative convention.
Artificial Intelligence[edit]
In artificial intelligence, a model describes the probability that something will occur. Parameters in a model are the weight of the various probabilities. Tiernan Ray, in an article on GPT-3, described parameters this way:
Engineering[edit]
In engineering (especially involving data acquisition) the term parameter sometimes loosely refers to an individual measured item. This usage is not consistent, as sometimes the term channel refers to an individual measured item, with parameter referring to the setup information about that channel.
"Speaking generally, properties are those physical quantities which directly describe the physical attributes of the system; parameters are those combinations of the properties which suffice to determine the response of the system. Properties can have all sorts of dimensions, depending upon the system being considered; parameters are dimensionless, or have the dimension of time or its reciprocal."[5]
The term can also be used in engineering contexts, however, as it is typically used in the physical sciences.
Environmental science[edit]
In environmental science and particularly in chemistry and microbiology, a parameter is used to describe a discrete chemical or microbiological entity that can be assigned a value: commonly a concentration, but may also be a logical entity (present or absent), a statistical result such as a 95 percentile value or in some cases a subjective value.
Linguistics[edit]
Within linguistics, the word "parameter" is almost exclusively used to denote a binary switch in a Universal Grammar within a Principles and Parametersframework.
Logic[edit]
In logic, the parameters passed to (or operated on by) an open predicate are called parameters by some authors (e.g., Prawitz, "Natural Deduction"; Paulson, "Designing a theorem prover"). Parameters locally defined within the predicate are called variables. This extra distinction pays off when defining substitution (without this distinction special provision must be made to avoid variable capture). Others (maybe most) just call parameters passed to (or operated on by) an open predicate variables, and when defining substitution have to distinguish between free variables and bound variables.
Music[edit]
In music theory, a parameter denotes an element which may be manipulated (composed), separately from the other elements. The term is used particularly for pitch, loudness, duration, and timbre, though theorists or composers have sometimes considered other musical aspects as parameters. The term is particularly used in serial music, where each parameter may follow some specified series. Paul Lansky and George Perle criticized the extension of the word "parameter" to this sense, since it is not closely related to its mathematical sense,[6] but it remains common. The term is also common in music production, as the functions of audio processing units (such as the attack, release, ratio, threshold, and other variables on a compressor) are defined by parameters specific to the type of unit (compressor, equalizer, delay, etc.).
Fibonacci sequence
In mathematics, the Fibonacci sequence is a sequence in which each number is the sum of the two preceding ones. Numbers that are part of the Fibonacci sequence are known as Fibonacci numbers, commonly denoted Fn . The sequence commonly starts from 0 and 1, although some authors start the sequence from 1 and 1 or sometimes (as did Fibonacci) from 1 and 2. Starting from 0 and 1, the first few values in the sequence are:[1]
- 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144.
The Fibonacci numbers were first described in Indian mathematics,[2][3][4] as early as 200 BC in work by Pingalaon enumerating possible patterns of Sanskrit poetry formed from syllables of two lengths. They are named after the Italian mathematician Leonardo of Pisa, also known as Fibonacci, who introduced the sequence to Western European mathematics in his 1202 book Liber Abaci.[5]
Fibonacci numbers appear unexpectedly often in mathematics, so much so that there is an entire journal dedicated to their study, the Fibonacci Quarterly. Applications of Fibonacci numbers include computer algorithms such as the Fibonacci search techniqueand the Fibonacci heap data structure, and graphs called Fibonacci cubes used for interconnecting parallel and distributed systems. They also appear in biological settings, such as branching in trees, the arrangement of leaves on a stem, the fruit sprouts of a pineapple, the flowering of an artichoke, and the arrangement of a pine cone's bracts, though they don't occur in all species.
Fibonacci numbers are also strongly related to the golden ratio: Binet's formula expresses the nth Fibonacci number in terms of n and the golden ratio, and implies that the ratio of two consecutive Fibonacci numbers tends to the golden ratio as n increases. Fibonacci numbers are also closely related to Lucas numbers, which obey the same recurrence relation and with the Fibonacci numbers form a complementary pair of Lucas sequences.
Definition[edit]
The Fibonacci numbers may be defined by the recurrence relation[6]
Under some older definitions, the value is omitted, so that the sequence starts with and the recurrence is valid for n > 2.[7][8]
The first 20 Fibonacci numbers Fn are:[1]
F0 F1 F2 F3 F4 F5 F6 F7 F8 F9 F10 F11 F12 F13 F14 F15 F16 F17 F18 F19 0 1 1 2 3 5 8 13 21 34 55 89 144 233 377 610 987 1597 2584 4181
History[edit]
India[edit]
The Fibonacci sequence appears in Indian mathematics, in connection with Sanskrit prosody.[3][9][10] In the Sanskrit poetic tradition, there was interest in enumerating all patterns of long (L) syllables of 2 units duration, juxtaposed with short (S) syllables of 1 unit duration. Counting the different patterns of successive L and S with a given total duration results in the Fibonacci numbers: the number of patterns of duration m units is Fm+1.[4]
Knowledge of the Fibonacci sequence was expressed as early as Pingala (c. 450 BC–200 BC). Singh cites Pingala's cryptic formula misrau cha ("the two are mixed") and scholars who interpret it in context as saying that the number of patterns for m beats (Fm+1) is obtained by adding one [S] to the Fm cases and one [L] to the Fm−1cases.[11] Bharata Muni also expresses knowledge of the sequence in the Natya Shastra (c. 100 BC–c. 350 AD).[12][2] However, the clearest exposition of the sequence arises in the work of Virahanka (c. 700 AD), whose own work is lost, but is available in a quotation by Gopala (c. 1135):[10]
Hemachandra (c. 1150) is credited with knowledge of the sequence as well,[2] writing that "the sum of the last and the one before the last is the number ... of the next mātrā-vṛtta."[14][15]
Europe[edit]
the indices from present to XII (months) as Latin ordinals and Roman numerals and the numbers (of rabbit pairs) as Hindu-Arabic numerals starting with 1, 2, 3, 5 and ending with 377.
The Fibonacci sequence first appears in the book Liber Abaci (The Book of Calculation, 1202) by Fibonacci[16][17] where it is used to calculate the growth of rabbit populations.[18][19] Fibonacci considers the growth of an idealized (biologically unrealistic) rabbit population, assuming that: a newly born breeding pair of rabbits are put in a field; each breeding pair mates at the age of one month, and at the end of their second month they always produce another pair of rabbits; and rabbits never die, but continue breeding forever. Fibonacci posed the puzzle: how many pairs will there be in one year?
- At the end of the first month, they mate, but there is still only 1 pair.
- At the end of the second month they produce a new pair, so there are 2 pairs in the field.
- At the end of the third month, the original pair produce a second pair, but the second pair only mate to gestate for a month, so there are 3 pairs in all.
- At the end of the fourth month, the original pair has produced yet another new pair, and the pair born two months ago also produces their first pair, making 5 pairs.
At the end of the nth month, the number of pairs of rabbits is equal to the number of mature pairs (that is, the number of pairs in month n – 2) plus the number of pairs alive last month (month n – 1). The number in the nth month is the nth Fibonacci number.[20]
The name "Fibonacci sequence" was first used by the 19th-century number theorist Édouard Lucas.[21]
Relation to the golden ratio[edit]
Closed-form expression [edit]
Like every sequence defined by a linear recurrence with constant coefficients, the Fibonacci numbers have a closed-form expression. It has become known as Binet's formula, named after French mathematician Jacques Philippe Marie Binet, though it was already known by Abraham de Moivre and Daniel Bernoulli:[22]
where
is the golden ratio, and ψ is its conjugate:[23]
Since , this formula can also be written as
To see the relation between the sequence and these constants,[24] note that φ and ψ are both solutions of the equation and thus so the powers of φ and ψ satisfy the Fibonacci recursion. In other words,
![{\displaystyle {\begin{aligned}\varphi ^{n}&=\varphi ^{n-1}+\varphi ^{n-2},\\[3mu]\psi ^{n}&=\psi ^{n-1}+\psi ^{n-2}.\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/282a0b620cb2eca72787ef444e6d07d39678d383)
It follows that for any values a and b, the sequence defined by
satisfies the same recurrence,
![{\displaystyle {\begin{aligned}U_{n}&=a\varphi ^{n}+b\psi ^{n}\\[3mu]&=a(\varphi ^{n-1}+\varphi ^{n-2})+b(\psi ^{n-1}+\psi ^{n-2})\\[3mu]&=a\varphi ^{n-1}+b\psi ^{n-1}+a\varphi ^{n-2}+b\psi ^{n-2}\\[3mu]&=U_{n-1}+U_{n-2}.\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/49e155e3c74b9d82976473240c1fc8f5db5d0f1f)
If a and b are chosen so that U0 = 0 and U1 = 1 then the resulting sequence Un must be the Fibonacci sequence. This is the same as requiring a and bsatisfy the system of equations:
which has solution
producing the required formula.
Taking the starting values U0 and U1 to be arbitrary constants, a more general solution is:
where
![{\displaystyle {\begin{aligned}a&={\frac {U_{1}-U_{0}\psi }{\sqrt {5}}},\\[3mu]b&={\frac {U_{0}\varphi -U_{1}}{\sqrt {5}}}.\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9b7a1f5df3f3d8a6b6e48e76a19c75a2b5c65645)
Computation by rounding[edit]
Since for all n ≥ 0, the number Fn is the closest integer to . Therefore, it can be found by rounding, using the nearest integer function:
In fact, the rounding error is very small, being less than 0.1 for n ≥ 4, and less than 0.01 for n ≥ 8. This formula is easily inverted to find an index of a Fibonacci number F:
Instead using the floor function gives the largest index of a Fibonacci number that is not greater than F:
Magnitude[edit]
Since Fn is asymptotic to , the number of digits in Fn is asymptotic to . As a consequence, for every integer d > 1 there are either 4 or 5 Fibonacci numbers with d decimal digits.
More generally, in the base b representation, the number of digits in Fn is asymptotic to
Limit of consecutive quotients[edit]
Johannes Kepler observed that the ratio of consecutive Fibonacci numbers converges. He wrote that "as 5 is to 8 so is 8 to 13, practically, and as 8 is to 13, so is 13 to 21 almost", and concluded that these ratios approach the golden ratio [27][28]
This convergence holds regardless of the starting values and , unless . This can be verified using Binet's formula. For example, the initial values 3 and 2 generate the sequence 3, 2, 5, 7, 12, 19, 31, 50, 81, 131, 212, 343, 555, ... . The ratio of consecutive terms in this sequence shows the same convergence towards the golden ratio.
In general, , because the ratios between consecutive Fibonacci numbers approaches .
Successive tilings of the plane and a graph of approximations to the golden ratio calculated by dividing each Fibonacci number by the previous
Decomposition of powers[edit]
Since the golden ratio satisfies the equation
this expression can be used to decompose higher powers as a linear function of lower powers, which in turn can be decomposed all the way down to a linear combination of and 1. The resulting recurrence relationships yield Fibonacci numbers as the linear coefficients:
These expressions are also true for n < 1 if the Fibonacci sequence Fn is extended to negative integers using the Fibonacci rule
Identification[edit]
Binet's formula provides a proof that a positive integer x is a Fibonacci number if and only if at least one of or is a perfect square.[29]This is because Binet's formula, which can be written as , can be multiplied by and solved as a quadratic equationin via the quadratic formula:
Comparing this to , it follows that
In particular, the left-hand side is a perfect square.
Matrix form[edit]
A 2-dimensional system of linear difference equations that describes the Fibonacci sequence is
which yields . The eigenvalues of the matrix A are and corresponding to the respective eigenvectors
Equivalently, the same computation may be performed by diagonalization of A through use of its eigendecomposition:
which again yields
The matrix A has a determinant of −1, and thus it is a 2 × 2 unimodular matrix.
This property can be understood in terms of the continued fraction representation for the golden ratio:
The Fibonacci numbers occur as the ratio of successive convergents of the continued fraction for φ, and the matrix formed from successive convergents of any continued fraction has a determinant of +1 or −1. The matrix representation gives the following closed-form expression for the Fibonacci numbers:
For a given n, this matrix can be computed in O(log(n)) arithmetic operations, using the exponentiation by squaring method.
Taking the determinant of both sides of this equation yields Cassini's identity,
Moreover, since AnAm = An+m for any square matrix A, the following identities can be derived (they are obtained from two different coefficients of the matrix product, and one may easily deduce the second one from the first one by changing n into n + 1),
In particular, with m = n,
![{\displaystyle {\begin{aligned}F_{2n-1}&={F_{n}}^{2}+{F_{n-1}}^{2}\\[5pt]F_{2n{\phantom {{}-1}}}&=(F_{n-1}+F_{n+1})F_{n}\\[5pt]&=(2F_{n-1}+F_{n})F_{n}\\[5pt]&=(2F_{n+1}-F_{n})F_{n}.\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/64b95fd1ea97b7a4315e8ee4ddb7f6b44c7681f0)
These last two identities provide a way to compute Fibonacci numbers recursively in O(log(n)) arithmetic operations and in time O(M(n) log(n)), where M(n) is the time for the multiplication of two numbers of n digits. This matches the time for computing the nth Fibonacci number from the closed-form matrix formula, but with fewer redundant steps if one avoids recomputing an already computed Fibonacci number (recursion with memoization).[30]
Combinatorial identities[edit]
Combinatorial proofs[edit]
Most identities involving Fibonacci numbers can be proved using combinatorial arguments using the fact that can be interpreted as the number of (possibly empty) sequences of 1s and 2s whose sum is . This can be taken as the definition of with the conventions , meaning no such sequence exists whose sum is −1, and , meaning the empty sequence "adds up" to 0. In the following, is the cardinality of a set:
In this manner the recurrence relation
In a similar manner it may be shown that the sum of the first Fibonacci numbers up to the nth is equal to the (n + 2)nd Fibonacci number minus 1.[31] In symbols:
This may be seen by dividing all sequences summing to based on the location of the first 2. Specifically, each set consists of those sequences that start until the last two sets each with cardinality 1.
Following the same logic as before, by summing the cardinality of each set we see that
... where the last two terms have the value . From this it follows that .
A similar argument, grouping the sums by the position of the first 1 rather than the first 2 gives two more identities:
A different trick may be used to prove
Symbolic method[edit]
The sequence is also considered using the symbolic method.[33] More precisely, this sequence corresponds to a specifiable combinatorial class. The specification of this sequence is . Indeed, as stated above, the -th Fibonacci number equals the number of combinatorial compositions (ordered partitions) of using terms 1 and 2.
It follows that the ordinary generating function of the Fibonacci sequence, , is the rational function
Induction proofs[edit]
Fibonacci identities often can be easily proved using mathematical induction.
For example, reconsider
and so we have the formula for
Similarly, add to both sides of
Binet formula proofs[edit]
The Binet formula is
For example, to prove that note that the left hand side multiplied by becomes
Other identities[edit]
Numerous other identities can be derived using various methods. Here are some of them:[34]
Cassini's and Catalan's identities[edit]
Cassini's identity states that
d'Ocagne's identity[edit]
More generally,[34]
or alternatively
Putting k = 2 in this formula, one gets again the formulas of the end of above section Matrix form.
Generating function[edit]
The generating function of the Fibonacci sequence is the power series
This series is convergent for any complex number satisfying and its sum has a simple closed form:[35]
This can be proved by multiplying by :
where all terms involving for cancel out because of the defining Fibonacci recurrence relation.
The partial fraction decomposition is given by
The related function is the generating function for the negafibonacci numbers, and satisfies the functional equation
Using equal to any of 0.01, 0.001, 0.0001, etc. lays out the first Fibonacci numbers in the decimal expansion of . For example,
Reciprocal sums[edit]
Infinite sums over reciprocal Fibonacci numbers can sometimes be evaluated in terms of theta functions. For example, the sum of every odd-indexed reciprocal Fibonacci number can be written as
and the sum of squared reciprocal Fibonacci numbers as
If we add 1 to each Fibonacci number in the first sum, there is also the closed form
and there is a nested sum of squared Fibonacci numbers giving the reciprocal of the golden ratio,
The sum of all even-indexed reciprocal Fibonacci numbers is[36]
So the reciprocal Fibonacci constant is[37]
Moreover, this number has been proved irrational by Richard André-Jeannin.[38]
Millin's series gives the identity[39]
Primes and divisibility[edit]
Divisibility properties[edit]
Every third number of the sequence is even (a multiple of ) and, more generally, every kth number of the sequence is a multiple of Fk. Thus the Fibonacci sequence is an example of a divisibility sequence. In fact, the Fibonacci sequence satisfies the stronger divisibility property[40][41]
In particular, any three consecutive Fibonacci numbers are pairwise coprime because both and . That is,
for every n.
Every prime number p divides a Fibonacci number that can be determined by the value of p modulo 5. If p is congruent to 1 or 4 modulo 5, then p divides Fp−1, and if p is congruent to 2 or 3 modulo 5, then, p divides Fp+1. The remaining case is that p = 5, and in this case p divides Fp.
These cases can be combined into a single, non-piecewise formula, using the Legendre symbol:[42]
Primality testing[edit]
The above formula can be used as a primality test in the sense that if
Fibonacci primes[edit]
A Fibonacci prime is a Fibonacci number that is prime. The first few are:[44]
- 2, 3, 5, 13, 89, 233, 1597, 28657, 514229, ...
Fibonacci primes with thousands of digits have been found, but it is not known whether there are infinitely many.[45]
Fkn is divisible by Fn, so, apart from F4 = 3, any Fibonacci prime must have a prime index. As there are arbitrarily long runs of composite numbers, there are therefore also arbitrarily long runs of composite Fibonacci numbers.
No Fibonacci number greater than F6 = 8 is one greater or one less than a prime number.[46]
The only nontrivial square Fibonacci number is 144.[47] Attila Pethő proved in 2001 that there is only a finite number of perfect power Fibonacci numbers.[48] In 2006, Y. Bugeaud, M. Mignotte, and S. Siksek proved that 8 and 144 are the only such non-trivial perfect powers.[49]
1, 3, 21, and 55 are the only triangular Fibonacci numbers, which was conjectured by Vern Hoggatt and proved by Luo Ming.[50]
No Fibonacci number can be a perfect number.[51] More generally, no Fibonacci number other than 1 can be multiply perfect,[52] and no ratio of two Fibonacci numbers can be perfect.[53]
Prime divisors[edit]
With the exceptions of 1, 8 and 144 (F1 = F2, F6 and F12) every Fibonacci number has a prime factor that is not a factor of any smaller Fibonacci number (Carmichael's theorem).[54] As a result, 8 and 144 (F6 and F12) are the only Fibonacci numbers that are the product of other Fibonacci numbers.[55]
The divisibility of Fibonacci numbers by a prime p is related to the Legendre symbol which is evaluated as follows:
If p is a prime number then
For example,
It is not known whether there exists a prime p such that
Such primes (if there are any) would be called Wall–Sun–Sun primes.
Also, if p ≠ 5 is an odd prime number then:[58]
Example 1. p = 7, in this case p ≡ 3 (mod 4) and we have:
Example 2. p = 11, in this case p ≡ 3 (mod 4) and we have:
Example 3. p = 13, in this case p ≡ 1 (mod 4) and we have:
Example 4. p = 29, in this case p ≡ 1 (mod 4) and we have:
For odd n, all odd prime divisors of Fn are congruent to 1 modulo 4, implying that all odd divisors of Fn (as the products of odd prime divisors) are congruent to 1 modulo 4.[59]
For example,
All known factors of Fibonacci numbers F(i ) for all i < 50000 are collected at the relevant repositories.[60][61]
Periodicity modulo n[edit]
If the members of the Fibonacci sequence are taken mod n, the resulting sequence is periodic with period at most 6n.[62] The lengths of the periods for various n form the so-called Pisano periods.[63] Determining a general formula for the Pisano periods is an open problem, which includes as a subproblem a special instance of the problem of finding the multiplicative order of a modular integer or of an element in a finite field. However, for any particular n, the Pisano period may be found as an instance of cycle detection.
Generalizations[edit]
The Fibonacci sequence is one of the simplest and earliest known sequences defined by a recurrence relation, and specifically by a linear difference equation. All these sequences may be viewed as generalizations of the Fibonacci sequence. In particular, Binet's formula may be generalized to any sequence that is a solution of a homogeneous linear difference equation with constant coefficients.
Some specific examples that are close, in some sense, to the Fibonacci sequence include:
- Generalizing the index to negative integers to produce the negafibonacci numbers.
- Generalizing the index to real numbers using a modification of Binet's formula.[34]
- Starting with other integers. Lucas numbers have L1 = 1, L2 = 3, and Ln = Ln−1 + Ln−2. Primefree sequences use the Fibonacci recursion with other starting points to generate sequences in which all numbers are composite.
- Letting a number be a linear function (other than the sum) of the 2 preceding numbers. The Pell numbers have Pn = 2Pn−1 + Pn−2. If the coefficient of the preceding value is assigned a variable value x, the result is the sequence of Fibonacci polynomials.
- Not adding the immediately preceding numbers. The Padovan sequence and Perrin numbers have P(n) = P(n − 2) + P(n − 3).
- Generating the next number by adding 3 numbers (tribonacci numbers), 4 numbers (tetranacci numbers), or more. The resulting sequences are known as n-Step Fibonacci numbers.[64]
Applications[edit]
Mathematics[edit]
The Fibonacci numbers occur as the sums of binomial coefficients in the "shallow" diagonals of Pascal's triangle:[65]
To see how the formula is used, we can arrange the sums by the number of terms present:
5 = 1+1+1+1+1 = 2+1+1+1 = 1+2+1+1 = 1+1+2+1 = 1+1+1+2 = 2+2+1 = 2+1+2 = 1+2+2
which is , where we are choosing the positions of k twos from n−k−1 terms.
These numbers also give the solution to certain enumerative problems,[66] the most common of which is that of counting the number of ways of writing a given number n as an ordered sum of 1s and 2s (called compositions); there are Fn+1 ways to do this (equivalently, it's also the number of domino tilings of the rectangle). For example, there are F5+1 = F6 = 8 ways one can climb a staircase of 5 steps, taking one or two steps at a time:
5 = 1+1+1+1+1 = 2+1+1+1 = 1+2+1+1 = 1+1+2+1 = 2+2+1 = 1+1+1+2 = 2+1+2 = 1+2+2
The figure shows that 8 can be decomposed into 5 (the number of ways to climb 4 steps, followed by a single-step) plus 3 (the number of ways to climb 3 steps, followed by a double-step). The same reasoning is applied recursively until a single step, of which there is only one way to climb.
The Fibonacci numbers can be found in different ways among the set of binary strings, or equivalently, among the subsets of a given set.
- The number of binary strings of length n without consecutive 1s is the Fibonacci number Fn+2. For example, out of the 16 binary strings of length 4, there are F6 = 8 without consecutive 1s – they are 0000, 0001, 0010, 0100, 0101, 1000, 1001, and 1010. Such strings are the binary representations of Fibbinary numbers. Equivalently, Fn+2 is the number of subsets S of {1, ..., n} without consecutive integers, that is, those S for which {i, i + 1} ⊈ S for every i. A bijection with the sums to n+1 is to replace 1 with 0 and 2 with 10, and drop the last zero.
- The number of binary strings of length n without an odd number of consecutive 1s is the Fibonacci number Fn+1. For example, out of the 16 binary strings of length 4, there are F5 = 5 without an odd number of consecutive 1s – they are 0000, 0011, 0110, 1100, 1111. Equivalently, the number of subsets S of {1, ..., n} without an odd number of consecutive integers is Fn+1. A bijection with the sums to n is to replace 1 with 0 and 2 with 11.
- The number of binary strings of length n without an even number of consecutive 0s or 1s is 2Fn. For example, out of the 16 binary strings of length 4, there are 2F4 = 6 without an even number of consecutive 0s or 1s – they are 0001, 0111, 0101, 1000, 1010, 1110. There is an equivalent statement about subsets.
- Yuri Matiyasevich was able to show that the Fibonacci numbers can be defined by a Diophantine equation, which led to his solving Hilbert's tenth problem.[67]
- The Fibonacci numbers are also an example of a complete sequence. This means that every positive integer can be written as a sum of Fibonacci numbers, where any one number is used once at most.
- Moreover, every positive integer can be written in a unique way as the sum of one or more distinct Fibonacci numbers in such a way that the sum does not include any two consecutive Fibonacci numbers. This is known as Zeckendorf's theorem, and a sum of Fibonacci numbers that satisfies these conditions is called a Zeckendorf representation. The Zeckendorf representation of a number can be used to derive its Fibonacci coding.
- Starting with 5, every second Fibonacci number is the length of the hypotenuse of a right triangle with integer sides, or in other words, the largest number in a Pythagorean triple, obtained from the formula The sequence of Pythagorean triangles obtained from this formula has sides of lengths (3,4,5), (5,12,13), (16,30,34), (39,80,89), ... . The middle side of each of these triangles is the sum of the three sides of the preceding triangle.[68]
- The Fibonacci cube is an undirected graph with a Fibonacci number of nodes that has been proposed as a network topology for parallel computing.
- Fibonacci numbers appear in the ring lemma, used to prove connections between the circle packing theorem and conformal maps.[69]
Computer science[edit]
The keys in the left spine are Fibonacci numbers.
- The Fibonacci numbers are important in computational run-time analysis of Euclid's algorithm to determine the greatest common divisor of two integers: the worst case input for this algorithm is a pair of consecutive Fibonacci numbers.[70]
- Fibonacci numbers are used in a polyphase version of the merge sort algorithm in which an unsorted list is divided into two lists whose lengths correspond to sequential Fibonacci numbers – by dividing the list so that the two parts have lengths in the approximate proportion φ. A tape-drive implementation of the polyphase merge sort was described in The Art of Computer Programming.
- A Fibonacci tree is a binary tree whose child trees (recursively) differ in height by exactly 1. So it is an AVL tree, and one with the fewest nodes for a given height — the "thinnest" AVL tree. These trees have a number of vertices that is a Fibonacci number minus one, an important fact in the analysis of AVL trees.[71]
- Fibonacci numbers are used by some pseudorandom number generators.
- Fibonacci numbers arise in the analysis of the Fibonacci heap data structure.
- A one-dimensional optimization method, called the Fibonacci search technique, uses Fibonacci numbers.[72]
- The Fibonacci number series is used for optional lossy compression in the IFF 8SVX audio file format used on Amiga computers. The number series compands the original audio wave similar to logarithmic methods such as μ-law.[73][74]
- Some Agile teams use a modified series called the "Modified Fibonacci Series" in planning poker, as an estimation tool. Planning Poker is a formal part of the Scaled Agile Framework.[75]
- Fibonacci coding
- Negafibonacci coding
Nature[edit]
Fibonacci sequences appear in biological settings,[76] such as branching in trees, arrangement of leaves on a stem, the fruitlets of a pineapple,[77] the flowering of artichoke, and the arrangement of a pine cone,[78] and the family tree of honeybees.[79][80] Kepler pointed out the presence of the Fibonacci sequence in nature, using it to explain the (golden ratio-related) pentagonal form of some flowers.[81] Field daisies most often have petals in counts of Fibonacci numbers.[82] In 1830, K. F. Schimper and A. Braun discovered that the parastichies (spiral phyllotaxis) of plants were frequently expressed as fractions involving Fibonacci numbers.[83]
Przemysław Prusinkiewicz advanced the idea that real instances can in part be understood as the expression of certain algebraic constraints on free groups, specifically as certain Lindenmayer grammars.[84]
A model for the pattern of florets in the head of a sunflower was proposed by Helmut Vogel in 1979.[85] This has the form
where n is the index number of the floret and c is a constant scaling factor; the florets thus lie on Fermat's spiral. The divergence angle, approximately 137.51°, is the golden angle, dividing the circle in the golden ratio. Because this ratio is irrational, no floret has a neighbor at exactly the same angle from the center, so the florets pack efficiently. Because the rational approximations to the golden ratio are of the form F( j):F( j + 1), the nearest neighbors of floret number n are those at n ± F( j) for some index j, which depends on r, the distance from the center. Sunflowers and similar flowers most commonly have spirals of florets in clockwise and counter-clockwise directions in the amount of adjacent Fibonacci numbers,[86] typically counted by the outermost range of radii.[87]
Fibonacci numbers also appear in the pedigrees of idealized honeybees, according to the following rules:
- If an egg is laid by an unmated female, it hatches a male or drone bee.
- If, however, an egg was fertilized by a male, it hatches a female.
Thus, a male bee always has one parent, and a female bee has two. If one traces the pedigree of any male bee (1 bee), he has 1 parent (1 bee), 2 grandparents, 3 great-grandparents, 5 great-great-grandparents, and so on. This sequence of numbers of parents is the Fibonacci sequence. The number of ancestors at each level, Fn, is the number of female ancestors, which is Fn−1, plus the number of male ancestors, which is Fn−2.[88] This is under the unrealistic assumption that the ancestors at each level are otherwise unrelated.
It has been noticed that the number of possible ancestors on the human X chromosome inheritance line at a given ancestral generation also follows the Fibonacci sequence.[89] A male individual has an X chromosome, which he received from his mother, and a Y chromosome, which he received from his father. The male counts as the "origin" of his own X chromosome (), and at his parents' generation, his X chromosome came from a single parent (). The male's mother received one X chromosome from her mother (the son's maternal grandmother), and one from her father (the son's maternal grandfather), so two grandparents contributed to the male descendant's X chromosome (). The maternal grandfather received his X chromosome from his mother, and the maternal grandmother received X chromosomes from both of her parents, so three great-grandparents contributed to the male descendant's X chromosome (). Five great-great-grandparents contributed to the male descendant's X chromosome (), etc. (This assumes that all ancestors of a given descendant are independent, but if any genealogy is traced far enough back in time, ancestors begin to appear on multiple lines of the genealogy, until eventually a population founder appears on all lines of the genealogy.)
Other[edit]
- In optics, when a beam of light shines at an angle through two stacked transparent plates of different materials of different refractive indexes, it may reflect off three surfaces: the top, middle, and bottom surfaces of the two plates. The number of different beam paths that have k reflections, for k > 1, is the th Fibonacci number. (However, when k = 1, there are three reflection paths, not two, one for each of the three surfaces.)[90]
- Fibonacci retracement levels are widely used in technical analysis for financial market trading.
- Since the conversion factor 1.609344 for miles to kilometers is close to the golden ratio, the decomposition of distance in miles into a sum of Fibonacci numbers becomes nearly the kilometer sum when the Fibonacci numbers are replaced by their successors. This method amounts to a radix 2 number register in golden ratio base φ being shifted. To convert from kilometers to miles, shift the register down the Fibonacci sequence instead.[91]
- The measured values of voltages and currents in the infinite resistor chain circuit (also called the resistor ladder or infinite series-parallel circuit) follow the Fibonacci sequence. The intermediate results of adding the alternating series and parallel resistances yields fractions composed of consecutive Fibonacci numbers. The equivalent resistance of the entire circuit equals the golden ratio.[92]
- Brasch et al. 2012 show how a generalized Fibonacci sequence also can be connected to the field of economics.[93] In particular, it is shown how a generalized Fibonacci sequence enters the control function of finite-horizon dynamic optimisation problems with one state and one control variable. The procedure is illustrated in an example often referred to as the Brock–Mirman economic growth model.
- Mario Merz included the Fibonacci sequence in some of his artworks beginning in 1970.[94]
- Joseph Schillinger (1895–1943) developed a system of composition which uses Fibonacci intervals in some of its melodies; he viewed these as the musical counterpart to the elaborate harmony evident within nature.[95] See also Golden ratio § Music.
Fibonacci
Fibonacci | |
|---|---|
 Statue of Fibonacci (1863) by Giovanni Paganucci in the Camposanto di Pisa[a] | |
| Born | c. 1170 |
| Died | c. 1250 (aged 79–80) Pisa, Republic of Pisa |
| Other names | Leonardo Fibonacci, Leonardo Bonacci, Leonardo Pisano |
| Occupation | Mathematician |
| Known for |
|
| Parent | Guglielmo "Bonacci" (father) |
Fibonacci (/ˌfɪbəˈnɑːtʃi/;[3] also US: /ˌfiːb-/,[4][5] Italian: [fiboˈnattʃi]; c. 1170 – c. 1240–50),[6] also known as Leonardo Bonacci, Leonardo of Pisa, or Leonardo Bigollo Pisano ('Leonardo the Traveller from Pisa'[7]), was an Italian mathematician from the Republic of Pisa, considered to be "the most talented Western mathematician of the Middle Ages".[8]
The name he is commonly called, Fibonacci, was made up in 1838 by the Franco-Italian historian Guillaume Libri[9][10] and is short for filius Bonacci ('son of Bonacci').[11][b] However, even earlier, in 1506, a notary of the Holy Roman Empire, Perizolo mentions Leonardo as "Lionardo Fibonacci".[12]
Fibonacci popularized the Indo–Arabic numeral system in the Western world primarily through his composition in 1202 of Liber Abaci (Book of Calculation).[13][14] He also introduced Europe to the sequence of Fibonacci numbers, which he used as an example in Liber Abaci.[15]
Biography
Fibonacci was born around 1170 to Guglielmo, an Italian merchant and customs official.[7] Guglielmo directed a trading post in Bugia (Béjaïa), in modern-day Algeria, the capital of the Hammadid empire.[16] Fibonacci travelled with him as a young boy, and it was in Bugia (Algeria) where he was educated that he learned about the Hindu–Arabic numeral system.[17][6]
Fibonacci travelled around the Mediterranean coast, meeting with many merchants and learning about their systems of doing arithmetic.[18] He soon realised the many advantages of the Hindu-Arabic system, which, unlike the Roman numerals used at the time, allowed easy calculation using a place-value system. In 1202, he completed the Liber Abaci (Book of Abacus or The Book of Calculation),[19] which popularized Hindu–Arabic numerals in Europe.[6]
Fibonacci was a guest of Emperor Frederick II, who enjoyed mathematics and science. A member of Frederick II's court, John of Palermo, posed several questions based on Arab mathematical works for Fibonacci to solve. In 1240, the Republic of Pisa honored Fibonacci (referred to as Leonardo Bigollo)[20] by granting him a salary in a decree that recognized him for the services that he had given to the city as an advisor on matters of accounting and instruction to citizens.[21][22]
Fibonacci is thought to have died between 1240[23] and 1250,[24] in Pisa.
Liber Abaci
In the Liber Abaci (1202), Fibonacci introduced the so-called modus Indorum (method of the Indians), today known as the Hindu–Arabic numeral system,[25][26] with ten digits including a zero and positional notation. The book showed the practical use and value of this by applying the numerals to commercial bookkeeping, converting weights and measures, calculation of interest, money-changing, and other applications. The book was well-received throughout educated Europe and had a profound impact on European thought. Replacing Roman numerals, its ancient Egyptian multiplication method, and using an abacus for calculations, was an advance in making business calculations easier and faster, which assisted the growth of banking and accounting in Europe.[27][28]
The original 1202 manuscript is not known to exist.[29] In a 1228 copy of the manuscript, the first section introduces the numeral system and compares it with others, such as Roman numerals, and methods to convert numbers to it. The second section explains uses in business, for example converting different currencies, and calculating profit and interest, which were important to the growing banking industry. The book also discusses irrational numbersand prime numbers.[29][27][28]
Fibonacci sequence
Liber Abaci posed and solved a problem involving the growth of a population of rabbits based on idealized assumptions. The solution, generation by generation, was a sequence of numbers later known as Fibonacci numbers. Although Fibonacci's Liber Abaci contains the earliest known description of the sequence outside of India, the sequence had been described by Indian mathematicians as early as the sixth century.[30][31][32][33]
In the Fibonacci sequence, each number is the sum of the previous two numbers. Fibonacci omitted the "0" and first "1" included today and began the sequence with 1, 2, 3, ... . He carried the calculation up to the thirteenth place, the value 233, though another manuscript carries it to the next place, the value 377.[34][35] Fibonacci did not speak about the golden ratio as the limit of the ratio of consecutive numbers in this sequence.
Legacy
In the 19th century, a statue of Fibonacci was set in Pisa. Today it is located in the western gallery of the Camposanto, historical cemetery on the Piazza dei Miracoli.[1][36]
There are many mathematical concepts named after Fibonacci because of a connection to the Fibonacci numbers. Examples include the Brahmagupta–Fibonacci identity, the Fibonacci search technique, and the Pisano period. Beyond mathematics, namesakes of Fibonacci include the asteroid 6765 Fibonacci and the art rock band The Fibonaccis.
Works
- Liber Abaci (1202), a book on calculations (English translation by Laurence Sigler, 2002)[25]
- Practica Geometriae (1220), a compendium of techniques in surveying, the measurement and partition of areas and volumes, and other topics in practical geometry (English translation by Barnabas Hughes, Springer, 2008).
- Flos (1225), solutions to problems posed by Johannes of Palermo
- Liber quadratorum ("The Book of Squares") on Diophantine equations, dedicated to Emperor Frederick II. See in particular congruum and the Brahmagupta–Fibonacci identity.
- Di minor guisa (on commercial arithmetic; lost)
- Commentary on Book X of Euclid's Elements (lost)
See also
Notes
References
- ^ a b "Fibonacci's Statue in Pisa". Epsilones.com. Archived from the original on 2014-02-22. Retrieved 2010-08-02.
- ^ Smith, David Eugene; Karpinski, Louis Charles (1911), The Hindu–Arabic Numerals, Boston and London: Ginn and Company, p. 128, archivedfrom the original on 2023-03-13, retrieved 2016-03-02.
- ^ "Fibonacci, Leonardo". Lexico UK English Dictionary. Oxford University Press. Archived from the original on 2021-05-12.
- ^ "Fibonacci series" Archived 2019-06-23 at the Wayback Machineand "Fibonacci sequence". Collins English Dictionary. HarperCollins. Archived from the original on 12 June 2012. Retrieved 23 June 2019.
- ^ "Fibonacci number". Merriam-Webster Dictionary. Retrieved 23 June2019.
- ^ a b c MacTutor, R. "Leonardo Pisano Fibonacci". www-history.mcs.st-and.ac.uk. Archived from the original on 2019-10-28. Retrieved 2018-12-22.
- ^ a b c Livio, Mario (2003) [2002]. The Golden Ratio: The Story of Phi, the World's Most Astonishing Number (First trade paperback ed.). New York City: Broadway Books. pp. 92–93. ISBN 0-7679-0816-3. Archived from the original on 2023-03-13. Retrieved 2018-12-19.
- ^ Eves, Howard. An Introduction to the History of Mathematics. Brooks Cole, 1990: ISBN 0-03-029558-0 (6th ed.), p. 261.
- ^ Devlin, Keith (2017). Finding Fibonacci: The Quest to Rediscover the Forgotten Mathematical Genius Who Changed the World. Princeton University Press. p. 24.
- ^ Colin Pask (7 July 2015). Great Calculations: A Surprising Look Behind 50 Scientific Inquiries. Prometheus Books. p. 35. ISBN 978-1-63388-029-0. Archived from the original on 13 March 2023. Retrieved 19 January 2020.
- ^ Keith Devlin, The Man of Numbers: Fibonacci's Arithmetic Revolution,A&C Black, 2012 p. 13.
- ^ Drozdyuk, Andriy; Drozdyuk, Denys (2010). Fibonacci, his numbers and his rabbits. Toronto: Choven Pub. p. 18. ISBN 978-0-9866300-1-9. OCLC 813281753. Archived from the original on 2020-02-17. Retrieved 2020-01-26.
- ^ "Fibonacci Numbers". www.halexandria.org. Archived from the original on 2019-10-13. Retrieved 2015-04-29.
- ^ Leonardo Pisano: "Contributions to number theory" Archived 2008-06-17 at the Wayback Machine. Encyclopædia Britannica Online, 2006. p. 3. Retrieved 18 September 2006.
- ^ Singh, Parmanand. "Acharya Hemachandra and the (so called) Fibonacci Numbers". Math. Ed. Siwan, 20(1):28–30, 1986. ISSN 0047-6269
- ^ G. Germano, New editorial perspectives in Fibonacci's Liber abaci, «Reti medievali rivista» 14, 2, pp. 157–173 Archived 2021-07-09 at the Wayback Machine.
- ^ Thomas F. Glick; Steven Livesey; Faith Wallis (2014). Medieval Science, Technology, and Medicine: An Encyclopedia. Routledge. p. 172. ISBN 978-1-135-45932-1. Archived from the original on 2023-03-13. Retrieved 2018-12-07.
- ^ In the Prologus of the Liber abacci he said: "Having been introduced there to this art with an amazing method of teaching by means of the nine figures of the Indians, I loved the knowledge of such an art to such an extent above all other arts and so much did I devote myself to it with my intellect, that I learned with very earnest application and through the technique of contradiction anything to be studied concerning it and its various methods used in Egypt, in Syria, in Greece, in Sicily, and in Provence, places I have later visited for the purpose of commerce" (translated by G. Germano, New editorial perspectives in Fibonacci's Liber abaci, «Reti medievali rivista» 14, 2, pp. 157–173 Archived 2021-07-09 at the Wayback Machine.
- ^ The English edition of the Liber abacci was published by L.E. Sigler, Leonardo Pisano's book of calculation, New York, Springer-Verlag, 2003
- ^ See the incipit of Flos: "Incipit flos Leonardi bigolli pisani..." (quoted in the MS Word document Sources in Recreational Mathematics: An Annotated Bibliography by David Singmaster, 18 March 2004 – emphasis added), in English: "Here starts 'the flower' by Leonardo the wanderer of Pisa..."
The basic meanings of "bigollo" appear to be "bilingual" or "traveller". A. F. Horadam contends a connotation of "bigollo" is "absent-minded" (see first footnote of "Eight hundred years young" Archived 2008-12-19 at the Wayback Machine), which is also one of the connotations of the English word "wandering". The translation "the wanderer" in the quote above tries to combine the various connotations of the word "bigollo" in a single English word. - ^ Keith Devlin (7 November 2002). "A man to count on". The Guardian. Archived from the original on 17 September 2016. Retrieved 7 June2016.
- ^ «Considerantes nostre civitatis et civium honorem atque profectum, qui eis, tam per doctrinam quam per sedula obsequia discreti et sapientis viri magistri Leonardi Bigolli, in abbacandis estimationibus et rationibus civitatis eiusque officialium et aliis quoties expedit, conferuntur; ut eidem Leonardo, merito dilectionis et gratie, atque scientie sue prerogativa, in recompensationem laboris sui quem substinet in audiendis et consolidandis estimationibus et rationibus supradictis, a Comuni et camerariis publicis, de Comuni et pro Comuni, mercede sive salario suo, annis singulis, libre xx denariorum et amisceria consueta dari debeant (ipseque pisano Comuni et eius officialibus in abbacatione de cetero more solito serviat), presenti constitutione firmamus». F. Bonaini, Memoria unica sincrona di Leonardo Fibonacci, novamente scoperta, «Giornale storico degli archivi toscani» 1, 4, 1857, pp. 239–246.
- ^ Koshy, Thomas (2011), Fibonacci and Lucas Numbers with Applications, John Wiley & Sons, p. 3, ISBN 9781118031315, archivedfrom the original on 2023-03-13, retrieved 2015-12-12.
- ^ Tanton, James Stuart (2005), Encyclopédia of Mathematics, Infobase Publishing, p. 192, ISBN 9780816051243, archived from the original on 2023-03-13, retrieved 2015-12-12.
- ^ a b Fibonacci's Liber Abaci, translated by Sigler, Laurence E., Springer-Verlag, 2002, ISBN 0-387-95419-8
- ^ Grimm 1973
- ^ a b "Fibonacci: The Man Behind The Math". NPR.org. Archived from the original on 2011-07-16. Retrieved 2015-08-29.
- ^ a b Devlin, Keith. "The Man of Numbers: Fibonacci's Arithmetic Revolution [Excerpt]". Scientific American. Archived from the original on 2014-06-18. Retrieved 2015-08-29.
- ^ a b Gordon, John Steele. "The Man Behind Modern Math". Archivedfrom the original on 2015-08-23. Retrieved 2015-08-28.
- ^ Singh, Pamanand (1985). "The so-called fibonacci numbers in ancient and medieval India". Historia Mathematica. 12 (3): 229–244. doi:10.1016/0315-0860(85)90021-7.
- ^ Goonatilake, Susantha (1998). Toward a Global Science. Indiana University Press. p. 126. ISBN 978-0-253-33388-9.
Virahanka Fibonacci.
- ^ Knuth, Donald (2006). The Art of Computer Programming: Generating All Trees – History of Combinatorial Generation; Volume 4. Addison-Wesley. p. 50. ISBN 978-0-321-33570-8. Archived from the original on 2023-03-13. Retrieved 2020-11-11.
- ^ Hall, Rachel W. Math for poets and drummers Archived 2012-02-12 at the Wayback Machine. Math Horizons 15 (2008) 10–11.
- ^ Sloane, N. J. A. (ed.). "Sequence A000045 (Fibonacci Numbers)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
- ^ Pisanus, Leonardus; Boncompagni, Baldassarre (1 January 1857). Scritti: Il Liber Abbaci. Tip. delle Scienze Fisiche e Matematiche. p. 231. Archived from the original on 13 March 2023. Retrieved 20 December2018 – via Google Books.
- ^ Devlin, Keith (2010). "The Man of Numbers: In Search of Leonardo Fibonacci" (PDF). Mathematical Association of America. pp. 21–28. Archived (PDF) from the original on 2015-09-07. Retrieved 2018-12-21.
Further reading
- Devlin, Keith (2012). The Man of Numbers: Fibonacci's Arithmetic Revolution. Walker Books. ISBN 978-0802779083.
- Goetzmann, William N. and Rouwenhorst, K.Geert (2005). The Origins of Value: The Financial Innovations That Created Modern Capital Markets. Oxford University Press Inc., US, ISBN 0-19-517571-9.
- Goetzmann, William N., Fibonacci and the Financial Revolution (October 23, 2003), Yale School of Management International Center for Finance Working Paper No. 03–28
- Grimm, R. E., "The Autobiography of Leonardo Pisano", Fibonacci Quarterly, Vol. 11, No. 1, February 1973, pp. 99–104.
- Horadam, A. F. "Eight hundred years young," The Australian Mathematics Teacher 31 (1975) 123–134.
- Gavin, J., Schärlig, A., extracts of Liber Abaci online and analyzed on BibNum [click 'à télécharger' for English analysis]
External links
- "Fibonacci, Leonardo, or Leonardo of Pisa." Complete Dictionary of Scientific Biography. 2008. Encyclopedia.com. (April 20, 2015). [1]
- Fibonacci at Convergence
- O'Connor, John J.; Robertson, Edmund F., "Leonardo Pisano Fibonacci", MacTutor History of Mathematics Archive, University of St Andrews
- Fibonacci (2 vol., 1857 & 1862) Il liber abaci and Practica Geometriae – digital facsimile from the Linda Hall Library
- Fibonacci, Liber abbaci Bibliotheca Augustana
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