The Factorial (!) in Mathematics and Statistics

In mathematics, symbols that have certain meanings in the English language can mean very specialized and different things. For example, consider the following expression: 3! No, we did not use the exclamation point to show that we’re excited about three, and we shouldn’t read the last sentence with emphasis. In mathematics, the expression 3! is read as three factorial and is really a shorthand way to denote the multiplication of several consecutive whole numbers. Since there are many places throughout mathematics and statistics where we need to multiply numbers together, the factorial is quite useful. Some of the main places where it shows up are combinatorics and probability calculus. Definition The definition of the factorial is that for any positive whole number n, the factorial: n! n x (n -1) x (n - 2) x . . . x 2 x 1 Examples for Small Values First we will look at a few examples of the factorial with small values of n: 1! 12! 2 x 1 23! 3 x 2 x 1 64! 4 x 3 x 2 x 1 245! 5 x 4 x 3 x 2 x 1 1206! 6 x 5 x 4 x 3 x 2 x 1 7207! 7 x 6 x 5 x 4 x 3 x 2 x 1 50408! 8 x 7 x 6 x 5 x 4 x 3 x 2 x 1 403209! 9 x 8 x 7 x 6 x 5 x 4 x 3 x 2 x 1 36288010! 10 x 9 x 8 x 7 x 6 x 5 x 4 x 3 x 2 x 1 3628800 As we can see the factorial gets very large very quickly. Something that may seem small, such as 20! actually has 19 digits. Factorials are easy to compute, but they can be somewhat tedious to calculate. Fortunately, many calculators have a factorial key (look for the ! symbol). This function of the calculator will automate the multiplications. A Special Case One other value of the factorial  and one for which the standard definition above does not hold is that of zero factorial. If we follow the formula, then we would not arrive at any value for 0!. There are no positive whole numbers less than 0. For several reasons, it is appropriate to define 0! 1. The factorial for this value shows up particularly in the formulas for combinations and permutations. More Advanced Calculations When dealing with calculations, it is important to think before we press the factorial key on our calculator. To calculate an expression such as 100!/98! there are a couple of different ways of going about this. One way is to use a calculator to find both 100! and 98!, then divide one by the other. Although this is a direct way to calculate, it has some difficulties associated with it. Some calculators cannot handle expressions as large as 100! 9.33262154 x 10157. (The expression 10157 is a scientific notation that means that we multiply by 1 followed by 157 zeros.) Not only is this number massive, but it is also only an estimate to the real value of 100! Another way to simplify an expression with factorials like the one seen here does not require a calculator at all. The way to approach this problem is to recognize that we can rewrite 100! not as 100 x 99 x 98 x 97 x . . . x 2 x 1, but instead as 100 x 99 x 98! The expression 100!/98! now becomes (100 x 99 x 98!)/98! 100 x 99 9900.

Notes from Underground Binding Limits Essay - 1521 Words

In Dostoyevsky’s Notes from Underground, the underground man struggles between two beliefs. The first acknowledges that his fictional existence is predetermined, subject to his author’s conduct. The second opposes that, insisting the underground man can only live in an undetermined world that extols free will, situating it within the human. For a remedy, the underground man turns to writing, hoping to probe into this duality and to not reject any truth that comes forth, horrifying or not. Through this, he understands that his self awareness gives him no control he has over his actions. Even though he doesn’t possess it, the underground man continues to believe in free will. The reader’s acknowledged on the book’s final page when the†¦show more content†¦Moreover, to believe two times two is four and that the outcome of four is unchangeable within the distance future, is something that the Underground man refutes. For one’s life to h ave meaning and validity, actions must be understood as non-robotic; an answer to this question must not only be unknown, but more importantly, undecided. He then goes on to say that â€Å"consciousness, for example, is infinitely higher than two times two† (pg 33). As we continue to read, the underground man’s consciousness which desires to rid itself from the notion of determinism produces an ironic situation. As readers, we understand through words and upcoming chapters, his story has already been decided. The underground man can not make conscious decisions because the author has already done so: the underground man is crafted solely from the author’s literary experience. This revelation marks the shift from part one of the book to part two. In part two the underground man writes about his past (20 years ago) as an experiment designed to gain understanding about his true nature and the logic that governs it. His hope is to realize where his path took a wrong turn and what, if anything, can be learned from it. In â€Å"Apropos of the Wet Snow,† the underground man describes his time spent reading as an attempt toShow MoreRelatedSelecting An Real Estate Agent Essay1919 Words   |  8 Pagesavoid getting into a legal or financial tangle during the transaction. Selecting an Real Estate Agent You need to select an agent to service you. Find one who is familiar with the location and type of property you have in mind. Make sure the agent is from a Listed Housing Agency and also a member of the Institute of Estate Agents (IEA). Discuss your requirements with him and be specific. This will allow him to advise and service you in a professional manner. Ownership Always ensure that the seller isRead MoreOrgan Trafficking Law2609 Words   |  11 Pagesinternational organ trade has been recognized as a significant health policy issue in the international community. 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Comparison of Pride and Prejudice with Sense and Sensibility Free Essays

The poem â€Å"Pied Beauty† was written by Hopkins in 1877 and the â€Å"Ode to Autumn† was penned down by Keats in 1820. These poems appreciate all the aspects and diversity of Nature positively leading to the praise of God. In Pied Beauty, this praise of the Creator is vivid and apparent as the poet magnanimously asserts: â€Å"He fathers-forth whose beauty is past change: Praise him. We will write a custom essay sample on Comparison of Pride and Prejudice with Sense and Sensibility or any similar topic only for you Order Now † However, in Ode to Autumn the praise is subtly highlighted through the description of exuberant bounties of Autumn. The overarching theme of â€Å"beauty beyond appearance† is witnessed in both the poetic works mentioned above. In â€Å"Pied Beauty† this theme encapsulates the idea that perfection of things lies behind their appearance. The poet accentuates the fact that the spirit and energy beyond the appearance is to be appreciated. Thereby, a metaphysical streak echoes throughout, emphasizing the true essence of the word, â€Å"beauty†. Similarly, this theme is observed in â€Å"Ode to Autumn† connoting that the season Autumn has its own beauty like other seasons. Generally, Autumn is associated with old age or the end of life but Keats has creatively portrayed it as a season of â€Å"mellow fruitfulness†, thus, denoting the beauty of Autumn afar from its appearance. He has artistically projected an implied meaning to the poem by illustrating that autumn is the time of life where everything eventually leads to its completion or to have acquired the ultimate m otive of life. The crux of Ode to Autumn is to symbolize the uncertainties of life the Universe is so prone. However, in Pied Beauty, a synonymous thought is presented in the subtle descriptions of Nature that juxtapose the opposites to underline the notion that life is unpredictable , non-static and full of ambiguities. Hopkins asserts: â€Å"And all the trades,their gear tackle trim. Whatever is ficke, freckled,(who knows how?) With swift, slow; sweet, sour;adazzle,dim; He fathers-forth whose beauty is past change† Here, the poet stresses on the diverse nature of the life on Universe that everything is in a flux. The only thing constant is change and therefore, in Ode to Autumn we notice that autumn has arrived and we should appreciate the various shades of Nature it offers because they are momentary. Consequently, the beauty and speciality of everything present should be appreciated. It is because every atom and molecule on earth gifts us with some meaning and purpose in life. Another eminent theme is of â€Å"Nature†. In â€Å"Pied Beauty† numerous aspects of Nature are delineated using compound words. Hopkins appreciates that â€Å"Glory be to God for dappled things†, â€Å"skies of couple-colours† the â€Å"rose-moles† on the trout, the â€Å"Fresh-firecoal chestnut-falls; finches’ wings† and also the â€Å"Landscape plotted and pieced -fold,fallow plough†. The poet is praisng God for everything that he has created and also for the sublime energies that Nature encapsulates beyond its physical appearance. Furthermore, this theme of Nature is also evident in â€Å"Ode to Autumn†. This ode is pregnant with admiration for different activities that take place in autumn. Keats enumerates that it is a â€Å"season of mists and mellow fruitfulness†, â€Å"load and bless with fruits the vines that round the thatch-eves run†, â€Å"fill all fruits with ripeness to the core†, †Å"swell the gourd and plump the hazel shells with sweet kernel†,†barred clouds bloom the soft-dying day†, â€Å"lambs loud bleat from hilly bourn† and the â€Å"Hedge-cricket sing†. All these descriptions stimulate our senses and lend a beautiful cinematographic element to the poem. Analogous to Hopkins, Keats also implies the stylistic device of compound words. However, the contrast lies in the fact that Keats uses compound words to make his poetry sensual whereas, Hopkins employs compound words to give different things a set shape and pattern. This is called instress and inscape technique that Hopkins uses. For example, he says â€Å"For skies of couple-colour as a brinded cow†. Here he is characterizing the sky’s colour and accentuating the inter relation and harmony of Nature. Like the brinded cow the sky is also spotted and patchy as the blue sky and white clouds are â€Å"coupled† together. The creative amalgamation of sky with creations on earth reflects the poet’s love of nature to a commendable degree. On the contrary, Keats utilizes compound words to give a corporeal effect. For instance, he calls autum the â€Å"bosom-friend of the maturing Sun† and also he asserts â€Å"half-reap’d furrow†, â€Å"bloo m the soft-dying day† and â€Å"full-grown lambs†. All these examples stir the senses. Animal and bird imagery have also been incorporated in these poems. Hopkins uses the image of the â€Å"brinded cow† to make the colour and pattern of the sky tangible, and also uses the image of a trout with rose-moles and the finches’ wings to signify the variety and diversity of Nature. Nonetheless, Keats uses the images of the â€Å"swallow† singing, â€Å"lambs† bleating , â€Å"hedge-cricket† singing and the red-breast,†garden-crofet† whistling to indicate that autumn has a life and activity of its own which must be appreciated instead waiting for other seasons to come. Alliteration in the â€Å"Pied Beauty† is used to heighten the auditory effect of the poem. It is said about Hopkins that â€Å"his poetry should not be read with eyes but ears† (Bridges). The alliteartion used in the compound word â€Å"couple-clouds† empahsizes on making the sky tangible whereas â€Å"Fresh-firecoal† denotes a paradox to enhance the colour of the chestnut that is falling off the tree and â€Å"fold, fallow† these descriptions of the landscape suggest the multiplicity of lands created by God. On the other hand, in â€Å"Ode to Autumn† the alliterations â€Å"winnowing wind†, â€Å"dying day† and â€Å"lambs loud† all of these are stressing upon the activities that take place in autumn. Thus, signifying that autumn is also lively and is not about the end of life rather it announces a new happy beginning that follows. Nonetheless, apart from a few stylistic contrasts, both poets share a common natural ground of ideas, that is , the love and intense adulation of nature. The imagery that these poets employ is far-fetched and typical of the Romantic school of thought. How to cite Comparison of Pride and Prejudice with Sense and Sensibility, Papers