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Sign in Create an account. Syntax Advanced Search. Numbers and Proofs. Useful methods of proof are illustrated in the context of studying problems concerning mainly numbers real, rational, complex and integers. An indispensable guide to all students of mathematics.

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The Fibonacci numbers are 0, 1, 1, 2, 3, 5, 8, 13,. There is a large amount of information at this site more than pages if it was printed , so if all you want is a quick introduction then the first link takes you to an introductory page on the Fibonacci numbers and where they appear in Nature.

The rest of this page is a brief introduction to all the web pages at this site on Fibonacci Numbers the Golden Section and the Golden String together with their many applications.

What s New? Please can you re-send your email if you ve had no reply - sorry! Fibonacci Numbers and Golden sections in Nature Fibonacci Numbers and Nature Fibonacci and the original problem about rabbits where the series first appears, the family trees of cows and bees, the golden ratio and the Fibonacci series, the Fibonacci Spiral and sea shell shapes, branching plants, flower petal and seeds, leaves and petal arrangements, on pineapples and in apples, pine cones and leaf arrangements.

All involve the Fibonacci numbers - and here s how and why. Fibonacci Numbers, the Golden section and the Golden String section appears in nature. Now with a Geometer s Sketchpad dynamic demonstration. The Puzzling World of Fibonacci Numbers A pair of pages with plenty of playful problems to perplex the professional and the parttime puzzler!

The Easier Fibonacci Puzzles page has the Fibonacci numbers in brick wall patterns, Fibonacci bee lines, seating people in a row and the Fibonacci numbers again, giving change and a game with match sticks and even with electrical resistance and lots more puzzles all involve the Fibonacci numbers! If you know the Fibonacci Jigsaw puzzle where rearranging the 4 wedge-shaped pieces makes an additional square appear, did you know the same puzzle can be rearranged to make a different shape where a square now disappears?

For these puzzles, I do not know of any simple explanations of why the Fibonacci numbers occur - and that s the real puzzle - can you supply a simple reason why?? The Intriguing Mathematical World of Fibonacci and Phi The golden section numbers are also written using the greek letters Phi and phi. The Mathematical Magic of the Fibonacci numbers looks at the patterns in the Fibonacci numbers themselves, the Fibonacci numbers in Pascal s Triangle and using Fibonacci series to generate all rightangled triangles with integers sides based on Pythagoras Theorem.

Impress your friends with a simple Fibonacci numbers trick! There are many investigations for you to do to find patterns for yourself as well as a complete list of. The first Fibonacci numbers. A Formula for the Fibonacci numbers Is there a direct formula to compute Fib n just from n? Yes there is! Fibonacci Numbers, the Golden section and the Golden String This page shows several and why they involve Phi and phi - the golden section numbers.

Fibonacci bases and other ways of representing integers We use base 10 decimal for written numbers but computers use base 2 binary. What happens if we use the Fibonacci numbers as the column headers? The Golden Section - the Number and Its Geometry The golden section is also called the golden ratio, the golden mean and the divine proportion. Two pages are devoted to its applications in Geometry - first in flat or two dimensional geometry and then in the solid geometry of three dimensions.

Fantastic Flat Phi Facts See some of the unexpected places that the golden section Phi occurs in Geometry and in Trigonometry: pentagons and decagons, paper folding and Penrose Tilings where we phind phi phrequently!

The Golden Geometry of the Solid Section or Phi in 3 dimensions The golden section occurs in the most symmetrical of all the threedimensional solids - the Platonic solids. What are the best shapes for fair dice?

Why are there only 5? Phi s Fascinating Figures - the Golden Section number All the powers of Phi are just whole multiples of itself plus another whole number. Did you guess that these multiples and the whole numbers are, of course, the Fibonacci numbers again? Each power of Phi is the sum of the previous two - just like the Fibonacci numbers too.

Introduction to Continued Fractions An optional page that expands on the idea of a continued fraction introduced in the Phi s Fascinating Figures page. Fibonacci Numbers, the Golden section and the Golden String Phigits and Base Phi Representations We have seen that using a base of the Fibonacci Numbers we can represent all integers in a binary-like way.

Here we show there is an interesting way of representing all integers in a binary-like fashion but using only powers of Phi instead of powers of 2 binary or 10 decimal. Fibonacci Rabbit Sequence There is another way to look at Fibonacci s Rabbits problem that gives an infinitely long sequence of 1s and 0s, which we will call the Fibonacci Rabbit sequence:1 0 1 1 0 1 0 1 1 0 1 1 0 1 0 1 1 0 1.

You can hear the Golden sequence as a Quicktime movie track too! The Fibonacci Rabbit sequence is an example of a fractal. Here is a brief biography of Fibonacci and his historical achievements in mathematics, and how he helped Europe replace the Roman numeral system with the "algorithms" that we use today.

Also there is a guide to some memorials to Fibonacci to see in Pisa, Italy. One that has been used a lot is based on a nice formula for calculating which angle has a given tangent, discovered by James Gregory. His formula together with the Fibonacci numbers can be used to compute pi. Fibonacci Numbers, the Golden section and the Golden String from scratch. Fibonacci Forgeries Sometimes we find series that for quite a few terms look exactly like the Fibonacci numbers, but, when we look a bit more closely, they aren t - they are Fibonacci Forgeries.

Since we would not be telling the truth if we said they were the Fibonacci numbers, perhaps we should call them Fibonacci Fibs!! The Lucas Numbers Here is a series that is very similar to the Fibonacci series, the Lucas series, but it starts with 2 and 1 instead of Fibonacci s 0 and 1.

It sometimes pops up in the pages above so here we investigate it some more and discover its properties. It ends with a number trick which you can use "to impress your friends with your amazing calculating abilities" as the adverts say.

It uses facts about the golden section and its relationship with the Fibonacci and Lucas numbers. The first Lucas numbers and their factors together with some suggestions for investigations you can do. This is a different page to those above, being concerned with speculations about where the golden section both does and does not occur in art, architecture and music.

All the other pages are factual and verifiable - the material here is a often a matter of opinion - but interesting nevertheless! Links and References Fibonacci, Phi and Lucas numbers Formulae A reference page of over formulae and equations showing the properties of these series.

Links and references Links to other sites on Fibonacci numbers and the Golden section together with references to books and articles. Check them out! The Knot a Braid of Links Project at Camel designated this page a cool math site of the week for November now available via in the Kabol Database search engine.

There are now more than visits each weekday to this Menu page alone. Knott surrey. This, the first, looks at the Fibonacci numbers and why they appear in various "family trees" and patterns of spirals of leaves and seeds. The second page then examines why the golden section is used by nature in some detail, including animations of growing plants.

Contents of this Page The line means there is a Things to do investigation at the end of the section. Fibonacci s Rabbits. Fibonacci s Rabbits The original problem that Fibonacci investigated in the year was about how fast rabbits could breed in ideal circumstances. The Fibonacci Numbers and Golden section in Nature - 1 Suppose a newly-born pair of rabbits, one male, one female, are put in a field. Rabbits are able to mate at the age of one month so that at the end of its second month a female can produce another pair of rabbits.

Suppose that our rabbits never die and that the female always produces one new pair one male, one female every month from the second month on. The puzzle that Fibonacci posed was. How many pairs will there be in one year? At the end of the first month, they mate, but there is still one only 1 pair. At the end of the second month the female produces a new pair, so now there are 2 pairs of rabbits in the field. At the end of the third month, the original female produces a second pair, making 3 pairs in all in the field.

At the end of the fourth month, the original female has produced yet another new pair, the female born two months ago produces her first pair also, making 5 pairs. The number of pairs of rabbits in the field at the start of each month is 1, 1, 2, 3, 5, 8, 13, 21, 34,.

The Fibonacci Numbers and Golden section in Nature - 1 Can you see how the series is formed and how it continues?

If not, look at the answer! The first Fibonacci numbers are here and some questions for you to answer. Now can you see why this is the answer to our Rabbits problem? If not, here s why. Another view of the Rabbit s Family Tree: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, , , , , The Rabbits problem is not very realistic, is it?

It seems to imply that brother and sisters mate, which, genetically, leads to problems. We can get round this by saying that the female of each pair mates with any male and produces another pair. Another problem which again is not true to life, is that each birth is of exactly two rabbits, one male and one female. In one of them he adapts Fibonacci s Rabbits to cows, making the problem more realistic in the way we observed above.

He gets round the problems by noticing that really, it is only the females that are interesting - er - I mean the number of females! He changes months into years and rabbits into bulls male and cows females in problem in his book puzzles and Curious Problems , Souvenir press : If a cow produces its first she-calf at age two years and after that produces another single she-calf every year, how many she-calves are there after 12 years, assuming none die?

This is a better simplification of the problem and quite realistic now. But Fibonacci does what mathematicians often do at first, simplify the problem and see what happens - and the series bearing his name does have lots of other interesting and practical applications as we see later. So let s look at another real-life situation that is exactly modelled by Fibonacci s series - honeybees.

Honeybees, Fibonacci numbers and Family trees There are over 30, species of bees and in most of them the bees live solitary lives. The one most of us know best is the honeybee and it, unusually, lives in a colony called a hive and they have an unusual Family Tree.

In fact, there are many unusual features of honeybees and in this section we will show how the Fibonacci numbers count a honeybee s ancestors in this section a "bee" will mean a "honeybee". First, some unusual facts about honeybees such as: not all of them have two parents!

In a colony of honeybees there is one special female called the queen. There are many worker bees who are female too but unlike the queen bee, they produce no eggs. There are some drone bees who are male and do no work.

Males are produced by the queen s unfertilised eggs, so male bees only have a mother but no father! All the females are produced when the queen has mated with a male and so have two parents. Females usually end up as worker bees but some are fed with a special substance called royal jelly which makes them grow into queens ready to go off to start a new colony when the bees form a swarm and leave their home a hive in search of a place to build a new nest.

Fibonacci Numbers And The Golden Section.(pdf)

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As the basis of equations and therefore problem-solving , linear algebra is the most widely taught sub-division of pure mathematics. Dr Allenby has used his experience of teaching linear algebra to write a lively book on the subject that includes The author provides a mixture of informal and formal material which Du kanske gillar. Linear Algebra Reg Allenby E-bok.


Useful methods of proof are illustrated in the context of studying problems concerning mainly numbers real, rational, complex and integers. An indispensable guide to all students of mathematics. Each proof is preceded by a discussion which is intended to show the reader the kind of thoughts they might have before any attempt proof is made. Established proofs which the student is in a better position to follow then follow.

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 - Зачем. Стратмор казался озадаченным. Он не привык, чтобы кто-то повышал на него голос, пусть даже это был его главный криптограф. Он немного смешался. Сьюзан напряглась как тигрица, защищающая своего детеныша.

Он так торопился, что не заметил побелевших костяшек пальцев, вцепившихся в оконный выступ. Свисая из окна, Беккер благодарил Бога за ежедневные занятия теннисом и двадцатиминутные упражнения на аппарате Наутилус, подготовившие его мускулатуру к запредельным нагрузкам. Увы, теперь, несмотря на силу рук, он не мог подтянуться, чтобы влезть обратно. Плечи его отчаянно болели, а грубый камень не обеспечивал достаточного захвата и впивался в кончики пальцев подобно битому стеклу. Беккер понимал, что через несколько секунд его преследователь побежит назад и с верхних ступеней сразу же увидит вцепившиеся в карниз пальцы. Он зажмурился и начал подтягиваться, понимая, что только чудо спасет его от гибели.

 Боль внизу нестерпима, - прошипел он ей на ухо. Колени у Сьюзан подкосились, и она увидела над головой кружащиеся звезды. ГЛАВА 80 Хейл, крепко сжимая шею Сьюзан, крикнул в темноту: - Коммандер, твоя подружка у меня в руках. Я требую выпустить меня отсюда. В ответ - тишина. Его руки крепче сжали ее шею. - Я сейчас ее убью.

Numbers and Proofs

Он впутал в это дело Сьюзан и должен ее вызволить. Голос его прозвучал, как всегда, твердо: - А как же мой план с Цифровой крепостью. Хейл засмеялся: - Можете пристраивать к ней черный ход - я слова не скажу.  - Потом в его голосе зазвучали зловещие нотки.

 Оно будет громадным, - застонал Джабба.  - Ясно, что это будет число-монстр. Сзади послышался возглас: - Двухминутное предупреждение. Джабба в отчаянии бросил взгляд на ВР. Последний щит начал рушиться.

Сейчас переключит. Мне не успеть. Но когда шестерни разомкнулись, чтобы включилась другая их пара, автобус слегка притормозил, и Беккер прыгнул. Шестерни сцепились, и как раз в этот момент его пальцы схватились за дверную ручку. Руку чуть не вырвало из плечевого сустава, когда двигатель набрал полную мощность, буквально вбросив его на ступеньки.

 - Посмотрим, что у них. Соши начала просматривать документ. Ей попалось описание нитрата мочевины, в десять раз более мощной взрывчатки, чем динамит.

Она смотрела на обмякшее тело коммандера и знала, о чем он думает. Рухнул не только его план пристроить черный ход к Цифровой крепости. В результате его легкомыслия АНБ оказалось на пороге крупнейшего в истории краха, краха в сфере национальной безопасности Соединенных Штатов. - Коммандер, вы ни в чем не виноваты! - воскликнула.  - Если бы Танкадо был жив, мы могли бы заключить с ним сделку, и у нас был бы выбор.

Пожилой человек вдруг поднялся и куда-то побежал, видимо, вызвать скорую. Танкадо явно терял последние силы, но по-прежнему совал кольцо прямо в лицо тучному господину. Тот протянул руку, взял Танкадо за запястье, поддерживая остававшуюся на весу руку умирающего.

 - Сядьте. Однако Беккер был слишком ошеломлен, чтобы понять смысл этих слов. - Sientate! - снова крикнул водитель. Беккер увидел в зеркале заднего вида разъяренное лицо, но словно оцепенел. Раздраженный водитель резко нажал на педаль тормоза, и Беккер почувствовал, как перемещается куда-то вес его тела.

Он подумал, дома ли Сьюзан. Куда она могла уйти.