Matematica Di Base

[Thomas Merino]

IMOOC (Massive Open Online Courses) sono strumenti di didattica online accessibili a chiunque in maniera libera e gratuita.

CISIA ha progettato e realizzato i MOOC per offrire opportunit di autoapprendimento valide per studenti e studentesse ma anche per chiunque voglia approfondire le proprie conoscenze e orientarsi. I contenuti dei corsi delle diverse discipline sono stati sviluppati da docenti delle universit consorziate.

Nella fase di progettazione, realizzazione ed erogazione dei MOOC, CISIA si avvale della collaborazione di Federica Web Learning.

I MOOC aiutano nella preparazione e nello studio delle materie di base necessarie per affrontare il percorso universitario. Se vuoi frequentare un MOOC perch devi sostenere un TOLC o un altro tipo di test di ammissione, ricorda che necessario organizzare lo studio partendo dalla lettura dei sillabi del test che ti interessa ed eventualmente completare la preparazione anche attraverso altre attivit. I MOOC non sono pensati per la preparazione al test ma per autovalutarsi e migliorare conoscenze e competenze di base.

Potrai imparare a utilizzare gli strumenti matematici e a esercitarti con il ragionamento scientifico. In particolare, durante le lezioni affronterai il linguaggio della matematica, alcuni rudimenti della teoria degli insiemi e alcune nozioni di base su:

Potrai avvicinarti ai principali temi della biologia, familiarizzando con gli argomenti che sono trattati nei primi anni dei corsi di laurea in scienze della vita. Questo MOOC rappresenta un valido strumento per apprendere i fondamenti della biologia, dai fenomeni di base del funzionamento degli organismi viventi alle applicazioni delle scienze biologiche nella vita quotidiana.

Return the greatest common divisor of the specified integer arguments.If any of the arguments is nonzero, then the returned value is the largestpositive integer that is a divisor of all arguments. If all argumentsare zero, then the returned value is 0. gcd() without argumentsreturns 0.

The IEEE 754 special values of NaN, inf, and -inf will behandled according to IEEE rules. Specifically, NaN is not consideredclose to any other value, including NaN. inf and -inf are onlyconsidered close to themselves.

Return the least common multiple of the specified integer arguments.If all arguments are nonzero, then the returned value is the smallestpositive integer that is a multiple of all arguments. If any of the argumentsis zero, then the returned value is 0. lcm() without argumentsreturns 1.

Return the IEEE 754-style remainder of x with respect to y. Forfinite x and finite nonzero y, this is the difference x - n*y,where n is the closest integer to the exact value of the quotient x /y. If x / y is exactly halfway between two consecutive integers, thenearest even integer is used for n. The remainder r = remainder(x,y) thus always satisfies abs(r)

Special cases follow IEEE 754: in particular, remainder(x, math.inf) isx for any finite x, and remainder(x, 0) andremainder(math.inf, x) raise ValueError for any non-NaN x.If the result of the remainder operation is zero, that zero will havethe same sign as x.

Return x with the fractional partremoved, leaving the integer part. This rounds toward 0: trunc() isequivalent to floor() for positive x, and equivalent to ceil()for negative x. If x is not a float, delegates to x.__trunc__, which should return an Integral value.

Return e raised to the power x, minus 1. Here e is the base of naturallogarithms. For small floats x, the subtraction in exp(x) - 1can result in a significant loss of precision; the expm1()function provides a way to compute this quantity to full precision:

Return x raised to the power y. Exceptional cases followthe IEEE 754 standard as far as possible. In particular,pow(1.0, x) and pow(x, 0.0) always return 1.0, evenwhen x is a zero or a NaN. If both x and y are finite,x is negative, and y is not an integer then pow(x, y)is undefined, and raises ValueError.

Return the complementary error function at x. The complementary errorfunction is defined as1.0 - erf(x). It is used for large values of x where a subtractionfrom one would cause a loss of significance.

CPython implementation detail: The math module consists mostly of thin wrappers around the platform Cmath library functions. Behavior in exceptional cases follows Annex F ofthe C99 standard where appropriate. The current implementation will raiseValueError for invalid operations like sqrt(-1.0) or log(0.0)(where C99 Annex F recommends signaling invalid operation or divide-by-zero),and OverflowError for results that overflow (for example,exp(1000.0)). A NaN will not be returned from any of the functionsabove unless one or more of the input arguments was a NaN; in that case,most functions will return a NaN, but (again following C99 Annex F) thereare some exceptions to this rule, for example pow(float('nan'), 0.0) orhypot(float('nan'), float('inf')).

Returns the result of adding any amount of blocks that have a number value together. Blocks with a number value include the basic number block, length of list or text, variables with a number value, etc. This block is a mutator and can be expanded to allow more numbers in the sum.

Use this block to generate repeatable sequences of random numbers. You can generate the same sequence of random numbers by first calling random set seed with the same value. This is useful for testing programs that involve random values.

Returns the given number rounded to the closest integer. If the fractional part is .5 it will be rounded up. If it is exactly equal to .5, numbers with an even whole part will be rounded down, and numbers with an odd whole part will be rounded up. (This method is called round to even.)

Formats a number as a decimal with a given number of places after the decimal point. The number of places must be a non-negative integer. The result is produced by rounding the number (if there were too many places) or by adding zeros on the right (if there were too few).

Takes a text string that represents a positive integer in one base and returns a string that represents the same number is another base. For example, if the input string is 10, then converting from base 10 to binary will produce the string 1010; while if the input string is the same 10, then converting from binary to base 10 will produce the string 2. If the input string is the same 10, then converting from base 10 to hex will produce the string A.

Mathematical induction proves that we can climb as high as we like on a ladder, by proving that we can climb onto the bottom rung (the basis) and that from each rung we can climb up to the next one (the step).

The method can be extended to prove statements about more general well-founded structures, such as trees; this generalization, known as structural induction, is used in mathematical logic and computer science. Mathematical induction in this extended sense is closely related to recursion. Mathematical induction is an inference rule used in formal proofs, and is the foundation of most correctness proofs for computer programs.[3]

Despite its name, mathematical induction differs fundamentally from inductive reasoning as used in philosophy, in which the examination of many cases results in a probable conclusion. The mathematical method examines infinitely many cases to prove a general statement, but it does so by a finite chain of deductive reasoning involving the variable n \displaystyle n , which can take infinitely many values. The result is a rigorous proof of the statement, not an assertion of its probability.[4]

In 370 BC, Plato's Parmenides may have contained traces of an early example of an implicit inductive proof,[5] however, the earliest implicit proof by mathematical induction was written by al-Karaji around 1000 AD, who applied it to arithmetic sequences to prove the binomial theorem and properties of Pascal's triangle. Whilst the original work was lost, it was later referenced by Al-Samawal al-Maghribi in his treatise al-Bahir fi'l-jabr (The Brilliant in Algebra) in around 1150 AD.[6][7]

Another important idea introduced by al-Karaji and continued by al-Samaw'al and others was that of an inductive argument for dealing with certain arithmetic sequences. Thus al-Karaji used such an argument to prove the result on the sums of integral cubes already known to Aryabhata [...] Al-Karaji did not, however, state a general result for arbitrary n. He stated his theorem for the particular integer 10 [...] His proof, nevertheless, was clearly designed to be extendable to any other integer. [...] Al-Karaji's argument includes in essence the two basic components of a modern argument by induction, namely the truth of the statement for n = 1 (1 = 13) and the deriving of the truth for n = k from that of n = k - 1. Of course, this second component is not explicit since, in some sense, al-Karaji's argument is in reverse; this is, he starts from n = 10 and goes down to 1 rather than proceeding upward. Nevertheless, his argument in al-Fakhri is the earliest extant proof of the sum formula for integral cubes.[8]

None of these ancient mathematicians, however, explicitly stated the induction hypothesis. Another similar case (contrary to what Vacca has written, as Freudenthal carefully showed)[10] was that of Francesco Maurolico in his Arithmeticorum libri duo (1575), who used the technique to prove that the sum of the first n odd integers is n2.

The induction hypothesis was also employed by the Swiss Jakob Bernoulli, and from then on it became well known. The modern formal treatment of the principle came only in the 19th century, with George Boole,[13] Augustus De Morgan, Charles Sanders Peirce,[14][15] Giuseppe Peano, and Richard Dedekind.[9]

In practice, proofs by induction are often structured differently, depending on the exact nature of the property to be proven.All variants of induction are special cases of transfinite induction; see below.

3a8082e126