In 1629, a Flemish mathematician, Albert Girard, published a book called L’invention nouvelle en l’ Algebre. In his book, he claimed that there were always n solutions for equations of degree n. However he did not assert that solutions are of the form a + bi, w... ... middle of paper ... ...eneral theorem on the existence of a minimum of a continuous function. Two years after Argand's proof, Gauss published a second proof of the Fundamental Theorem of Algebra. Gauss used Euler's approach but instead of operating with roots, Gauss operated with indeterminates. This proof was considered complete and correct.
By the time Euclid's Elements appeared in about 300 BC, several important results about primes had been proved. In Book IX of the Elements, Euclid proves that there are infinitely many prime numbers. This is one of the first proofs known which uses the method of contradiction to establish a result. Euclid also gives a proof of the Fundamental Theorem of Arithmetic: Every integer can be written as a product of primes in an essentially unique way. Euclid also showed that if the number 2n - 1 is prime then the number 2n-1(2n - 1) is a perfect number.
Fittingly, recurring decimals fall under the elegant category of number theory in mathematics, called the “queen of mathematical studies” by Gauss. We have learned much about modular arithmetic and its useful applications; my investigation will revolve around the relationship between modular arithmetic and recurring decimals. Euler’s totient function, which is just a generalized form of Fermat’s Little Theorem, appears to parallel the period of recurring decimals, the number of digits the decimal expansion goes before repeating once more. The totient function can be defined as the following: it is the number of positive integers 2 less than or equal to a number “n” that are relatively prime to “n”. For example, the number 7 has a totient of 6 because 1,2,3,4,5, and 6 are the only numbers that satisfy the conditions.
Whether Q is a prime number or a composite number, the original list is incomplete. As long as the list of prime numbers is finite, it is always possible to find one more new prime number. Therefore, if prime numbers are a finite list, it will always be complete. This then contradicts the original assumption that there are finitely many primes and hence, proves Euclid’s theorem, that the set of prime numbers is
Designing a Random Number Generator Introduction Random Number Generator is a computational routine or a physical device that produces numbers which don’t have any pattern in them. Although using computational algorithms involves adding pattern to the resulting sequence of numbers. We focussed on generating uniformly distributed random numbers between (0,1) since the same distribution can be used to get numbers from different ranges. We all know that LCG is well known method for giving a sequence of randomized numbers using linear equation but it still has some issues. We have done 9 different tests on our RNG and have come up with quantitative results.
A prime number is an integer with only positive divisors one and itself. The ancient Greeks proved that there where infinitely many primes and that they where irregularly spaced. Mersenne examined prime numbers and wanted to discover a formula that would represent all primes. The formula is (2p-1) where p is a known prime number. Mersenne claimed that if a number n=(2p-1) is prime then p=2,3,5,7,13,17,31,67,127, and 257, but composite for the other forty-four primes smaller than or equal to 257.
The concept of impossible constructions in mathematics draws in a unique interest by Mathematicians wanting to find answers that none have found before them. For the Greeks, some impossible constructions weren’t actually proven at the time to be impossible, but merely so far unachieved. For them, there was excitement in the idea that they might be the first one to do so, excitement that lay in discovery. There are a few impossible constructions in Greek mathematics that will be examined in this chapter. They all share the same criteria for constructability: that they are to be made using solely a compass and straightedge, and were referred to as the three “classical problems of antiquity”.
My second task is to check that if these are correct: f(7x4) = f(7) x f(4) and f(6x4) = f(6) x f(4) then create some of my own and check that if they are equal to each other or not. Part 1 ====== (1) f(3) = 2 The factors of f(3) are: 1 and 3. The integers, which are less than 3, are 1 and 2. The table below shows the integers, factors and whether it fits into the expression the number of positive integers less than n which have no factor (other than 1) in common (are co-prime) with n. Integers Factors Does it fit into expression? Yes or No 1 1 yes 2 1,2 yes This table shows you that the number of positive integers less than three and has no other common factor other than 1 is two integers: 1 and 2.